Abstract
We extend profound results in pluripotential theory on Kähler manifolds (Darvas in arXiv:1902.01982, 2019) to Sasaki setting via its transverse Kähler structure. As in Kähler case, these results form a very important piece to solve the existence of Sasaki metrics with constant scalar curvature in terms of properness of \(\mathcal {K}\)-energy, considered by the first named author in He (arXiv:1802.03841, 2019). One main result is to generalize Darvas’ theory on the geometric structure of the space of Kähler potentials in Sasaki setting. Along the way we extend most of corresponding results in pluripotential theory to Sasaki setting via its transverse Kähler structure.
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1 Introduction
Sasaki manifolds have gained their prominence in physics, algebraic geometry, and Riemannian geometry [13]. There are tremendous work in the last two decades in Sasaki geometry, in particular on Sasaki–Einstein manifolds, see [13, 14, 27, 37, 39, 50, 54] and reference therein. On the other hand, Sasaki geometry is an odd- dimensional analog of Kähler geometry and almost all results in Kähler geometry have their counterparts in Sasaki geometry. Calabi’s extremal metric [17, 18] (and csck) has played a very important role in Kähler geometry and it has a direct adaption in Sasaki setting [16]. In 1997, Donaldson [34] proposed an extremely fruitful program to approach existence of csck (extremal metrics) on a compact Kähler manifold with a fixed Kähler class. Donaldson’s program has also been extended to Sasaki setting, see [42, 46] for example.
A major problem in Kähler geometry is to characterize exactly when a Kähler class contains a csck (extremal). The analytic part for existence of csck is to solve a fourth-order highly non-linear elliptic equation, the scalar curvature-type equation. This problem is regarded as a very hard problem in the field. Recently Chen and Cheng [22,23,24] have solved a major conjecture that existence of csck is equivalent to well-studied conditions such as properness of Mabuchi’s K-energy, or geodesic stability. The first named author [49] proved the following counterpart in Sasaki setting,
Theorem 1
[49] There exists a Sasaki metric with constant scalar curvature if and only if the \(\mathcal {K}\)-energy is reduced proper with respect to \(\text {Aut}_0(\xi , J)\), the identity component of automorphism group which preserves the Reeb vector field and transverse complex structure.
The proof of Theorem 1 is an adaption of recent breakthrough of Chen–Cheng [24] on the existence of csck in Kähler setting to Sasaki setting. Technically, the arguments consist of two major parts: a priori estimates of non-linear PDE and pluripotential theory. Building up on previous development of pluripotential theory, Darvas [28, 29] has developed profound theory to study the geometric structure of space of Kähler potentials. Among others, he introduced a Finsler metric \(d_1\), and proved very effective estimates of distance function \(d_1\) in terms of well-studied energy functionals such as Aubin’s I-functional. Darvas’s results turn out to be very useful to understand the geometric structure of space of Kähler potentials, in particular in the study of csck [6, 24, 32]. In this paper, we extend many results in pluripotential theory on Kähler manifolds, notably in [28, 29, 44] to Sasaki setting. These results play an important role in the proof of Theorem 1. To prove these results, we need to explore the geometric structures of Sasaki manifolds, in particular the Kähler cone structure and transverse Kähler structure.
Let (M, g) be a compact Riemannian manifold of dimension \(2n+1\), with a Riemannian metric g. Sasaki manifolds have very rich geometric structures and have many equivalent descriptions. A probably most straightforward formulation is as follows: its metric cone
is a Kähler cone. Hence there exists a complex structure J on X such that \(( g_X, J)\) defines a Kähler structure. We identify M with its natural embedding \(M\rightarrow \{r=1\}\subset X\). The 1-form \(\eta \) is given by \(\eta =J(r^{-1}\mathrm{{d}}r)\) and it defines a contact structure on M. The vector field \(\xi :=J(r\partial _r)\) is a nowhere vanishing, holomorphic Killing vector field and it is called the Reeb vector field when it is restricted on M. The integral curves of \(\xi \) are geodesics, and give rise to a foliation on M, called the Reeb foliation. Then there is a Kähler structure on the local leaf space of the Reeb foliation, called the transverse Kähler structure. A standard example of a Sasaki manifold is the odd-dimensional round sphere \(S^{2n+1}\). The corresponding Kähler cone is \(\mathbb {C}^{n+1}\backslash \{0\}\) with the flat metric and its transverse Kähler structure descends to \(\mathbb {CP}^n\) with the Fubini-Study metric.
We can also formulate Sasaki geometry, in particular the transverse Kähler structure via its contact bundle \(\mathcal {D}=\text {Ker}(\eta )\subset TM\). The complex structure J on the cone descends to the contact bundle via \(\Phi :=J|_\mathcal {D}\). The Sasaki metric can be written as follows,
where \(g^T\) is the transverse Kähler metric, given by \(g^T:=2^{-1}\mathrm{{d}}\eta (\Phi \otimes \mathbb {I})\). The transverse Kähler form is denoted by \(\omega ^T=2^{-1}\mathrm{{d}}\eta \). We shall study the transverse Kähler geometry of Sasaki metrics, with the Reeb vector field \(\xi \) and transverse complex structure (equivalently the complex structure J on the cone) both fixed. This means that we fix the basic Kähler class \([\omega ^T]\) with \(\omega ^T=2^{-1}\mathrm{{d}}\eta \) and study the Sasaki structures induced by the space of transverse Kähler potentials,
where \(C_B^\infty (M)\) is the space of smooth basic functions. The main result in the paper is:
Theorem 2
\((\mathcal {E}_p(M, \xi , \omega ^T), \mathrm{{d}}_p)\) is a complete geodesic metric space for \(p\in [1, \infty )\), which is the metric completion of \((\mathcal {H}, \mathrm{{d}}_p)\). For any \(u, v\in \mathcal {E}_p(M,\xi ,\omega ^T)\), \(\mathrm{{d}}_p(u, v)\) is realized by a unique finite-energy geodesic in \(\mathcal {E}_p(M,\xi ,\omega ^T)\) connecting u and v. There exists a uniform constant \(C=C(n, p)>1\) such that
where the energy functional \(I_p\) is given by
Moreover, we have
We refer to Sect. 3 for notions such as \(\mathcal {E}_p(M,\xi ,\omega ^T), \mathrm{{d}}_p\). Theorem 2 is the counterpart of main results in [28] in Sasaki setting. An important notion in the study of csck is the convexity of \(\mathcal {K}\)-energy along \(C^{1, \bar{1}}\) geodesics [3] (see also [25]), which was generalized to Sasaki setting by [51, 58]. Given the results above, one can then extend \(\mathcal {K}\)-energy to \(\mathcal {E}_1\)-class and keep its convexity along finite energy geodesics as in [7]. Moreover, this allows to define precisely the properness of \(\mathcal {K}\)-energy in terms of the distance \(d_1\). One can then prove Theorem 1 using a priori estimates of scalar curvature-type equation together with properness assumption, where the effective estimates of \(d_1\) in Theorem 2 play an important role; for details, see [49].
Along the way to prove Theorem 2, it is necessary to extend results as in [30, 44] to Sasaki setting. Certainly the essential ideas lie in results in Kähler setting and many results are rather straightforward extensions from Kahler setting; we refer to Darvas’ lecture notes [30] for an excellent reference. However, we should also emphasize that in Sasaki setting, there are several substantial new difficulties when the Reeb foliation is irregular. To overcome these difficulties, the Sasaki structure (the Kähler cone structure and transverse Kähler structure) plays an essential role. Lemma 3.1 is an extension of Blocki–Kolodziej’s approximation of plurisubharmonic functions by smooth decreasing sequence. For this proof we construct explicit holomorphic charts on the Kähler cone out of its transverse Kähler structure, see Lemma 2.1. This very explicit relation between the holomorphic charts and foliation charts of transverse Kähler structure seems to appear in literature for the first time, to the authors’ knowledge. This explicit construction of holomorphic charts builds a very straightforward relation between plurisubharmonic functions on cone and (transverse) plurisubharmonic functions via transverse Kähler structure. Lemmas 3.2 and 3.3 give Darvas’ volume partition formula for rooftop construction. This decomposition is a very important technical result for Darvas’ theory and the proof in Kähler setting does not carry over for irregular Sasaki structures. We overcome this difficulty using Type-I deformation (see Theorem 6.1, Lemmas 6.1 and 6.2). (Of course there are many other places that there are substantial new difficulties; for example, the geodesic equation solved by Guan–Zhang is harder.) For completeness we include the details of almost all arguments, even in the case when the proof follows rather straightforwardly from the Kähler setting. The pluripotential theory in Sasaki setting has few references (see [51, 58] for example) and we hope that our presentation is helpful.
We organize the paper as follows. In Sect. 2 we introduce basic notations and concepts of Sasaki geometry. We study the geometric structure of the space of transverse Kähler potentials using geodesic equation and pluripotential theory in Sect. 3. In Sect. 4 we prove the main theorem. We include a brief discussion of Sasaki-extremal metric in Sect. 5. Appendix contains various topics in pluripotential theory, including complex Monge–Ampere operator and various energy functionals on \(\mathcal {E}_1\); we prove various results which are stated in [49, Section 2.2].
2 Preliminary on Sasaki Geometry
A good reference on Sasaki geometry can be found in the monograph [13] by Boyer–Galicki. Let M be a compact differentiable manifold of dimension \(2n+1 (n\ge 1)\). A Sasaki structure on M is defined to be a Kähler cone structure on \(X=M\times \mathbb {R}_{+}\), i.e., a Kähler metric \((g_X, J)\) on X of the form
where \(r>0\) is a coordinate on \(\mathbb {R}_{+}\), and g is a Riemannian metric on M. We call \((X, g_X, J)\) the Kähler cone of M. We also identify M with the link \(\{r=1\}\) in X if there is no ambiguity. Because of the cone structure, the Kähler form on X can be expressed as
We denote by \(r\partial _r\) the homothetic vector field on the cone, which is easily seen to be a real holomorphic vector field. A tensor \(\alpha \) on X is said to be of homothetic degree k if
In particular, \(\omega \) and g have homothetic degree two, while J and \(r\partial _r\) has homothetic degree zero. We define the Reeb vector field
Then \(\xi \) is a holomorphic Killing field on X with homothetic degree zero. Let \(\eta \) be the dual one-form to \(\xi \):
We also use \((\xi , \eta )\) to denote the restriction of them on (M, g). Then we have
-
\(\eta \) is a contact form on M, and \(\xi \) is a Killing vector field on M which we also call the Reeb vector field;
-
\(\eta (\xi )=1, \iota _{\xi } \mathrm{{d}}\eta (\cdot )=\mathrm{{d}}\eta (\xi , \cdot )=0\);
-
the integral curves of \(\xi \) are geodesics.
The Reeb vector field \(\xi \) defines a foliation \(\mathcal {F}_\xi \) of M by geodesics. There is a classification of Sasaki structures according to the global property of the leaves. If all the leaves are compact, then \(\xi \) generates a circle action on M, and the Sasaki structure is called quasiregular. In general, this action is only locally free, and we get a polarized orbifold structure on the leaf space. If the circle action is globally free, then the Sasaki structure is called regular, and the leaf space is a polarized Kähler manifold. If \(\xi \) has a non-compact leaf, the Sasaki structure is called irregular.
One can also understand Sasaki structure through contact metric structure. There is an orthogonal decomposition of the tangent bundle
where \(L\xi \) is the trivial bundle generated by \(\xi \), and \(\mathcal {D}=\text {Ker}(\eta )\). The metric g and the contact form \(\eta \) determine a (1, 1) tensor field \(\Phi \) on M by
\(\Phi \) restricts to an almost complex structure on \(\mathcal {D}\):
Since both g and \(\eta \) are invariant under \(\xi \), there is a well-defined Kähler structure \((g^T, \omega ^T, J^T)\) on the local leaf space of the Reeb foliation. We call this a transverse Kähler structure. In the quasiregular case, this is the same as the Kähler structure on the quotient. Clearly, \(\omega ^T=2^{-1}\mathrm{{d}}\eta . \) The upper script T is used to denote both the transverse geometric quantity, and the corresponding quantity on the bundle \(\mathcal {D}\). For example, we have on M
From the above discussion it is not hard to see that there is an intrinsic formulation of a Sasaki structure as a compatible integrable pair \((\eta , \Phi )\), where \(\eta \) is a contact one-form and \(\Phi \) is an almost CR structure on \(\mathcal {D}=\text {Ker}\eta \). Here “compatible” means first that \(\mathrm{{d}}\eta (\Phi U, \Phi V)=\mathrm{{d}}\eta (U, V)\) for any \(U, V\in \mathcal {D}\), and \(\mathrm{{d}}\eta (U, \Phi U)>0\) for any non- zero \(U\in \mathcal {D}\). Further, we require \(\mathcal {L}_{\xi }\Phi =0\), where \(\xi \) is the unique vector field with \(\eta (\xi )=1\), and \(\mathrm{{d}}\eta (\xi , \cdot )=0\). \(\Phi \) induces a splitting
with \(\overline{\mathcal {D}^{1,0}}=\mathcal {D}^{0,1}\). “Integrable” means that \([\mathcal {D}^{0,1}, \mathcal {D}^{0,1}]\subset \mathcal {D}^{0,1}\). This is equivalent to that the induced almost complex structure on the local leaf space of the foliation by \(\xi \) is integrable. For more discussions on this, see [13, Chapter 6].
Definition 2.1
A p-form \(\theta \) on M is called basic if
Let \(\Lambda ^p_B\) be the bundle of basic p-forms and \(\Omega ^p_B=\Gamma (S, \Lambda ^p_B)\) the set of sections of \(\Lambda ^p_B\).
The exterior differential preserves basic forms. We set \(\mathrm{{d}}_B=d|_{\Omega ^p_B}\). Thus the subalgebra \(\Omega _{B}(\mathcal {F}_\xi )\) forms a subcomplex of the de Rham complex, and its cohomology ring \(H^{*}_{B}(\mathcal {F}_\xi )\) is called the basic cohomology ring. When \((M, \xi , \eta , g)\) is a Sasaki structure, there is a natural splitting of \(\Lambda ^p_B\otimes \mathbb {C}\) such that
where \(\Lambda ^{i, j}_B\) is the bundle of type (i, j) basic forms. We thus have the well-defined operators
Then we have \(\mathrm{{d}}_B=\partial _B+\bar{\partial }_B\). Set \(\mathrm{{d}}^c_B=\frac{1}{2}\sqrt{-1}\left( \bar{\partial }_B-\partial _B\right) .\) It is clear that
We shall recall the transverse complex (Kähler) structure on local coordinates. Let \(U_\alpha \) be an open covering of M and \(\pi _\alpha : U_\alpha \rightarrow V_\alpha \subset \mathbb {C}^n\) submersions such that
is biholomorphic when \(U_\alpha \cap U_\beta \) is not empty. One can choose local coordinate charts \((z_1, \ldots , z_n)\) on \(V_\alpha \) and local coordinate charts \((x, z_1, \ldots , z_n)\) on \(U_\alpha \subset M\) such that \(\xi =\partial _x\), where we use the notations
The map \(\pi _\alpha : (x, z_1, \ldots , z_n)\rightarrow (z_1, \ldots , z_n)\) is then the natural projection. There is an isomorphism, for any \(p\in U_\alpha \),
Hence the restriction of g on \(\mathcal {D}\) gives an Hermitian metric \(g^T_\alpha \) on \(V_\alpha \) since \(\xi \) generates isometries of g. One can verify that there is a well-defined Kähler metric \(g_\alpha ^T\) on each \(V_\alpha \) and
gives an isometry of Kähler manifolds \((V_\alpha , g^T_\alpha )\). The collection of Kähler metrics \(\{g^T_\alpha \}\) on \(\{V_\alpha \}\) can be used as an alternative definition of the transverse Kähler metric. The (local) transverse holomorphic (Kähler) structure is essential for us and we shall use these charts enormously. We summarize as follows:
Definition 2.2
(Local foliation charts) We can choose the open covering \(\{U_\alpha \}\) of M such that there exists a local product structure for each \(\alpha \), determined by its foliation structure and transverse complex structure. That is, there are charts
where \(W_\alpha = (-\delta , \delta ) \times V_\alpha .\) For a point \(p\in W_\alpha \), we write \(p=(x, z_1, \ldots , z_n)\) with \(\xi =\partial _x\) and \(V_\alpha = B_r(0)\subset \mathbb {C}^n\) for \(0<r\) . We assume that \(\delta , r\) are sufficiently small depending only on \((M, \xi , \eta , g)\), and \(\omega ^T_\alpha \) is uniformly equivalent to an Euclidean metric on each \(V_\alpha =B_r\subset \mathbb {C}^n\),
In Sasaki geometry, it is often mostly convenient to work with these charts when we need to consider the Sasaki structure locally. For each \(U_\alpha \), we assume it is contained in the geodesic normal neighborhood of its “center,” \(\Psi _\alpha ^{-1}(0, 0, \ldots , 0)\), by choosing \(\delta , r\) small enough. We call these charts foliation charts. The existence of foliation charts is well known in the subject, see [40]; in particular, any Sasaki metric g can be locally expressed in terms of a real function of 2n variables. Given a foliation chart \(W_\alpha =(-\delta , \delta )\times V_\alpha \), for \((x, z_1, \ldots , z_n)\in U_\alpha \), locally there exists a strictly plurisubharmonic function \(h: V_\alpha \rightarrow \mathbb {R}\), and the Sasaki structure reads
If we consider a Sasaki structure induced by a transverse Kähler potential \(\phi \), then locally we have \(h\rightarrow h+\phi \). In particular, we have
We shall also use holomorphic charts on its Kähler cone X. There exist indeed holomorphic charts on the Kähler cone X which are closely related to foliation charts on M. This seems to be much less well known and we shall describe them now.
Lemma 2.1
(Holomorphic coordinates on the Kähler cone) For a Sasaki structure locally generated by a plurisubharmonic function \(h: V_\alpha \rightarrow \mathbb {R}\) in foliation charts on M, then the following gives a local holomorphic structure on its Kähler cone X, for \(w=(w_0, \ldots , w_n)\in \tilde{U}_\alpha \subset \mathbb {C}\times V_\alpha \),
The holomorphic structure J is given by the holomorphic coordinates \(w=(w_0, \ldots , w_n)\),
Proof
Given (2.1), it is straightforward to check that (2.2) gives a holomorphic chart satisfying (2.3). \(\square \)
Remark 2.1
These holomorphic charts would be very useful for us later; in particular, when we consider plurisubharmonic functions on X and transverse plurisubharmonic functions on M. The explicit holomorphic charts given above seem to appear in literature first time to our knowledge, while the foliation charts are well known.
When the Reeb vector field \(\xi \) is irregular, the local foliation charts satisfy cocycle condition but they do not give a manifold (or orbifold) structure of the quotient \(M/\mathcal {F}_\xi \). We shall recall Type-I deformation defined in [15]. Let \((M, \xi _0, \eta _0, g_0)\) be a compact Sasaki manifold, denote its automorphism group by \(\text {Aut}(M, \xi _0, \eta _0, g_0)\). We fix a torus
Definition 2.3
(Type-I deformation) Let \((M, \xi _0, \eta _0, g_0)\) be a T-invariant Sasaki structure. For any \(\xi \in \mathfrak {t}\) such that \(\eta _0(\xi )>0\). We define a new Sasaki structure on M explicitly as
Note that under Type-I deformation, the essential change is the Reeb vector field \(\xi _0 \leftrightarrow \xi \) and this construction can be done vice versa.
3 The Space of Transverse Kähler Potentials
In this section, we consider the space of transverse Kähler potentials on a compact Sasaki manifold through its transverse Kähler structure. It turns out to be necessary to consider these objects not only from point of view of PDE, but also from the point of view of pluripotential theory. Geometric pluripotential theory on Kähler manifolds turns out to be one crucial piece in the proof of properness conjecture [6, 24]. We refer [30, 44] and references therein for details of pluripotential theory. We extend these results to Sasaki manifolds. These results would form a crucial piece for existence of constant scalar curvature (cscs) on Sasaki manifolds as well, see [49] for details.
Using the transverse Kähler structure of a Sasaki structure, many of the extensions of pluripotential theory on Kähler manifolds to Sasaki manifolds are rather straightforward, and the proofs are a direct adaption of Kähler setting with some necessary modifications. On the other hand, there are several exceptions that would need essential inputs from the Sasaki structure. And the proofs are new and substantially different, compared with the Kähler setting. We summarize the main differences as follows. The first is Lemma 3.1, where we will prove a counterpart of an approximation result of plurisubharmonic functions as in Kähler setting by Blocki–Kolodziej [10]. One can apply Blocki–Kolodziej approximation locally to transverse Kähler structure and obtain a local approximation, but such construction has trouble to patch together when the Sasaki structure is irregular. Instead, we need to do the construction on the Kähler cone, and the holomorphic chart on the cone (Lemma 2.1) plays a substantial role in our construction. The second main difference is Lemma 3.2, where we will prove an important property of the rooftop envelop construction \(P(u_0, u_1)\) on the non-contact set; this result plays a very important role in Darvas’s theory. The proof as in Kähler setting again does not work directly to Sasaki setting when the Sasaki structure is irregular. Instead, we need to apply a Type-I deformation carefully (Theorem 6.1, Lemma 6.1) to bypass the difficulty.
3.1 The Quasiplurisubharmonic Functions on Sasaki Manifolds
Denote \(\mathcal {H}=\{\phi \in C^\infty _B(M): \omega _\phi =\omega ^T+\sqrt{-1}\partial _B\bar{\partial } _B \phi >0\}\), the space of transverse Kähler potentials on a Sasaki manifold \((M, \xi , \eta , g)\). Given \(\phi \in \mathcal {H}\), it defines a new Sasaki structure, \((M, \xi , \eta _\phi , g_{\eta _\phi })\) as follows,
The most relevant results in pluripotential theory for us lie in [44, 5, Section 2], [45] and [30]. Part of them has been done by van Covering [58, Section 2], including the Monge–Ampere operator and weak convergence, with main focus on \(L^\infty \) and \(C^0\) potentials. We shall need most of the results on the energy classes \(\mathcal {E}\) and \(\mathcal {E}_p\) (defined below).
Given a Sasaki structure \((M, \xi , \eta , g)\), we recall the following definition,
Definition 3.1
An \(L^1\), upper semicontinuous (usc) function \(u: M\rightarrow \mathbb {R}\cup \{-\infty \}\) is called a transverse \(\omega ^T\)-plurisubharmonic (TPSH for short) if u is invariant under the Reeb flow, and u is \(\omega ^T\)-plurisubharmonic on each local foliation chart \(V_\alpha \), that is \(\omega ^T_\alpha +\sqrt{-1}\partial _B\bar{\partial }_B u\ge 0\) as a (1, 1)-current on \(V_\alpha \).
It is apparent that the definition above does not depend on the choice of foliation charts. Indeed, u is invariant along the flow of \(\xi \) and we extend u trivially in the cone direction to a function on cone. Using the holomorphic structure on the cone (see Lemma 2.1), u is a TPSH if and only if \(\omega ^T+\sqrt{-1}\partial \bar{\partial } u\ge 0\) is a closed, positive (1,1) current on the cone X. We use the notation,
One of the cornerstones of Bedford–Taylor theory [2] is to associate a complex Monge–Ampere measure to a bounded psh function. Their construction generalizes to bounded Kähler potentials in a straightforward manner [44] and it has direct adaption to Sasaki setting. We refer to [58, Section 2] and Sect. 1 for definition of complex Monge–Ampere measures \(\omega _u^n\wedge \eta \) for \(u\in \text {PSH}(M,\xi ,\omega ^T)\cap L^{\infty }\) on Sasaki manifolds, which is a direct adaption of Bedford–Taylor theory [2].
Proposition 3.1
Suppose that the sequence \(u_j \in \text {PSH}(M,\xi ,\omega ^T)\cap L^{\infty }\) decreases to \(u \in \text {PSH}(M,\xi ,\omega ^T)\cap L^{\infty }\). Then for \(k=1, \ldots , n\), we have the following weak convergences of complex Monge–Ampere measures,
Proof
By applying a partition of unity subordinated to covering by foliation charts, we need to show that for \(f\in C^\infty \), supported on a foliation chart \(W_\alpha =(-\delta , \delta )\times V_\alpha \)
We should emphasize that f is not a basic function in general. The weak convergence in Kähler setting implies that for each \(x\in (-\delta , \delta )\)
Note that for each x, f is supported on \(V_\alpha \). Taking integration with respect to \(\mathrm{{d}}x\), this leads to (3.2), since on \(W_\alpha \), \(\omega _{u}^k\wedge (\omega ^T)^{n-k} \wedge \eta =\omega _{u}^k\wedge (\omega ^T)^{n-k} \wedge \mathrm{{d}}x\) as a product measure. \(\square \)
The following Bedford–Taylor identity in Sasaki setting would be used numerously:
Proposition 3.2
For \(u, v\in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty \),
Proof
We only need to prove this in foliation charts. Recall for each foliation chart \(W_\alpha =(-\delta , \delta )\times V_\alpha \), \(V_\alpha =B_r(0)\subset \mathbb {C}^n\) gives the local transverse complex structure. For a point \(p\in W_\alpha \), we write \(p=(x, z)\) with \(\xi =\partial _x\). Given \(u\in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty \) it defines a Kähler current \(\omega _u^n\) on \(V_\alpha \). Since both u and v are basic functions, u, v are independent of x in \(W_\alpha \). Hence on \(W_\alpha \cap \{u>v\}=(-\delta , \delta )\times \{z\in V_\alpha : u>v\}.\) Note that \(\omega ^T_u\wedge \eta \) is invariant along the Reeb direction, and it coincides with the product measure \(\mathrm{{d}}x\wedge \omega _u^n\) on \(W_\alpha =(-\delta , \delta )\times V_\alpha \). On each \(W_\alpha \), we have
To prove (3.3), it reduces to show that
This is just the Bedford–Taylor identity [2]. \(\square \)
It is possible to generalize the Bedford–Taylor constructions to a much larger class on a compact Kähler manifold, see Guedj–Zeriahi [44]. The reference [30, Section 2] is sufficient for our purpose. These constructions in Kähler setting have a direct extension to Sasaki setting, where Proposition 3.2 plays an important role. First we prove the following well-known result in pluripotential theory.
Proposition 3.3
There exists \(C=C(M, g)\) such that for any \(u\in \text {PSH}(M, \xi , \omega ^T)\),
Proof
When u is \(C^2\) this is obvious by the fact that \(\Delta _g u+n\ge 0\). In general, we can prove this using sub-mean value property of plurisubharmonic functions, similar as in [30, Lemma 3.45]. In this proof, we can either use foliation charts on M or Kähler cone structure on \(X=C(M)\). We use foliation charts in this argument.
We assume \(\sup _M u=0\) and want to show that the integration of u is uniformly bounded below. We can cover M by finitely many nested foliation charts \(U_k\subset W_k\subset M (1\le k \le N)\) such that there exist diffeomorphisms \(\varphi _k: B(0, 4)\times (-2\delta , 2\delta )\rightarrow W_k\) with \(\varphi _k (B(0, 1)\times (-\delta , \delta ))=U_k\), where \(\delta \) is a fixed positive constant and \(B(0, 1)\subset B(0, 4)\subset \mathbb {C}^n\) are Euclidean balls centered at the origin in \(\mathbb {C}^n\). We assume that \((z, x)\in B(0, 4)\times (-2\delta , 2\delta )\) such that \(z\in B(0, 4)\) represents transverse holomorphic charts and \(x\in (-2\delta , 2\delta )\) represents the Reeb direction (i.e., \(\xi =\partial _x\)). On each \(W_k\), there exists a smooth basic function \(\psi _k=\psi _k(z)\) such that \(\omega ^T=\sqrt{-1}\partial _z\bar{\partial }_z \psi _k\). Note that we only need to show that, there exists a uniformly bounded constant \(C>0\), such that
Note that u is basic, we have
where \(\mathrm{{d}}\mu _{x, z}\) and \(\mathrm{{d}}\mu _z\) are Euclidean measure on \(\mathbb {C}^n\times \mathbb {R}\) and \(\mathbb {C}^n\), respectively. Hence we only need to show that
Note that by our construction, \((\psi _k+u)\circ \varphi _k\) is independent of x and is plurisubharmonic on B(0, 4) for each k. As u is usc, its supremum is realized at some point \(p_1\in M\) such that \(u\le u(p_1)=0\). Since \(U_k\) covers M, we can assume \(p_1\in U_1\) with the coordinate \(\varphi _1(z_1, x_1)=p_1\) for some \((z_1, x_1)\in B(0, 1)\times (-\delta , \delta )\). Note that since u is basic, hence it is independent of x-coordinate we can also take \(x_1=0\). Since \(B(z_1, 2)\subset B(0, 4)\), we have the following sub-mean value property for \((\psi _1+u)\circ \varphi _1\),
Since \(u\le 0\) and \(B(0, 1)\subset B(z_1, 2)\), there exists \(C_1>0\), independent of u, such that
Since \(\{U_k\}_k\) covers M, we can assume \(U_1\) intersects \(U_2\). We can choose \(r_2>0\), such that \(\varphi _2(B(z_2, r_2)\times (\delta _1, \delta _2))\subset U_1\cap U_2\) for some \(B(z_2, r_2)\subset B(0, 1)\) and \(-\delta<\delta _1<\delta _2<\delta \). Since \(u\le 0\), it follows that there exists \(\tilde{C}_1>0\), independent of u (\(\tilde{C}_1\) depends only on \(C_1\), \(r_2\), and \(\psi _2\)), such that
Since \((u+\psi _2) \circ \varphi _2\) is plurisubharmonic in B(0, 4), we can obtain that
Since \(u\le 0\) and \(B(0, 1)\subset B(z_2, 2)\), we obtain for some \(C_2>0\)
We continue this process to consider that \(U_1\cup U_2\) intersects a member, say \(U_3\). After at most \(N-2\) step, we prove (3.4). \(\square \)
As a direct consequence, we know the following (see [33, Proposition I.5.9]):
Proposition 3.4
The set \(\mathcal {C}=\{u\in \text {PSH}(M, \xi , \omega ^T): \sup _M u\le C\}\) is bounded in \(L^1\) and it is precompact in \(L^1(\mathrm{{d}}\mu _g)\) topology.
Proof
By the above-mentioned proposition, we know that \(\sup _M u\) bounded above implies that \(\int _M |u|\mathrm{{d}}\mu _g\) is uniformly bounded. By the Motel property of subharmonic functions and plurisubharmonic functionals [33, Propositions I.4.21, I.5.9], \(\mathcal {C}\) is precompact with respect to \(L^1(\mathrm{{d}}\mu _g)\) topology. Note that in Sasaki setting we apply the compactness of plurisubharmonic functions to nested foliations charts \(U_k\subset W_k\) as above for \(\omega ^T_k\)-plurisubharmonic functions locally, that \(\mathcal {C}\) is precompact in \(L^1\) topology in each \(U_k\). After passing by subsequence if necessary, we can then get weak compactness of \(\mathcal {C}\) with respect to \(L^1(\mathrm{{d}}\mu _g)\) topology. \(\square \)
Let \(v\in \text {PSH}(M, \xi , \omega ^T)\). For \(h\in \mathbb {R}\), we denote \(v_h=\max \{v, -h\}\) to be the canonical cutoffs of v. It is evident that \(v_h\) is invariant under the Reeb flow and hence \(v_h\in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty \). If \(h_1<h_2\), then Proposition 3.2 implies that
Hence \(\chi _{\{v>-h\}}\omega ^n_{v_{h}}\wedge \eta \) is an increasing sequence of Borel measure on M with respect to h. This leads to the following definition:
Definition 3.2
We define
We shall emphasize that by the definition above, we have for any Borel set \(B\subset M\),
Hence the convergence in (3.6) is a stronger than the weak convergence of measures.
To proceed, we need the following approximation of TPSH functions. Our proof uses the Kähler cone structure and builds up on Blocki–Kolodziej [10].
Lemma 3.1
Given \(u\in \text {PSH}(M, \xi , \omega ^T)\), there exists a decreasing sequence \(\{u_k\}_{k \in \mathbb {N}}\subset \mathcal {H}\) such that \(u_k\) converges to u.
Proof
First we assume that u has zero Lelong number. Recall X is the Kähler cone and we identify M with the link \(\{r=1\}\subset X\). For \(u\in \text {PSH}(M, \xi , \omega ^T)\), we extend u to be a function on X such that \(u(r, p)=u(p)\), for any \(r>0\). We recall that \(\omega ^T=\frac{1}{2}\mathrm{{d}}\eta =\mathrm{{dd}}^c(\log r)=\sqrt{-1}\partial \bar{\partial } (\log r)\). Hence for \(u\in \text {PSH}(M, \xi , \omega ^T)\), we have the following,
In other words, \(v=u+\log r\) is a plurisubharmonic function on X. This is transparent in foliations charts and corresponding holomorphic charts as in Lemma 2.1. Let \(h_\alpha \) be a local potential for \(\omega ^T\) in a foliation chart \(V_\alpha \), and we write \(h=h(w_1, \bar{w}_1, \ldots , w_n, \bar{w}_n)\) in the holomorphic chart on cone, then \(\log r=h_\alpha +\text {Re}(w_0)\). Denote \(\omega _X\) to be the Kähler form on X. Since u has zero Lelong number, applying Blocki–Kolodziej [10, Theorem 2], we get a sequence of smooth functions \(v_k\) converges to u, decreasing in k, such that on \(X^{'}=\{2^{-1} \le r \le 2\}\subset X\)
We can assume in addition that \(v_k\) is invariant under the flow of \(\xi \), by taking average with respect to the torus action generated by \(\xi \in \text {Aut}(\xi , \eta , g)\). We define a basic function \(u_k\) on M such that, by taking \(r=1\), \(u_k=v_k|_{r=1}\).
Now for any point on \(X^{'}\), we choose holomorphic charts \(\tilde{U}_\alpha \) as in Lemma 2.1 to cover \(X^{'}\). We write the function in a holomorphic chart as
We recall the relation between the holomorphic charts and the foliation charts,
Note we assume that \(v_k\) is invariant under the flow of \(\xi \), hence \(v_k\) is independent of \(x=\text {Im}(w_0)\). We write \(v_k\) as follows, using (3.9),
Locally, this gives
The tangent space \(T_pX\) is given by, in terms of coordinate \((r, x, z_1, \ldots , z_n)\),
Note that the contact bundle \(D_p=\text {span}\{X_i, X_{\bar{i}}, i=1, \ldots , n\}\). For \(p\in M\subset X\), we can assume that \(h(z, \bar{z})=\partial h=\bar{\partial } h=0, h_{i\bar{j}}=\delta _{i\bar{j}}\) at p, and hence
By (3.8), we compute (at p),
where t stands for the first argument of \(v_k\). This is equivalent to the following, on M we have,
It is clear that \(u_k\) converges to u, deceasing in k. Without loss of generality, we can assume that \(u\le -1\) and \(u_k\le 0\). It follows that \(k(k+2)^{-1}u_k\in \mathcal {H}\) such that \(k(k+2)^{-1}u_k\) converges to u, decreasing in k. This completes the proof when u has zero Lelong number.
Now suppose \(u\in \text {PSH}(M, \xi , \omega ^T)\). We consider the canonical cutoffs \(u_j=\max \{u, -j\}\in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty \). By the above statements, we know that for each j, there exists a sequence of smooth functions \(\{v_j^k\}_k\subset \mathcal {H}\) which decreases to \(u_j\). By adding a small constant \(k^{-1}\) to each \(v^k_j\), we can assume that \(\{v^k_j\}_k\) strictly decreases (for each j). Then for each k, we can find \(k_{j+1}\) such that
Indeed we consider the open set \(U^l:=\{x\in M: v_{j+1}^l<v^k_j\}\). Clearly \(\{U^l\}_l\) is an increasing sequence of open sets such that \(\cup _l U^l=M\), since
Since M is compact, there exists \(k_{j+1}\) such that \(U^{k_{j+1}}=M\). By (3.12), we can find a sequence \(\{v_{j}^{k_j}\}_j \subset \mathcal {H}\) inductively such that \(v_j^{k_j}\searrow u\). This completes the proof. \(\square \)
Remark 3.1
The Kähler cone structure, in particular, the relation between holomorphic charts and foliation charts as in Lemma 2.1, plays a very important role in Sasaki setting. If the Reeb vector field is irregular, the approximation from transverse Kähler structure can produce local approximation. But it seems to be hard to patch such a local construction together when the Reeb vector field is irregular. Instead we do approximation on the Kähler cone. We shall mention that in (3.12), the assumption that each sequence \(\{v_j^k\}_k\) strictly decreases is necessary. For example, we can take \(u=1\) over [0, 1], \(v=0\) over [0, 1) and \(v(1)=1\). We can choose \(u_k=1\) for each k, and \(v_k(x)=x^k+k^{-1}\). Then \(v\le u\) and \(\{u_k\}_k\) decreases to u and \(v_k\) (strictly) decreases to v. But for \(\{u_k\}_k\) and \(\{v_k\}_k\), (3.12) does not hold: given \(u_k\), there does not exist l such that \(v_{l}\le u_k\) since \(v_{l}(1)>1\) for all l.
As a direct consequence, we have the following (just as in Kähler setting, see [30, Lemma 2.2]),
Proposition 3.5
For \(u\in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty \),
Proof
By Lemma 3.1, we can choose a smooth sequence \(u_k\) converges to u as a decreasing sequence. It then follows from Bedford–Taylor theory (see Proposition 3.1) that \(\omega _{u_k}^n\wedge \eta \) converges to \(\omega _u^n\wedge \eta \) weakly, we obtain (3.13). \(\square \)
It is then clear that, given (3.6), we have only \(\int _M \omega ^n_v\wedge \eta \le \text {Vol}(M)\) for \(v\in \text {PSH}(M, \xi , \omega ^T)\).
Definition 3.3
We define the full-mass elements in \(\text {PSH}(M, \xi , \omega ^T)\) as
As in Kähler case, many of the properties that hold for bounded TPSH functions hold for elements of \(\mathcal {E}(M, \xi , \omega ^T)\) as well. We include the comparison principle, monotonicity property, and generalized Bedford–Taylor identity as follows. These properties are proved in [44] for Kähler setting. Given (3.3) and (3.13), our proof follows almost identical as in Kähler setting (see [44, Theorem 1.5, Proposition 1.6, Corollary 1.7]). Nevertheless, we include the details.
Proposition 3.6
(Comparison principle) Suppose \(u, v\in \mathcal {E}(M, \xi , \omega ^T)\). Then
Proof
Our proof is similar to Kähler case, see [30, Proposition 2.3]. First we show (3.15) for u, v bounded. Using Propositions 3.2 and 3.5, we write
Replacing v by \(v+\epsilon \), we have
Recall that
Hence (3.15) for bounded potentials follows immediately by letting \(\epsilon \rightarrow 0\).
In general, let \(u_l=\max \{u, -l\}, v_{k}=\max \{v, -k\}, l, k\in \mathbb {N}\) be the canonical cutoffs of u, v respectively. We apply (3.15) for bounded potentials to get
Together with the inclusions \( \{v_l<u\}\subset \{v_l< u_k\}\subset \{v<u_k\} \), we have
Letting \(l\rightarrow \infty \), using the definition (3.6) on \(\omega ^n_{v_l}\wedge \eta \) and \(\{v<u\}=\cup _{l \in \mathbb {N}}\{v_l <u\}\), (3.16) gives
Letting \(k\rightarrow \infty \), using the definition (3.6) on \(\omega ^n_{u_k}\wedge \eta \) and \(\{v\le u\}=\cap _{k\in \mathbb {N}}\{v<u_k\}\), we get
The replacing v by \(v+\epsilon \) in the above inequality, we can then argue as in the bounded case, taking the limit \(\epsilon \rightarrow 0\) yields (3.15). \(\square \)
Proposition 3.7
(Monotonicity property) Suppose \(u\in \mathcal {E}(M, \xi , \omega ^T)\) and \(v\in \text {PSH}(M, \xi , \omega ^T)\). If \(u\le v\), then \(v\in \mathcal {E}(M, \xi , \omega ^T)\).
Proof
This is proved in [44, Proposition 1.6] in Kähler case and the Sasaki case is almost identical. First we show that \(\psi =v/2\in \mathcal {E}(M, \xi , \omega ^T)\). We can assume that \(u\le v<-2\), hence \(\psi <-1\). This normalization gives the following inclusions for the canonical cutoffs \(u_j, v_j, \psi _j\),
By Proposition 3.15 and the inclusions above, we have
Note that we have
Applying Proposition 3.2 to \(\max \{u_{2j}, -j\}=u_{j}\) on the set \(\{u_{2j}>-j\}=\{u_j>-j\}\), we have
It then follows that
By definition of \(u\in \mathcal {E}(M, \xi , \omega ^T)\), it follows that, as \(j\rightarrow \infty \),
Hence \(\psi =v/2\in \mathcal {E}(M, \xi , \omega ^T)\). To show that \(v\in \mathcal {E}(M, \xi , \omega ^T)\), we observe that \(\{v\le -2j\}=\{\psi \le -j\}\) and \(\omega _{\psi _j}\ge \omega _{v_{2j}}/2\), hence
By letting \(j\rightarrow \infty \), we can then conclude that \(v\in \mathcal {E}(M, \xi , \omega ^T)\). \(\square \)
Proposition 3.8
(Generalized Bedford–Taylor identity) For \(u\in \mathcal {E}(M, \xi , \omega ^T)\), \(v\in \text {PSH}(M, \xi , \omega ^T)\), then \(\max \{u, v\}\in \mathcal {E}(M, \xi , \omega ^T)\) and
Proof
Our argument is identical to the Kähler setting; see [44, Corollary 1.7] and [30, Lemma 2.5]. Proposition 3.7 implies that \(w:=\max \{u, v\}\in \mathcal {E}(M, \xi , \omega ^T)\). Now observe that \(\max \{u_j, v_{j+1}\}=\max \{u, v, -j\}=w_j\). Since the cutoffs are bounded we have
By 3.7, we know that \(\chi _{\{u>v\}}\omega ^n_{u_j}\wedge \eta \rightarrow \chi _{\{u>v\}}\omega ^n_{u}\wedge \eta \) and \(\chi _{\{u>v\}}\omega ^n_{w_j}\wedge \eta \rightarrow \chi _{\{u>v\}}\omega ^n_{w}\wedge \eta \) as \(j\rightarrow \infty \) (we also use the fact that \(u, w\in \mathcal {E}(M, \xi , \omega ^T)\)). Since
it follows that
Similarly since
we also obtain that
By taking limit in (3.18) together with the limit facts above, we get the desired result. \(\square \)
Next we introduce finite-energy class on Sasaki manifolds, following [44]. By considering Young weights \(\chi \in \mathcal {W}^+_p\) (see [30, Chapter 1] for a short introduction to Young weights), one can introduce various finite-energy subclasses of \(\mathcal {E}(M, \xi , \omega ^T)\),
where \(E_\chi \) is the \(\chi \)-energy defined by
Of special importance are the weights \(\chi ^p(t)=|t|^p/p\) and the associated classes \(\mathcal {E}_p(M, \xi , \omega ^T)\). For theses weights it is clear that \(\mathcal {E}_p(M, \xi , \omega ^T) \subset \mathcal {E}_1(M, \xi , \omega ^T)\) for \(p \ge 1\). We will need the following straightforward fact:
Proposition 3.9
For any \(u\in \mathcal {E}_1(M, \xi , \omega ^T)\), u has Lelong number zero at every point.
Proof
For similar results in Kähler case, see [44, Corollary 1.8]. This is straightforward. We can assume \(\sup u=0\). For \(u\in \mathcal {E}_1(M, \xi , \omega ^T)\), we have
We consider locally \((0, 0)\in W_\alpha =(-\delta , \delta )\times V_\alpha \) in a foliation chart. Then we have
This implies that u has Lelong number zero at (0, 0). \(\square \)
The following result implies that to test membership in \(\mathcal {E}_\chi (M, \xi , \omega ^T)\) it is enough to test the finiteness condition \(E_\chi (u)<\infty \) on canonical cutoffs.
Proposition 3.10
Suppose \(u\in \mathcal {E}(M, \xi , \omega ^T)\) with canonical cutoffs \(\{u_k\}_{k\in \mathbb {N}}\). If \(h: \mathbb {R}_+\rightarrow \mathbb {R}_+\) is continuous and increasing, then
Moreover, if the above condition holds, then
Proof
Our proof is similar to the Kähler case, see [30, Proposition 2.6]. Without loss of generality we can assume that \(u \le 0\). If \(\limsup _{k\rightarrow \infty }\int _M h(|u_k|)\omega _{u_k}^n\wedge \eta <\infty \), we obtain that the sequence of Radon measures \(h(|u_k|)\omega ^n_{u_k} \wedge \eta \) is weakly compact. Hence there exists a subsequence \(h(|u_{k_j}|)\omega _{u_{k_j}}^n \wedge \eta \) converging weakly to a Radon measure \(\mu \). Recall that \(h(|u_{k_j}|)\) is an increasing sequence of lower semicontinuous functions converging to h(|u|) and \(\omega _{u_{k_j}}^n \wedge \eta \xrightarrow {w} \omega _u^n \wedge \eta \), this yields that \(h(|u|)\omega _u^n \wedge \eta \le \mu \) as measure. In particular \(\int _M \omega _u^n \wedge \eta \le \mu (M)<\infty \).
Now assume \(\int _M h(|u|)\omega _{u}^n \wedge \eta < \infty \). If \(\lim \nolimits _{t \rightarrow +\infty }h(t)=+\infty \), we have
It follows from Propositions 3.5, 3.8 and Definition 3.3 that
Then by Propositions 3.5, 3.8 and Definition 3.3 again we have
It follows that \(\int _M h(|u_k|)\omega _{u_k}^n\wedge \eta \) is bounded and \(\int _M h(|u|)\omega _u^n\wedge \eta =\lim \nolimits _{k\rightarrow \infty }\int _M h(|u_k|)\omega _{u_k}^n\wedge \eta \).
If \(\lim \nolimits _{t \rightarrow +\infty } h(t)=L<\infty \), it follows from Proposition 3.5 that \(\int _M h(|u_k|)\omega _{u_k}^n\wedge \eta \) is bounded. Moreover for any \(\epsilon >0\) there exists \(N>0\) such that \(0<L-h(t) <\epsilon \) for all \(t>N\). By Propositions 3.5, 3.8 and Definition 3.3 we have
for \(k>N\). It yields that \(\int _M h(|u|)\omega _u^n\wedge \eta =\lim _{k\rightarrow \infty }\int _M h(|u_k|)\omega _{u_k}^n\wedge \eta \). \(\square \)
With the proposition above, we can then prove the so-called fundamental estimate.
Proposition 3.11
(Fundamental estimate) Suppose \(\chi \in \mathcal {W}^+_p\) and \(u, v\in \mathcal {E}_\chi (M, \xi , \omega ^T)\) such that \(u\le v\le 0\). Then
Proof
The proof is similar to the Kähler case, see [44, Lemma 3.5]. First of all we assume that \(u, v \in \text {PSH}(M,\xi ,\omega ^T) \cap L^{\infty }\). For \(0 \le j \le n-1\), we have
Recall that \(\chi '(l) \le 0\) for \(l<0\). Using integration by parts, we have
Recall that \(\chi '(l) \le 0\) for \(l<0\) and \(l\chi '(l) \le p \chi (l)\) for \(l \ge 0\). Using the integration by parts repeatedly, we have
Combine the inequalities above we obtain
It follows that
In the general case \(u, v \in \mathcal {E}_{\chi }(M,\xi ,\omega ^T)\), we have \(E_{\chi }(v_k) \le (p+1)^nE_{\chi }(u_k)\) for the canonical cutoffs \(u_k, v_k\) of u, v. It follows from Proposition 3.10 that \(E_{\chi }(v) \le (p+1)^n E_{\chi }(u)\). \(\square \)
As a direct consequence, we obtain the monotonicity property for \(\mathcal {E}_\chi (M, \xi , \omega ^T).\)
Proposition 3.12
Suppose \(u\in \mathcal {E}_\chi (M, \xi , \omega ^T)\) and \(v\in \text {PSH}(M, \xi , \omega ^T)\). If \(u\le v\), then \(v\in \mathcal {E}_\chi (M, \xi , \omega ^T).\)
Proof
Without loss of generality we can assume that \(u \le v \le 0\). The monotonicity property implies that \(v \in \mathcal {E}(M,\xi ,\omega ^T)\). We have \(u \le v_k\) for the canonical cutoffs \(v_k\) of v, then \(E_{\chi }(v_k) \le (p+1)^nE_{\chi }(u)\) according to the Proposition 3.11. It follows from Proposition 3.10 that \(E_{\chi }(v) \le (p+1)^nE_{\chi }(u)\) and \(v \in \mathcal {E}_{\chi }(M,\xi ,\omega ^T)\). \(\square \)
We also have the following,
Proposition 3.13
Suppose \(u, v\in \mathcal {E}_\chi (M, \xi , \omega ^T)\) for \(\chi \in \mathcal {W}^+_p\). If \(u, v\le 0\), then
Proof
For similar result is Kähler case, see [44, Proposition 3.6]. For \(\delta >0\), we have \(\tilde{\chi }(t)=\chi (t)+\delta |t| \in \mathcal {W}^+_p\). Assume that \(t>0\), it is obvious \(\tilde{\chi }(t),\tilde{\chi }'(t)>0\). Recall that \(\epsilon ^p\tilde{\chi }(t) \le \tilde{\chi }(\epsilon t)\) and \(t\tilde{\chi }'(t) \le p\tilde{\chi }(t)\) for \(\tilde{\chi } \in \mathcal {W}_p^+\) and \(0< \epsilon <1\), hence we have \(\tilde{\chi }(2t) \le 2^p \tilde{\chi }(t)\). It follows from the convexity of the function \(\tilde{\chi }(t)\) that \(\frac{\tilde{\chi }(t)}{t} \le \tilde{\chi }'(t)\). Then
Then \(\delta \rightarrow 0\) implies that \(\chi '(2t) \le p 2^{p-1}\chi '(t)\) for \(t>0\).
By Proposition 3.6 and \(\{ |u|>2t\} \subset \{ u<v-t\} \cup \{ v < -t\}\), we have
\(\square \)
Proposition 3.14
Suppose \(u\in \mathcal {E}_\chi (M, \xi , \omega ^T), \chi \in \mathcal {W}^+_p\). Then there exists \(\tilde{\chi }\in \mathcal {W}^{+}_{2p+1}\) such that \(\chi (t)\le \tilde{\chi }(t), \chi (t)/\tilde{\chi }(t)\rightarrow 0\) as \(t\rightarrow \infty \) and \(u\in \mathcal {E}_{\tilde{\chi }}(M, \xi , \omega ^T)\).
Proof
This construction borrows from similar results in Kähler case, see [30, Lemma 2.10]. Take \(\chi _0=\chi \), recall that \(\lim \limits _{t \rightarrow \infty } \chi _0(t)=\infty \) and \(u \in \mathcal {E}_{\chi }(M,\xi ,\omega ^T)\), we have
Then one can choose \(t_1>0\) such that \(\int _{\{|u|>t_1\}} \chi (|u|)\omega _u^n\wedge \eta <\frac{1}{2^2}\). We define \(\chi _1:\mathbb {R}^+ \rightarrow \mathbb {R}^+\) by the formula:
Then it is easy to verify that
-
(1)
\(\chi _0(t) \le \chi _1(t)\);
-
(2)
\(\lim \nolimits _{t \rightarrow \infty } \frac{\chi _1(t)}{\chi _0(t)}=2\);
-
(3)
\(E_{\chi _1}(u) \le E_{\chi _0}(u)+\frac{1}{2}\);
-
(4)
\(\sup \nolimits _{t>0} \frac{|t\chi '_1(t)|}{|\chi _1(t)|} \le \sup \limits _{t>0}\frac{2|t\chi '_0(t)|}{|\chi _0(t)|}<2p+1\);
-
(5)
\(\lim \nolimits _{t \rightarrow \infty } \frac{t\chi '_1(t)}{\chi _1(t)} \le p\).
These properties imply that for \(t_2>t_1\) big enough, the function \(\chi _2:\mathbb {R}^+ \rightarrow \mathbb {R}^+\)
satisfies
-
(1)
\(\chi _1(t) \le \chi _2(t)\);
-
(2)
\(\lim \nolimits _{t\rightarrow \infty } \frac{\chi _2(t)}{\chi _1(t)}=2\);
-
(3)
\(E_{\chi _2}(u) \le E_{\chi _1}(u)+\frac{1}{2^2}\);
-
(4)
\(\sup \nolimits _{t>0} \frac{|t\chi '_2(t)|}{|\chi _2(t)|} <2p+1\);
-
(5)
\(\lim \nolimits _{t \rightarrow \infty } \frac{t\chi '_2(t)}{\chi _2(t)} \le p\).
Continuing the above construction we can obtain an increasing sequence \(\{\chi _k\}_k\) and the limit weight \(\tilde{\chi }(t)=\lim \nolimits _{k \rightarrow \infty } \chi _k(t)\) will satisfy the requirements of the proposition. \(\square \)
Proposition 3.15
Assume that \(\{\psi _k\}_{k\in \mathbb {N}}, \{\phi _k\}_{k\in \mathbb {N}}, \{v_k\}_{k\in \mathbb {N}}\subset \mathcal {E}_\chi (M, \xi , \omega ^T)\) decrease (increase a. e) to \(\phi , \psi , v\in \mathcal {E}_\chi (M, \xi , \omega ^T),\) respectively. Suppose
-
(1)
\(\psi _k\le \phi _k\) and \(\psi _k\le v_k\).
-
(2)
\(h: \mathbb {R}\rightarrow \mathbb {R}\) is continuous with \(\limsup _{|l|\rightarrow \infty }|h(l)|/\chi (l)\le C\) for some \(C\ge 0\).
Then we have the weak convergence of
Proof
For similar results in Kähler case, see [30, Proposition 2.11]. Without loss of generality one can assume all the functions \(\phi _k,\phi ,\psi _k,\psi , v, v_k\) are negative. We will only prove the proposition for decreasing sequences, the case of increasing sequences can be proved similarly.
First of all we suppose that the functions involved are uniformly bounded below, that is, there exists \(L>1\) such that \(-L \le \phi _k,\phi ,\psi _k,\psi , v_k, v \le 0\). Given \(\epsilon >0\), it follows from Theorem 6.3 that there exists an open subset \(O_{\epsilon } \subset M\) such that \(\text {cap}(O_{\epsilon })<\epsilon \) and \(\phi _k,\phi ,\psi _k,\psi , v_k, v\) are continuous on \(M-O_{\epsilon }\). Then \(\phi _k \rightarrow \phi \) and \(\psi _k \rightarrow \psi \) uniformly on \(M-O_{\epsilon }\). Hence there exists N such that for \(k>N\) we have \(|h(\phi _k-\psi _k)-h(\phi -\psi )|<\epsilon \) on \(M-O_{\epsilon }\) and the term
is bounded by \(2\epsilon L^n \max \limits _{|l| \le L}|h(l)| +\epsilon \text {Vol}(M)\). Hence
Given \(\epsilon >0\), it follows from Theorem 6.3 that there exists an open subset \(\tilde{O}_{\epsilon }\) such that \(\text {cap}(\tilde{O}_{\epsilon }) <\epsilon \) and \(\phi ,\psi \) are continuous on \(M-\tilde{O}_{\epsilon }\). By the Tietze’s extension theorem, the function \(h(\phi -\psi )|_{M-\tilde{O}_{\epsilon }}\) can be extended to a continuous function \(\alpha \) on M bounded by \(\max \limits _{|l|\le L}|h(l)|\). By Proposition 3.1 we have \(\omega _{v_k}^n\wedge \eta \rightarrow \omega _v^n\wedge \eta \) weakly. Then there exists a constant N such that for \(k>N\) we have \(|\int _M \alpha \omega _{v_k}^n\wedge \eta -\int _M\alpha \omega _v^n\wedge \eta |<\epsilon \) and the term
is bounded by \(4\epsilon L^n \max \nolimits _{ |l|\le L}|h(l)|+\epsilon \). Hence
It follows from (3.20) and (3.21) that \(h(\phi _k-\psi _k)\omega _{v_k}^n\wedge \eta \rightarrow h(\phi -\psi )\omega _v^n\wedge \eta \).
Now consider the general case when \(\phi _k,\phi ,\psi _k,\psi , v_k, v\) are unbounded. Let \(\phi _k^l,\phi ^l,\psi _k^l,\psi ^l, v_k^l, v^l\) be the canonical cutoffs of the corresponding potentials, then we only have to show that
and
as \(l\rightarrow \infty \) uniformly with respect to k.
By Proposition 3.14 there exists \(\tilde{\chi } \in \mathcal {W}_{2p+1}^+\) such that \(\chi \le \tilde{\chi },\lim \limits _{t \rightarrow \infty } \frac{\chi (t)}{\tilde{\chi }(t)}=0\) and \(\psi \in \mathcal {E}_{\tilde{\chi }}(M,\xi ,\omega ^T)\). Then \(\psi _k,\phi _k,\phi , v_k, v \in \mathcal {E}_{\tilde{\chi }}(M,\xi ,\omega ^T)\) according to Proposition 3.12.
Recall that there exists \(L>0\) such that \(\chi (L) \ge 1\) and \(|h(t)| \le (C+1)\chi (t)\) for \(|t|>L\). Take \(\tilde{C}= \max \{C+1,\frac{\max \limits _{0 \le l \le L}|h(l)|}{\chi (L)}\}\), then we have
for \(l_2 \le -L\) and \(l_2 \le l_1 \le 0\). Using Propositions 3.8, 3.11, and 3.13, we have
for \(l>L\) and the statement (3.22) follows. We also have
for \(l>L\) and the statement (3.23) follows. This completes the proof. \(\square \)
Proposition 3.16
Suppose \(\chi \in \mathcal {W}^+_p\) and \(\{u_k\}_{k\in \mathbb {N}}\subset \mathcal {E}_\chi (M, \xi , \omega ^T)\) is a decreasing sequence converging to \(u\in \text {PSH}(M, \xi , \omega ^T)\). If \(\sup _k E_\chi (u_k)<\infty \), then \(u\in \mathcal {E}_\chi (M, \xi , \omega ^T)\) and
Proof
For similar results in Kähler case, see [44, Proposition 5.6]. Without loss of generality, we assume that \(u_1\le 0\). The canonical cutoffs \(u_k^l=\max \{u_k,-l\}\) decreases to the canonical cutoff \(u^l=\max \{u,-l\}\). As \(-l \le u^l \le u_k^l \le 0\), Propositions 3.15 and 3.11 imply that
By Proposition 3.10, \(u \in \mathcal {E}_{\chi }(M,\xi ,\omega ^T)\). Applying the previous proposition in the case \(\psi _k=v_k=u_k,\phi _k=0\) gives that \(E_{\chi }(u)=\lim \nolimits _{k \rightarrow \infty } E_{\chi }(u_k)\). \(\square \)
A very important notion in pluripotential theory is the envelop construction, which we shall describe below. In our setting on a compact Sasaki manifold, given a usc function \(f\in M\rightarrow [-\infty , \infty )\) such that f is invariant under the Reeb flow, we consider the envelop
As in Kähler setting, we have the following
Proposition 3.17
The envelop construction \(P(f)\in \text {PSH}(M, \xi , \omega ^T)\).
Proof
This statement is local in nature, hence we only need to argue in foliations charts \(W_\alpha =(-\delta , \delta )\times V_\alpha \), where \(V_\alpha \subset \mathbb {C}^n\) give a transverse holomorphic charts. Since P(f) is invariant under the Reeb flow, its usc regularization \(P(f)^{*}\) is invariant under the Reeb flow. Hence by \(P(f)^*\) is \(\omega ^T_\alpha \)-psh on each \(V_\alpha \), see [12, Theorem 1.2.3 (viii)]. Since f is usc, hence \(P(f)^*\le f^*=f\). Hence \(P(f)^*\) is a candidate in the definition of P(f), gives that \(P(f)^*\le P(f)\). This implies that \(P(f)=P(f)^*\) and \(P(f)\in \text {PSH}(M, \xi , \omega ^T)\). \(\square \)
We also introduce the notion rooftop envelop, for usc functions \(f_1, \ldots , f_n\) which are invariant under the Reeb flow,
We have the following,
Theorem 3.1
Given \(f\in C^\infty _B(M)\), then we have the following estimate
Moreover, if \(u_1, \ldots , u_k\in \mathcal {H}_\Delta \), where we use the notation
then \(P(u_1, \ldots , u_k)\in \mathcal {H}_\Delta \).
We shall prove Theorem 3.1 in Appendix. The following result (for similar result in Kähler case, see [2, Corollary 9.2]) would be very essential for the rooftop envelop \(P(u_0, u_1)\):
Lemma 3.2
For \(u_0, u_1\in \mathcal {H}_\Delta \), then
on the non-contact set \(\Gamma =\{P(u_0, u_1)<\min (u_0, u_1)\}\).
Proof
First we assume \(\xi \) is regular or quasiregular, then the proof follows similarly as in Kähler setting. We sketch the proof briefly. We consider the quotient Kähler manifold (orbifold) \((Z=M/\mathcal {F}_\xi , \omega _Z)\) such that \(\omega ^T=\pi ^*\omega _Z\), where \(\pi : M\rightarrow Z\) is the natural quotient map. Since \(u_0, u_1\), and \(P(u_0, u_1)\) are all basic functions, and they descend to Z to define the functions on Z, which we still denote as \(u_0, u_1\), and \(P(u_0, u_1)\). We only need to show that \((\omega _Z+\sqrt{-1}\partial \bar{\partial } P(u_0, u_1))^n=0\) on \(\Gamma _Z:=\{z\in Z: P(u_0, u_1)<\min (u_0, u_1)\}\). Note that \(\Gamma _Z=\pi (\Gamma )\). This simply follows from [2, Corollary 9.2].
Now we deal with the case when \(\xi \) is irregular. We need to use a Type-I deformation to approximate \((M, \xi , \eta , g, \Phi )\), as in Theorem 6.1. Denote \(T^k\) to be the torus in \(\text {Aut}(\xi , \eta , g)\) with the Lie algebra \(\mathfrak {t}\). Take \(\rho _i\in \mathfrak {t}\) such that \(\rho _i\rightarrow 0\) (convergence is smooth with respect to a fixed metric g). We can take \(\rho _i\) such that \(\xi _i=\xi +\rho _i\) is quasiregular. Consider the Type-I deformation \((M, \xi _i, \eta _i, g_i, \Phi _i)\) as in Definition 2.3. Given \(u_0, u_1\in \mathcal {H}_\Delta \) and we know that \(P(u_0, u_1)\in \mathcal {H}_\Delta \) (see Theorem 3.1), by Lemma 6.1, there exists \(\epsilon _i\rightarrow 0\) such that \((1-\epsilon _i) u_0, (1-\epsilon _i)u_1, (1-\epsilon _i)P(u_0, u_1)\in \text {PSH}(M, \xi _i, \omega ^T_i)\). Define
Since \((1-\epsilon _i)P(u_0, u_1)\in \text {PSH}(M, \xi _i, \omega ^T_i)\) and \((1-\epsilon _i)P(u_0, u_1)\le (1-\epsilon _i)u_0, (1-\epsilon _i) u_1\), hence \((1-\epsilon _i)P(u_0, u_1)\le P_i\). On the other hand, we apply Lemma 6.1 and we know there exists \(\varepsilon _i\rightarrow 0\), such that \((1-\varepsilon _i)P_i\in \text {PSH}(M, \xi , \omega ^T)\). It follows that
By Theorem 3.1, we know that \(|\mathrm{{d}}\Phi d P_i|\) is uniformly bounded and hence \(P_i\rightarrow P(u_0, u_1)\) in \(C^{1, \alpha }\). For any compact subset \( K\subset \Gamma =\{P(u_0, u_1)<\min (u_0, u_1)\}\), we can choose i sufficiently large, such that \(P_i<\min \{(1-\epsilon _i)u_0, (1-\epsilon _i) u_1\}\). Since \(\xi _i\) is quasiregular, by (3.26), we can then get that
Taking \(i\rightarrow \infty \), by Lemma 6.2, we get that
This completes the proof. \(\square \)
As a consequence, we get a volume partition formula for \(\omega ^n_{P(u_0, u_1)}\wedge \eta \) as follows:
Lemma 3.3
For \(u_0, u_1\in \mathcal {H}_\Delta \), denote \(\Lambda _{u_0}=\{P(u_0, u_1)=u_0\}\) and \(\Lambda _{u_1}=\{P(u_0, u_1)=u_1\}\). Then we have the following
Proof
The proof is similar to the Kähler case, see [29, Proposition 2.2]. The previous lemma implies that the measure \(\omega ^n_{P(u_0,u_1)}\wedge \eta \) is supported on the set \(\Lambda _{u_0} \cup \Lambda _{u_1}\). It follows from Theorem 3.1 that \(P(u_0,u_1)\) has bounded Laplacian, hence all second partial derivatives of \(P(u_0,u_1)\) are in \(L^p(M)\) for all \(p>1\). Then all the second-order partial derivatives of \(P(u_0,u_1)\) and \(u_0\) coincide on \(\Lambda _{u_0}\) almost everywhere, all the second-order partial derivatives of \(P(u_0,u_1)\) and \(u_1\) coincide on \(\Lambda _{u_1}\) almost everywhere. Recall the definition of Monge–Ampere operators on psh functions belong to \(W^{2,n}\), we can write:
\(\square \)
Lemma 3.4
Suppose \(\chi \in \mathcal {W}^+_p\) and \(u_0, u_1\in \mathcal {E}_\chi (M, \xi , \omega ^T)\). Then \(P(u_0, u_1)\in \mathcal {E}_\chi (M, \xi , \omega ^T)\). If \(u_0, u_1\le 0\), then the following estimates hold
Proof
The proof is similar to the Kähler case, see [29, Lemma 3.4]. Without loss of generality we can assume \(u_0,u_1<0\). It follows from Lemma 3.1 that there exist negative transverse Kähler potentials \(u_0^k, u_1^k \in \mathcal {H}\) deceasing to \(u_0,u_1\) respectively. By Theorem 3.1, the rooftop envelopes \(P(u_0^k, u_1^k) \in \mathcal {H}_{\triangle }\) decreases to \(P(u_0,u_1)\). And we have the following inequality by Lemma 3.3:
Then
By Proposition 3.16 we have \(P((u_0,u_1)) \in \mathcal {E}_{\chi }(M,\xi ,\omega ^T)\) and the required inequality holds. \(\square \)
As a corollary we know that \(\mathcal {E}_\chi (M, \xi , \omega ^T)\) is convex.
Corollary 3.1
If \(u_0, u_1\in \mathcal {E}_\chi (M, \xi , \omega ^T)\), then \(tu_0+(1-t)u_1\in \mathcal {E}_\chi (M, \xi , \omega ^T)\) for any \(t\in [0, 1]\).
Proof
By the previous lemma we have \(P(u_0,u_1) \in \mathcal {E}_\chi (M, \xi , \omega ^T)\). Notice that \(P(u_0,u_1) \le tu_0+(1-t)u_1\) for \(t \in [0,1]\), then the monotonicity property of \(\mathcal {E}_\chi (M, \xi , \omega ^T)\) implies that \(tu_0+(1-t)u_1 \in \mathcal {E}_\chi (M, \xi , \omega ^T)\). \(\square \)
To finish this subsection, we establish a domination principle which will be needed later.
Lemma 3.5
Let \(U \subset M\) be a Borel set with \((\omega ^T)^n\wedge \eta (U)>0\) and \(u \in \mathcal {E}_1(M, \xi , \omega ^T)\). Then there exists \(\varphi \in \mathcal {E}_1(M,\xi , \omega ^T)\) with \(\varphi \le u\) and \(\omega _{\varphi }^n\wedge \eta (U) >0\).
Proof
The proof is similar to the Kähler case, see [30, Lemma 2.22]. Without loss of generality we can assume that \(u<0\). Then we can choose a sequence \(u_k \in \mathcal {H}\) decreasing to u with \(u_k <0\). For a constant \(\tau >0\), we have \(\{P(u_k+\tau , 0)=u_k+\tau \} \subset \{ u_k \le -\tau \}\). It follows from Proposition 3.3 that
The sequence \(P(u_k+\tau , 0) \in \mathcal {E}_1(M,\xi ,\omega ^T)\) decreases to \(P(u+\tau , 0) \in \mathcal {E}_1(M, \xi , \omega ^T)\). It follows from Proposition 3.15 that
Hence we have
It follows from \(\omega _{P(u+\tau ,0)}^n\wedge \eta (M)=(\omega ^T)^n\wedge \eta (M)=\text {Vol}(M)\) that
and \(\omega _{P(u+\tau ,0)}^n\wedge \eta (U)>0\) for \(\tau \) big enough. Then \(\varphi =P(u+\tau , 0) -\tau \) satisfies the requirements. \(\square \)
Lemma 3.6
(The domination principle) If \(u, v\in \mathcal {E}_1(M,\xi ,\omega ^T)\) and \(u \le v\) almost everywhere with respect to the measure \(\omega _v^n\wedge \eta \). Then \(u\le v\).
Proof
The proof is similar to the Kähler case, see [30, Proposition 2.21]. We only have to prove \(u \le v\) almost everywhere with respect to \((\omega ^T)^n\wedge \eta \) for \(u, v<0\).
Suppose that \((\omega ^T)^n\wedge \eta (\{u>v\}) >0\). The previous lemma implies that there exists \(\varphi \in \mathcal {E}_1(M, \xi , \omega ^T)\) with \(\varphi \le u\) and \(\omega _{\varphi }^n\wedge \eta (\{u>v\}) >0\). It follows from Corollary 3.1 that \(t \varphi +(1-t) u \in \mathcal {E}_1(M, \xi , \omega ^T)\) for \(t \in [0,1]\). Using the fact \(\omega _{t\varphi +(1-t)u}^n\wedge \eta \ge t^n \omega _{\varphi }^n\wedge \eta \) , the Comparison principle (3.15) and \(\{v< t\varphi +(1-t)u\} \subset \{v < u\}\), we have
and \(\omega _{\varphi }^n\wedge \eta (\{v < t\varphi +(1-t)u\})=0\) for \(t \in (0,1]\). Then
This leads to a contradiction. \(\square \)
3.2 The Space of Transverse Kähler Potentials and \((\mathcal {H}, d_2)\)
The Riemannian structure on \(\mathcal {H}\) has been studied extensively, notably by Guan–Zhang [42]. Guan–Zhang proved that for any two points \(\phi _1, \phi _2\in \mathcal {H}\), there exists a unique \(C^{1, \bar{1}}_B\) geodesic which realizes the distance of \((\mathcal {H}, d_2)\) and \((\mathcal {H}, d_2)\) is a metric space. The Riemannian structure would play a very central role, as in Chen’s result [20] in Kähler setting.
We shall recall these results. For \(\psi _1,\psi _2 \in T_{\phi } \mathcal {H}=C_B^{\infty }(M)\), define a \(L^2\) inner product on this tangent space
and the length \(||\psi ||_{\phi }\) of a vector \(\psi \in T_{\phi } \mathcal {H}\) is
where we use the notation
For a smooth path \(\phi _t \in \mathcal {H}\), the length of the path is defined to be
This is a direct adaption of Mabuchi’s metric [53] on the space of Kähler potentials to Sasaki setting. The Levi-Civita connection \(\nabla \) is torsion free and satisfies
for any smooth vector fields \(u_t, v_t\) along the path \(\phi _t\) in \(\mathcal {H}\). Let \(u_t\in C^\infty _B(M)\) be smooth vector fields along a smooth curve \(\phi _t\) in \(\mathcal {H}\), then
The geodesic equation can be written as
Given \(\phi _0, \phi _1\in \mathcal {H}\), to solve the geodesic equation, Guan–Zhang [42] introduced the following perturbation equation, for a path \(\phi _t: M\times [0, 1]\rightarrow \mathbb {R}\),
Define a function \(\psi \) on \(M\times [1, 3/2]\), as a subset of the cone X,
Set a (1, 1) form by,
Guan–Zhang wrote an equivalent form of (3.32) in terms of a complex Monge–Ampere equation on \(\psi \) of the following form (with \(f=r^2, \epsilon \in (0, 1] \)),
Guan–Zhang proved the following results regarding (3.33):
Theorem 3.2
(Guan–Zhang) Fix a Sasaki structure \((M, \xi , \eta , g)\) on a compact manifold M. For any positive basic function f and any two points \(\phi _0, \phi _1\in \mathcal {H}\), there exists a unique smooth solution of \(\psi \) to (3.33), satisfying the following estimates: \(\psi \) is basic and there exists a constant \(C>0\), depending only on \(\Vert f^{\frac{1}{n}}\Vert _{C^{2}(M\times [1, \frac{3}{2}])}, \Vert \phi _0\Vert _{C^{2, 1}}, \Vert \phi _1\Vert _{C^{2, 1}}\) such that
Denote the corresponding solution of (3.32) by \(\phi _t^\epsilon \), then \(\phi ^\epsilon _t\) is called a \(\epsilon \)-geodesic (smooth) connecting \(\phi _0, \phi _1\) satisfying
When \(\epsilon \rightarrow 0\), there exists a unique (weak \(C^2_w\)) limit \(\phi _t\) of \(\phi ^\epsilon _t: M\times [0, 1]\rightarrow \mathbb {R}\) connecting \(\phi _0, \phi _1\) such that \(\Omega _{\phi ^\epsilon +4log r}\) is positive. The later is equivalent to
As a consequence, \((\mathcal {H}, d_2)\) is a metric space.
Remark 3.2
The constant 1 / 4 appears in the geodesic equation
This constant is insignificant. In Kähler setting, some authors write the constant as 1 / 2 and some write as 1, depending on the gradient \(\nabla \) is interpreted as real or complex; they differ by a constant 2. The constant 1 / 4 appears in Sasaki setting in [42] since the authors use the real gradient and use the space of Sasaki potentials (transverse Kähler potentials) defined as
In the following, we shall write the geodesic equation as
where we use complex gradient, and our choice space of transverse Kähler potentials is as
To prove \((\mathcal {H}, d_2)\) is a metric space, Guan–Zhang [42, Lemma 14, Proof of Theorem 2] proved the following triangle inequality,
Lemma 3.7
(Guan–Zhang) Let \(\psi (s): [0, 1]\rightarrow \mathcal {H}\) be a smooth curve, \(\phi \in \mathcal {H}\backslash \psi ([0, 1])\). Fix \(\epsilon \in (0, 1]\). Let \(u^{\epsilon }\in C^\infty _B([0, 1]\times [0, 1]\times M)\) be the function such that \(u^\epsilon _t(\cdot , s)\) is the \(\epsilon \)-geodesic connecting \(\phi \) and \(\psi _s\), for \(t\in [0, 1]\). Then the following estimate holds:
where \(C=C(\phi , \psi , g)\) is a uniform constant, independent of \(\epsilon \).
There are several estimates which are not explicitly stated or not proved in [42]. We include these estimates below since we shall need them below. Regarding (3.32), first we have the following comparison principle,
Lemma 3.8
Suppose we have two solutions \(\varphi , \phi \) with boundary datum \(\varphi _0, \varphi _1\) and \(\phi _0, \phi _1\), respectively,
then we have the following
Proof
This is a standard comparison principle. We sketch the proof for completeness. Denote the operator
The \(\epsilon \)-geodesic equation can be written as \(F(D^2\phi )=\epsilon \). Now suppose \(F(D^2\phi )=F(D^2\varphi )=\epsilon >0\), then (3.38) holds. Otherwise suppose at some interior point
Hence \(\phi -\varphi +at(1-t)\) obtains its maximum at an interior point p for some \(a>0\). Denote \(v=\phi +at(t-1)\). Then on one hand,
On the other hand at p, \(D^2 v\le D^2 \varphi \). It follows from the concavity of F, we have at p,
where \(\mathcal {L}_{F_v}\) is the linearized operator of F at v. Contradiction. \(\square \)
One can actually be more precise about the estimate (3.35) [and (3.34)]. For simplicity, we state the result for (3.32).
Lemma 3.9
The \(\epsilon \) geodesic \(\phi ^\epsilon _t\) connecting \(\phi _0, \phi _1\in \mathcal {H}\) satisfies the following estimate,
where C depends only on \(\phi _0, \phi _1\). Moreover, we have
Proof
The first estimate follows from \(\ddot{\phi }^\epsilon _t>0\) and the following \(C^0\) estimate (3.41), which can be proved similarly using the concavity of F. First there exists \(a>0\) such that
The right-hand side is a direction consequence of \(\ddot{\phi }^\epsilon _t>0\), while the left-hand side can be argued as follows. Denote \(U^a=at(t-1)+(1-t)\phi _0+t\phi _1\); we know \(\phi ^\epsilon _t\) agrees with \(U^a\) on the boundary. Hence if \(\phi ^\epsilon _t<U^a\), then \(\phi ^\epsilon _t-U^a\) takes its minimum at some interior point p. At p, we know \(D^2 \phi ^\epsilon \ge D^2 U^a\). By concavity of F, we get (at p)
That is \(F(D^2 U^a)\le \log \epsilon \). This is a contradiction when \(a>0\) is sufficiently large. Indeed, a direct computation shows that if \(a\ge C\max |\nabla (\phi _1-\phi _0)|^2+\epsilon \), then \(F(D^2 U^a)>\log \epsilon \). Hence for such choice of a, (3.41) holds. By convexity in t direction, we know that
It is evident to show that
Hence (3.39) follows. The gradient estimate \(|\nabla \phi ^\epsilon _t|\) is given by [42, Proposition 2]. The estimate on \(\Delta _g \phi ^\epsilon _t\), depending only on \(\phi _0, \phi _1\) up to second-order derivative, was proved for Kähler setting by the first named author [47, Theorem 1.1] (for \(\epsilon =0\), it was proved earlier in [8] using pluripotential theory). The method in [47] is to deal with Eq. (3.32) directly, and it can be carried over to prove the interior estimate of \(\Delta _g \phi ^\epsilon \) word by word (since in Sasaki setting, this estimate only involves transverse Kähler structure and basic functions). For completeness, we sketch the proof. Denote \(\phi ^\epsilon =\phi \) for simplicity. We write the equations as
using the transverse Kähler metric \(g_{i\bar{j}}^T\). For any basic function h, we denote
where \((g^{i\bar{j}}_\phi )\) is the inverse of the transverse Kähler metric \((g_{i\bar{j}}^T+\phi _{i\bar{j}})\). Proceeding exactly as in the computation in the Kähler setting (see [47, (2.4)–(2.19)]), we compute
where C depends only on the background transverse Kähler metric \(g^T\) and n. Hence we have
where we have used the elementary inequality
Hence it follows that either \(\log (n+\Delta \phi )-C\phi +t^2\) achieves its maximum on the boundary, or at an interior maximum point P,
This gives the desired bound
\(\square \)
By taking \(\epsilon \rightarrow 0\), we have the following,
Lemma 3.10
Suppose \(\phi \) is the weak geodesic connecting \(\phi _0, \phi _1\in \mathcal {H}\), then for some positive constant \(C=C(M, g, \Vert \phi _0\Vert _{C^2}, \Vert \phi _1\Vert _{C^2})\), we have
As a consequence, when \(\phi _0\rightarrow \phi _1\) in \(\mathcal {H}\), then \(d_2(\phi _0, \phi _1)\rightarrow 0\).
Remark 3.3
One can get a much sharper estimate,
using the uniqueness and comparison for the generalized solutions of complex Monge–Ampere in the sense of Bedford–Taylor, see [30, Lemma 3.5] for Kähler setting. We shall prove this sharper version below.
Using Lemmas 3.7 and 3.10, it follows that the distance function \(d_2(\phi _0, \phi _1)\) is realized by the weak geodesic \(\phi \) connecting \(\phi _0, \phi _1\). In particular,
Lemma 3.11
Given \(\phi _0, \phi _1\in \mathcal {H}\), we have,
Proof
Let \(\phi ^\epsilon _t\) be the \(\epsilon \) geodesic connecting \(\phi _0, \phi _1\). Then we compute
Since \(|\dot{\phi }^\epsilon _t|\) is uniformly bounded, letting \(\epsilon \rightarrow 0\), we get that
This proves (3.46). In particular if \(\phi _0\ne \phi _1\), \(\dot{\phi }_t\) is not identically zero for any t. Moreover, if \(\epsilon \) is small enough, depending on \(\phi _0\ne \phi _1\), then \(\dot{\phi }^\epsilon _t\) is not identically zero for any \(t\in [0, 1]\). This follows from (3.47) and it is easy to see that \(\int _M |\dot{\phi }^\epsilon _t|^2 (\omega _{\phi ^\epsilon _t})^n\wedge \eta \) has a positive lower bound for any t (say \(l(\phi ^\epsilon _t)/2\)), if \(\epsilon \) is sufficiently small. \(\square \)
We also have the following
Theorem 3.3
(Guan–Zhang, Theorem 2). For \(u, v, w\in \mathcal {H}\),
3.3 The Orlicz–Finsler Geometry on Sasaki Manifolds
The Orlicz–Finsler geometry on the space of Kähler potentials was introduced by Darvas [28] and it has played an important role in problems regarding csck and Calabi’s extremal metric in Kähler geometry. In particular, the Finsler metric \(d_1\) will play an important role and it is used to define the properness of \(\mathcal {K}\)-energy. In this section, we discuss the Orlicz–Finsler geometry on Sasaki manifolds. We prove the following theorem, which is the counterpart of Darvas’s [28, Theorem 1] in Sasaki setting.
Theorem 3.4
If \(\chi \in \mathcal {W}_p^{+}, p\ge 1\), then \((\mathcal {H}, \mathrm{{d}}_\chi )\) is a metric space and for any \(u_0, u_1\in \mathcal {H}\), the \(C^{1, \bar{1}}_B\) geodesic \(t\rightarrow u_t\) connecting \(u_0, u_1\) satisfies
Theorem 3.4 is the generalization for \(d_2\) to general Young weights. This important result in Darvas’s theory says that, the same \(C^{1, \bar{1}}_B\) geodesic (with respect to \(d_2\)) is “length minimizing” for all \(\mathrm{{d}}_\chi \) metric structures and this holds in Sasaki setting. The proof of Theorem 3.4 pretty much follows Darvas’s proof [30, Theorem 3.4], with minor modifications adapted to Sasaki setting. The main point is that only transverse Kähler structure is involved, and hence this is essentially the same as in Kähler setting. We include the details for completeness.
Following Darvas (see [30, Chapter 3]), we define the Orlicz–Finsler length of \(v\in T_u\mathcal {H}=C^\infty _B(M)\) for any weight \(\chi \in \mathcal {W}^+_p\):
For simplicity, we shall assume \(\text {Vol}(M)=1\) in this section. Given a smooth curve \(\gamma : t\in [0, 1]\rightarrow \mathcal {H}\), its length is computed by the formula
Furthermore, the distance \(\mathrm{{d}}_\chi (u_0, u_1)\) between \(u_0, u_1\in \mathcal {H}\) is defined to be
First we have the following,
Proposition 3.18
Suppose \(\chi \in \mathcal {W}_p^+ \cap C^{\infty }(\mathbb {R})\). For a smooth curve \(u_t (t \in [0,1])\) in \(\mathcal {H}\) and a vector field \(f_t \in C_B^{\infty }(M)\) along this curve with \(f_t \not \equiv 0\), we have
Proof
This works as in [28, Proposition 3.1] word by word. We skip the details. \(\square \)
Lemma 3.12
Suppose \(\chi \in \mathcal {W}_p^+ \cap C^{\infty }(\mathbb {R})\) and \(u_0,u_1 \in \mathcal {H},u_0 \ne u_1\). Then the \(\epsilon \)-geodesics \([0,1] \ni t \rightarrow u_{t}^{\epsilon } \in \mathcal {H} \) connecting \(u_0,u_1\) satisfies the following estimate:
for all \(t \in [0,1]\), where \(C:=C(\chi ,||u_0||_{C^2(M)},||u_1||_{C^2(M)})\).
Proof
This follows exactly as in Kähler setting [30, Lemma 3.8], by a direct computation and the convexity of \(\chi \). \(\square \)
Lemma 3.13
Suppose \(\chi \in \mathcal {W}_p^+ \cap C^{\infty }(\mathbb {R})\) and \(u_0,u_1 \in \mathcal {H},u_0 \ne u_1\). Then there exists a constant \(\epsilon _0\) that depends on \(u_0, u_1\) such that for all \(\epsilon \in (0, \epsilon _0]\) the \(\epsilon \)-geodesic \([0,1] \ni t \rightarrow u_{t}^{\epsilon } \in \mathcal {H} \) connecting \(u_0,u_1\) satisfies:
Proof
If we choose \(\epsilon _0>0\) sufficiently small, then \(\dot{u}^\epsilon _t\) is not identically zero for any \(t\in [0, 1]\), if \(\epsilon \in (0, \epsilon _0]\), given \(u_0\ne u_1\), see (3.46). Then the results follows from Proposition 3.18. \(\square \)
We have the following, similar to (3.46) (for \(d_2\)),
Proposition 3.19
Suppose \(\chi \in \mathcal {W}_p^+ \cap C^{\infty }(\mathbb {R})\) and \(u_0,u_1 \in \mathcal {H},u_0 \ne u_1\). Then there exists \(\epsilon _0>0\) such that for any \(\epsilon \in (0,\epsilon _0]\) the \(\epsilon \)-geodesic \([0,1] \ni t \rightarrow u_{t}^{\epsilon } \in \mathcal {H} \) connecting \(u_0,u_1\) satisfies
-
(i)
\(||\dot{u}_t^{\epsilon }||_{\chi ,u_t^{\epsilon }}>R_0,t \in [0,1]\);
-
(ii)
\(|\frac{\mathrm{{d}}}{\mathrm{{d}}t} ||\dot{u}_t^{\epsilon }||_{\chi ,u_t^{\epsilon }}| \le \epsilon R_1,t \in [0,1]\),
where \(\epsilon _0,R_0,R_1\) depend on upper bounds for \(||u_0||_{C^2(M)},||u_1||_{C^2(M)}\) and lower bounds for \(||\chi (u_1-u_0)||_{L^1((\omega ^T)^n \wedge \eta )},\frac{\omega _{u_0}^n \wedge \eta _{u_0}}{ (\omega ^T)^n\wedge \eta }\) and \(\frac{\omega _{u_1}^n \wedge \eta _{u_1}}{ (\omega ^T)^n\wedge \eta }\).
Proof
-
(i)
Recall Eq. (1.11) in [30]
$$\begin{aligned} ||f||_{\chi ,\mu } \ge \min \left\{ \frac{\int _{\Omega } \chi (f)\mathrm{{d}}\mu }{\chi (1)} ,\left( \frac{\int _{\Omega } \chi (f)\mathrm{{d}}\mu }{\chi (1)}\right) ^{\frac{1}{p}}\right\} \end{aligned}$$and Lemma 3.12, the estimate in (i) follows immediately.
-
(ii)
Choose \(\epsilon _0\) small so that Lemma 3.13 applies. Recall the Young identity
$$\begin{aligned} \chi (a)+\chi ^*(\chi '(a))=a\chi '(a),a, b \in \mathbb {R},\chi '(a) \in \partial \chi (a) \end{aligned}$$Then we have
$$\begin{aligned} \left| \frac{\mathrm{{d}}}{\mathrm{{d}}t} \right| |\dot{u}_t^{\epsilon }||_{\chi , u_t^{\epsilon }}|&=\epsilon \frac{\left| \int _M \chi '\left( \frac{\dot{u}_t^{\epsilon }}{||\dot{u}_t^{\epsilon }||_{\chi ,\dot{u}_t^{\epsilon }}}\right) (\omega ^T)^n \wedge \eta \right| }{\int _M \frac{\dot{u}_t^{\epsilon }}{||\dot{u}_t^{\epsilon }||_{\chi ,\dot{u}_t^{\epsilon }}} \chi '\left( \frac{\dot{u}_t^{\epsilon }}{||\dot{u}_t^{\epsilon }||_{\chi ,\dot{u}_t^{\epsilon }}}\right) \omega _{u_t^{\epsilon }} \wedge \eta _{u_t^{\epsilon }} } \nonumber \\&=\epsilon \frac{\left| \int _M \chi '\left( \frac{\dot{u}_t^{\epsilon }}{||\dot{u}_t^{\epsilon }||_{\chi ,\dot{u}_t^{\epsilon }}}\right) (\omega ^T)^n \wedge \eta \right| }{\chi (1)+\int _M \chi ^*\left( \chi '\left( \frac{\dot{u}_t^{\epsilon }}{||\dot{u}_t^{\epsilon }||_{\chi ,\dot{u}_t^{\epsilon }}}\right) \right) \omega _{u_t^{\epsilon }} \wedge \eta _{u_t^{\epsilon }} } \nonumber \\&\le \frac{\epsilon }{\chi (1)} \left| \int _M \chi '\left( \frac{\dot{u}_t^{\epsilon }}{||\dot{u}_t^{\epsilon }||_{\chi ,\dot{u}_t^{\epsilon }}}\right) (\omega ^T)^n \wedge \eta \right| . \end{aligned}$$(3.55)Then the estimates (ii) follows from (i) and the fact that \(\dot{u}_t^{\epsilon }\) is uniformly bounded in terms of \(||u_0||_{C^2(M)},||u_1||_{C^2(M)}\).
\(\square \)
Remark 3.4
The estimate (i) in Proposition 3.19 holds for general weights \(\chi \in \mathcal {W}_p^+\). Recall that \(\dot{u}_t^{\epsilon }\) is uniformly bounded in terms of \(||u_0||_{C^2(M)},||u_1||_{C^2(M)}\). We can choose smooth weights \(\chi _k \in \mathcal {W}_{p_k}^+ \cap C^{\infty }(\mathbb {R})\) which approximate \(\chi \) uniformly on compact subsets of \(\mathbb {R}\). Moreover we have \(\lim \limits _{k \rightarrow \infty }||\dot{u}_t^{\epsilon }||_{\chi _k,u_t^{\epsilon }}=||\dot{u}_t^{\epsilon }||_{\chi ,u_t^{\epsilon }}\) [30, Section 1]. It follows that the estimates (i) hold for \(\chi \).
Next we are ready to prove the triangle inequality, as in Lemma 3.7 for \(d_2\) and [28, Proposition 3.4] in Kähler setting.
Proposition 3.20
Suppose \(\chi \in \mathcal {W}_p^+ \cap C^{\infty }(\mathbb {R})\), \(\psi _s \in \mathcal {H}\) is a smooth curve, \(\phi \in \mathcal {H}\backslash {\psi ([0,1])}\), and \(\epsilon >0\). \(u^{\epsilon } \in C^{\infty }([0,1] \times [0,1] \times M)\) is the smooth function for which \(t \rightarrow u_t^{\epsilon }(\cdot ,s)=u^{\epsilon }(t,s,\cdot )\) is the \(\epsilon \)-geodesic connecting \(\phi \) and \(\psi _s\). Then there exists \(\epsilon _0(\phi ,\psi )>0\) such that for any \(\epsilon \in (0,\epsilon _0)\) the following holds:
for some \(R(\phi ,\psi ,\chi ,\epsilon _0)>0\) independent of \(\epsilon \).
Proof
Fix \(s \in [0,1]\). By Propositions 3.18 and 3.19, there exists a constant \(\epsilon _0(\phi ,\psi )>0\) such that for \(\epsilon \in (0,\epsilon _0)\)
Moreover we have
It follows from Proposition 3.19 that \(||\dot{u}||_{\chi ,t}\) is uniformly bounded away from zero and both \(\nabla _{\dot{u}}\dot{u}\mathrm{{d}}\mu _{u_t}\) and \(\frac{\mathrm{{d}}}{\mathrm{{d}}t} ||\dot{u}||_{\chi , u}\) are uniformly bounded by the form \(\epsilon R\), where R is uniformly bounded. Moreover \(\dot{u},\frac{\mathrm{{d}}u}{\mathrm{{d}}s}\) are uniformly bounded independent of \(\epsilon \) [42, Lemma 14]. Hence
where R is uniform bounded independent of \(\epsilon \).
Recall that \(\chi ^{*'}(\chi '(l))=l\) for \(l \in \mathbb {R}\), the expression
is a term of type \(\epsilon R\). Hence we can write
where the last line follows from the Young inequality
The integration of the above inequality with respect to \(s \in [0,1]\) yields the desired inequality. \(\square \)
Now we are ready to prove Theorem 3.4. Certainly the proof follows closely Darvas’s result in Kähler setting [28, Section 3].
Proof
First we show that for \(u_0,u_1 \in \mathcal {H}\) and the weak \(C_B^{1,\overline{1}}\)-geodesic \(u_t\) connecting \(u_0,u_1\)
We assume \(u_0 \ne u_1\). Recall that, by Guan–Zhang [42], \(\epsilon \)-geodesics \(u_t^{\epsilon }\) connecting \(u_0,u_1\) converge to the weak \(C^{1,\overline{1}}_B\) geodesic \(u_t\) in \(C^{1,\alpha }\). Hence \(\dot{u}_t^{\epsilon }\) converges uniformly to \(\dot{u}_t\).
Recall that \(\dot{u}_t^{\epsilon }\) is uniformly bounded in terms of \(||u_0||_{C^2(M)},||u_1||_{C^2(M)}\). Combine with the remark after Proposition 3.19, there exist constants \(0<C_1<C_2\) such that for sufficiently small \(\epsilon >0\)
Take a cluster point N of \(\{||\dot{u}_t^{\epsilon }||_{\chi , u_t^{\epsilon }}\}_{\epsilon >0}\), after taking a subsequence, we can assume that \(||\dot{u}_t^{\epsilon }||_{\chi , u_t^{\epsilon }}\rightarrow N\) as \(\epsilon \rightarrow 0\). Then \(\frac{\dot{u}_t^{\epsilon }}{||\dot{u}_t^{\epsilon }||_{\chi , u_t^{\epsilon }}}\) converges to \(\frac{\dot{u}_t}{N}\) uniformly. Moreover, we have \(\omega _{u_t^{\epsilon }}^n \wedge \eta \) converges to \(\omega _{u_t}^n \wedge \eta \) weakly.
Recall \(||f||_{\chi ,\mu } =\alpha >0\) if and only if \(\int _{\Omega } \chi (\frac{f}{\alpha })\mathrm{{d}}\mu =\chi (1)\)[30, Section 1]. We have
and \(N=||\dot{u}_t||_{\chi , u_t}\). Hence \(||\dot{u}_t||_{\chi , u_t}\) is the only possible cluster point of \(\{||\dot{u}_t^{\epsilon }||_{\chi , u_t^{\epsilon }}\}_{\epsilon >0}\). It means that
as \(\epsilon \rightarrow 0\). Then by the dominated convergence theorem we have
and \(\mathrm{{d}}_{\chi }(u_0,u_1) \le l_{\chi }(u_t)\).
Then Eq. (3.57) follows if we can prove
for all smooth curves \(\phi _t\) in \(\mathcal {H}\) connecting \(u_0,u_1\).
First we consider the case \(\chi \in \mathcal {W}_p^+ \cap C^{\infty }(\mathbb {R})\). We can assume that \(u_1 \notin \phi ([0,1))\) and take \(h \in [0,1)\). Applying Proposition 3.20 to the case \(\phi =u_1\) and \(\psi _s=\phi |_{[0,h]}\), letting \(\epsilon \rightarrow 0\), we can obtain
where \(u_{t}\) is the \(C_B^{1,\bar{1}}\) geodesic connecting \(u_1, u_0\), and \(w_t^h\) is the \(C_B^{1,\bar{1}}\) geodesic connecting \(u_1,\phi _h\). By Lemma 3.9, \(l_{\chi }(w_t^h) \rightarrow 0\) as \(h \rightarrow 1\). Hence \(l_{\chi }(\phi _t) \ge l_{\chi }(u_t)\).
For the general weight \(\chi \in \mathcal {W}_p^+\), we need to do approximation as in [28, Proposition 2.4]. There exists sequence \( \chi _k \in \mathcal {W}_{p_k}^+ \cap C^{\infty }(\mathbb {R})\) such that \(\chi _k\) converges to \(\chi \) uniformly on compact subsets. Then we have
and \(||\dot{\phi }_t||_{\chi _k,\phi _t} \rightarrow ||\dot{\phi }_t||_{\chi ,\phi _t},||\dot{u}_t||_{\chi _k,u_t} \rightarrow ||\dot{u}_t||_{\chi , u_t}\) [30, Section 1]. Moreover,\(\dot{u}_t,\dot{\phi }_t\) are uniformly bounded. By the dominated convergence theorem, \(l_{\chi }(\phi _t) \ge l_{\chi }(u_t)\). This completes the proof of 3.57.
Recall \(l_{\chi }(u_t)=\int _0^1 ||\dot{u}_t||_{\chi ,u_t} \mathrm{{d}}t\) and by Lemma 3.14, we have
Suppose \(u_0 \ne u_1 \in \mathcal {H}\), take \(\epsilon \rightarrow 0\) in the estimate Lemma 3.12 we obtain \(\dot{u}_0 \not \equiv 0\) and \(\mathrm{{d}}_{\chi }(u_0,u_1)=||\dot{u}_0||_{\chi ,u_0} >0\). This implies that \((\mathcal {H},\mathrm{{d}}_{\chi })\) is a metric space. \(\square \)
Lemma 3.14
Let \(u_t\) be the weak \(C^{1, \bar{1}}_B\) geodesic connecting \(u_0,u_1\). Then for any \(\chi \in \mathcal {W}_p^+\) and \(t_0, t_1 \in [0,1]\), the following holds
Proof
It had been shown that for \(\epsilon \)-geodesics \(u^{\epsilon }_t\) joining \(u_0,u_1\), we have
as \(\epsilon \rightarrow 0\). Proposition 3.19 implies that
Then taking \(\epsilon \rightarrow 0\) we have \(||\dot{u}_{t_0}||_{\chi , u_{t_0}} =||\dot{u}_{t_1}||_{\chi , u_{t_1}}\). \(\square \)
Finally, we have the following triangle inequality,
Lemma 3.15
For \(u, v, w\in \mathcal {H}\), \(\chi \in \mathcal {W}^+_p, p\ge 1\),
4 The Metric Space \((\mathcal {E}_p(M, \xi , \omega ^T), \mathrm{{d}}_p)\)
In this section, we prove Theorem 2. We shall follow the Kähler setting closely as in [28, Section 4], but we shall only consider \(\mathrm{{d}}_p\) distance. Given \(u_0, u_1\in \mathcal {E}_p(M, \xi , \omega ^T), p\ge 1\), by Lemma 3.1 there exists decreasing sequences \(\{u_0^k\}_{k \in \mathbb {N}}, \{u_1^k\}_{k \in \mathbb {N}}\subset \mathcal {H}\) such that \(u_0^k\searrow u_0\) and \(u_1^k\searrow u_1\). We shall prove that the following formula for distance \(\mathrm{{d}}_p\) is well defined,
and the definition in (4.1) coincides with (3.51) (we only consider \(\chi (l)=|l|^p/p\)). We will prove that
Theorem 4.1
\((\mathcal {E}_p(M,\xi ,\omega ^T), \mathrm{{d}}_p)\) is a complete geodesic metric space extending \((\mathcal {H}, \mathrm{{d}}_p)\).
We start with the notion of generalized solution of complex Monge–Ampere in the sense of Bedford–Taylor in Sasaki setting, which was considered by van Coevering in [58], by adapting the complex Monge–Ampere operator for basic functions in \(\text {PSH}(M, \xi , \omega ^T)\cap L^\infty \) to Sasaki setting. van Coevering discussed in particular weak solution in \(\text {PSH}(M, \xi , \omega ^T)\cap C^0(M)\) [58, Section 2.4]. Let \(S=[0, 1]\times S^1\) be the cylinder and \(N=M\times S\). Then N is a manifold of dimension \(2n+3\) with boundary and N has a transverse holomorphic structure, simply the product structure of transverse holomorphic structure on M, and holomorphic structure on S. A path \(\phi : [0, 1]\rightarrow C^\infty _B(M)\) corresponds to an \(S^1\)-invariant function \(\Phi _w\) on N. If \(\phi _t\) is a smooth path in \(\mathcal {H}\) then a direct computation gives
Note that this choice of complexification [see van Coevering (4.2)] is different with the choice of Guan–Zhang (3.33). It seems that (4.2) would be more natural to discuss weak solutions. By (4.2), a smooth geodesic then corresponds to a solution of homogeneous complex Monge–Ampere for basic function \(\Phi : N\rightarrow \mathbb {R}\),
We define a weak geodesic between \(u_0, u_1\in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty \) as follows, for \(\Phi (\cdot , w)=\Phi (\cdot , t)\in \text {PSH}({N}^{\circ }, \xi , \pi ^*\omega ^T)\cap L^\infty \), \((t=\text {Re}(w))\), it satisfies
We have the following strong maximum principle, see [58, Theorem 2.5.3], [11, Theorem 21], and [30, Theorem 3.2].
Lemma 4.1
Let \(u, v\in \text {PSH}({N}^{\circ }, \xi , \pi ^*\omega ^T)\cap L^\infty (N)\). Suppose that
and \(\lim _{x\rightarrow \partial N} (u-v)(x)\ge 0\), then \(u\ge v\) on N.
Proof
Our proof is similar to Kähler case, see [30, Theorem 3.2]. Fix \(\epsilon >0\) and \(v_\epsilon :=\max \{u, v-\epsilon \}\in \text {PSH}(N^\circ , \xi , \omega ^T)\cap L^\infty \). Then \(v_\epsilon =u\) near the boundary \(\partial N=M\times S^1\times \{t=0\}\cup M\times S^1\times \{t=1\}\). Hence it is enough to show that \(u=v_\epsilon \) on N.
We write \(N=M\times S\) and \(\omega _u=\pi ^*\omega ^T+\mathrm{{dd}}^c_B u\), etc. Note that on each foliation chart \(W_\alpha =(-\delta , \delta )\times V_\alpha \) of M, we have the following inequality on \(V_\alpha \times S\) for complex Monge–Ampere measure [12, Theorem 2.2.10]
It follows that on N, we have
Then we have the following
Using integration by parts, we obtain that
By an induction argument as in [30, Theorem 3.2], we can prove that
For \(k=n\), this shows that
Writing \(\rho =u-v_\epsilon \), this reads
Hence \(\partial _t\rho =0\). Since \(\rho =0\) near the boundary \(\partial N=M\times S^1\times \{t=0\}\cup M\times S^1\times \{t=1\}\), this shows that \(\rho =0\). It completes the proof. \(\square \)
Remark 4.1
One can certainly formulate a general version of comparison principle as in [30, Theorem 3.2]. But one would need certainly a (transverse) Kähler form. Note that \(\pi ^*\omega ^T\) is not transverse Kähler (it is zero along S-direction). Here we use the product structure of N.
With this maximum principle for bounded TPSH, we have the following,
Lemma 4.2
Given \(u_0, u_1\in \mathcal {H}\), let \(u_t: [0, 1]\rightarrow \mathcal {H}\) be the unique \(C^{1, \bar{1}}_B\) geodesic connecting \(u_0, u_1\). Then we have the following,
Proof
Note that this gives a much sharper estimate than Lemma 3.10. The proof follows the Kähler setting [30, Lemma 3.5]. Denote \(C=\max |u_0- u_1|\). By the convexity of u in t-variable, we know that
Note that \(v_t=u_0-Ct\) is a smooth geodesic connecting \(u_0\) and \(u_0-C\). Hence its complexification gives a solution to (4.3). By Lemma 4.1, we know that \(v_t\le u_t\), for \(t\in [0, 1]\), since \(u_0-C\le u_1\). It follows that \(-C\le \dot{u}_0\). Similarly, one can prove that \(\dot{u}_1\le C\), by considering \(\tilde{v}_t=u_0+Ct\). \(\square \)
Remark 4.2
The upper envelop construction was used to construct bounded weak geodesic segment in Kähler setting by Berndtsson [9], where he proved that Lemma 4.2 holds for \(u_0, u_1\in \text {PSH}(M, \omega )\) (when \((M, \omega )\) is Kähler). A direct adaption to Sasaki setting using Lemma 4.1 would lead to an extension of Berndtsson’s result to Sasaki setting.
In general, \(\Phi (\cdot , w)\in \text {PSH}(N^\circ , \xi , \pi ^*\omega ^T)\) will be called weak subgeodesic, if \(\Phi (\cdot , )=\Phi (\cdot , \text {Re}(w))\), \((t=\text {Re}(w))\). For \(u_0, u_1\in \text {PSH}(M, \xi , \omega ^T)\), we define
We have the following:
Proposition 4.1
\(u\in \text {PSH}(N^\circ , \xi , \pi ^*\omega ^T)\). Denote \(u_t=u(\cdot , t)\). We refer \(t\rightarrow u_t\) to the weak geodesic segment connecting \(u_0, u_1\).
Proof
Note that usc \(u^*\) is basic, and \(u^*\in \text {PSH}(N^\circ , \xi , \pi ^*\omega ^T)\). Since \(\Phi \) is convex in t direction, it follows that \(\Phi (\cdot , t)\le (1-t)u_0+tu_1\). Hence \(u_t\le (1-t)u_0+t u_1\). It follows that
In other words, \(u^*\le u\) by definition. It follows that \(u^*=u\). \(\square \)
Proposition 4.2
If \(u_0, u_1 \in \text {PSH}(M, \xi , \omega ^T)\cap L^{\infty }\), u is defined by (4.5) and \(u_t=u(\cdot , t)\) is the weak geodesic. Let C be a constant \(\ge ||u_1-u_0||_{L^{\infty }}\).
-
(1)
We have
$$\begin{aligned} \max (u_0 -Ct, u_1-C(1-t)) \le u_t \le (1-t) u_0+tu_1. \end{aligned}$$(4.6) -
(2)
\(u_t \in \text {PSH}(M, \xi , \omega ^T) \cap L^{\infty }\) and u is the unique solution of (4.3).
-
(3)
\(u_t\) is uniformly Lipschitz continuous with respect to t:
$$\begin{aligned} |u_t-u_s| \le C|s-t|. \end{aligned}$$for \(s, t \in [0,1]\).
-
(4)
The derivatives \(\dot{u}_0, \dot{u}_1\) exists and
$$\begin{aligned} |\dot{u}_0|\le C, \quad |\dot{u}_1| \le C. \end{aligned}$$
Proof
-
(1)
It is obvious that \(u_0-Ct, u_1-C(1-t)\) are weak subgeodesics. It follows from the definition of \(u_t\) (4.5) that
$$\begin{aligned} \max (u_0-Ct, u_1-C(1-t)) \le u_t. \end{aligned}$$The other half of the inequality comes from the convexity of \(u_t\) with respect to t.
-
(2)
By the inequality (4.6) we have \(u_t \in \text {PSH}(M, \xi , \omega ^T) \cap L^{\infty }\) and \(\lim \limits _{t \rightarrow 0,1} u_t=u_{0,1}\). Then \(u \in \text {PSH}({N}^{\circ }, \xi , \pi ^*\omega ^T)\cap L^\infty \). Using the classical Perron-Bremmerman argument, we have \((\pi ^*\omega ^T+\sqrt{-1}\partial _B\overline{\partial }_B u)^{n+1}\wedge \eta =0\). Hence u is a solution of (4.3). The uniqueness of the solution of (4.3) follows from the strong maximum principle.
-
(3)
If one of s, t equals to 0 or 1, the required inequality is a direct consequence of (4.6). If \(0<s<t<1\), by the convexity of \(u_t\) with respect to t we have
$$\begin{aligned} \frac{t-s}{s}(u_s-u_0) \le u_t-u_s \le \frac{t-s}{1-s}(u_1-u_s) \end{aligned}$$and the inequality follows from the case \(t=0,1\) we have proved.
-
(4)
By the convexity of \(u_t\), we have
$$\begin{aligned} \frac{u_{t_1}-u_0}{t_1} \le \frac{u_{t_2}-u_0}{t_2} \end{aligned}$$for \(0<t_1 <t_2\). These quantities are uniformly bounded by C. Hence \(\dot{u}_0\) exists and \(|\dot{u}_0| \le C\). The case of \(\dot{u}_1\) follows by a similar argument.
\(\square \)
Remark 4.3
If \(u_0, u_1 \in \mathcal {H}\), the weak geodesic \(u_t\) coincides with the \(C_B^{1,\bar{1}}\) geodesic.
The weak geodesic \(u_t\) connecting \(u_0,u_1 \in \text {PSH}(M,\xi , \omega ^T)\) has the advantage of admitting some homogeneous structures and offering a new interpretation of rooftop envelope. Moreover, it is closed for class \(\mathcal {E}_p(M,\xi ,\omega ^T)\): the weak geodesic \(u_t\) connecting \(u_0, u_1 \in \mathcal {E}_p(M,\xi ,\omega ^T)\) stays in the same class. It is called the finite-energy geodesic in \(\mathcal {E}_p(M,\xi ,\omega ^T)\).
Proposition 4.3
Let \(u_0^k,u_1^k\in \text {PSH}(M,\xi ,\omega ^T)\) be sequences decreasing to \(u_0,u_1\in \text {PSH}(M,\xi ,\omega ^T)\), respectively. Suppose that \(u_t^k,u_t \in \text {PSH}(M,\xi ,\omega ^T)\) be the weak geodesic connecting \(u_0^k,u_1^k\) and \(u_0,u_1\), respectively. Then
-
(1)
\(u_t^k\) decreases to \(u_t\) for \(t \in [0,1]\);
-
(2)
For any \(t_1,t_2\in [0,1]\), \([0,1] \ni t \rightarrow u_{(1-t)t_1+tt_2} \in \text {PSH}(M, \xi , \omega ^T)\) is the weak geodesic connecting \(u_{t_1}\) and \(u_{t_2}\).
Proof
-
(1)
By the definition of \(u^k_t\) (4.5) it is obvious that \(\{u_t^k\}_{k\in \mathbb {N}}\) is decreasing and \(v_t = \lim \limits _{k\rightarrow \infty } u_t^k \in \text {PSH}(M, \xi , \omega ^T)\). Again by the definition of \(u_t^k, u_t\) (4.5) we have \(u^k_t \ge u_t\), hence \(v_t \ge u_t\).
Recall that \(u_t^k\) is convex with respect to t. Then \(u_t^k \le (1-t)u_0^k+t u_1^k\) and \(v_t \le (1-t)u_0+t u_1\). It follows from the definition of \(u_t\) (4.5) that \(v_t \le u_t\).
Consequently the sequence \(\{u_t^k\}_{k \in \mathbb {N}}\) decreases to \(u_t\) for \(t \in [0,1]\).
-
(2)
Recall that \(u_0, u_1\) are the decreasing limits of their canonical cutoffs, it follows from part (1) that we only have to prove the proposition for \(u_0, u_1\) in \(L^{\infty }\). \(v_t=u_{(1-t)t_1+tt_2}\) is a path connecting \(u_{t_1}, u_{t_2}\). By Proposition 4.2 we have \(\lim \limits _{t\rightarrow 0,1} v_t=u_{t_1,t_2}\) and \(\Phi (\cdot ,t)=v_t\) is a solution of Eq. ((4.3)) with initial data \(u_{t_1},u_{t_2}\). Then it follows from Proposition 4.2(2) that \(v_t=u_{(1-t)t_1+tt_2}\) is the weak geodesic connecting \(u_{t_1}, u_{t_2}\).
\(\square \)
Lemma 4.3
Suppose \(u_0, u_1\in \text {PSH}(M, \xi , \omega ^T)\) and \(t\rightarrow u_t\) is the weak geodesic segment connecting \(u_0, u_1\).
-
(1)
For any \(\tau \in \mathbb {R}\) we have
$$\begin{aligned} \inf _{t\in (0, 1)} (u_t-t\tau )=P(u_0, u_1-\tau ), \tau \in \mathbb {R}. \end{aligned}$$(4.7) -
(2)
If \(u_0,u_1 \in \text {PSH}(M,\xi ,\omega ^T) \cap L^{\infty }\), then
$$\begin{aligned} \{\dot{u}_0\ge \tau \}=\{P(u_0, u_1-\tau )=u_0\}. \end{aligned}$$(4.8) -
(3)
If \(u_0, u_1\in \mathcal {E}_p(M, \xi , \omega ^T)\), then \(u_t\in \mathcal {E}_p(M, \xi , \omega ^T)\) for \(t \in [0,1]\).
Proof
-
(1)
First note that \(t\rightarrow v_t=u_t-\tau t\) is the weak geodesic connecting \(u_0, u_1-\tau \), hence the proof can be reduced to the particular case \(\tau =0\). By definition \(P(u_0, u_1)\le u_0, u_1\). As a result, the constant weak subgeodesic \(t\rightarrow h_t:=P(u_0, u_1)\) is a candidate for definition of \(u_t\), hence \(h_t\le u_t, t\in [0, 1]\). It follows that \(P(u_0, u_1)\le \inf _{t\in [0, 1]} u_{t}\).
For the other direction, we use Kiselman minimum principle [33, Chapter I, Theorem 7.5], which asserts that \(w:=\inf _{t\in [0, 1]} u_t\in \text {PSH}(M, \xi , \omega ^T)\) (note that \(u_t\) is a genuine plurisubharmonic function on foliation charts, for each t and \(u_t\) is convex in t variable; hence Kiselman minimum principle applies, as in Kähler setting). Note that \(u_t\le (1-t)u_0+t u_1\), it follows that w is a candidate for \(P(u_0, u_1)\) and hence \(w\le P(u_0, u_1)\). This completes the proof.
-
(2)
For \(x \in M\) we have \(P(u_0,u_1-\tau )(x)=u_0(x)\) if and only if \(\inf \limits _{t \in [0,1]}(u_t(x)-t\tau )=u_0(x)\). By the convexity of \(u_t\) in the t variable, it is equivalent to \(\dot{u}_0(x) \ge \tau \).
-
(3)
By Lemma 3.4, we have \(P(u_0,u_1) \in \mathcal {E}_p(M,\xi ,\omega ^T)\). Notice that \(P(u_0,u_1) \le u_0,u_1\). It follows from (1) that \(P(u_0,u_1) \le u_t\). By Proposition 3.11 we have \(u_t \in \mathcal {E}_p(M,\xi ,\omega ^T)\) for \(t \in [0,1]\).
\(\square \)
Now we prove Theorem 4.1, through a series of propositions and lemmas, following [28, Section 4] (and in particular [30, Section 3]).
First of all, the \(\mathrm{{d}}_p\) distance between comparable smooth potentials behaves well.
Lemma 4.4
Suppose \(u, v\in \mathcal {H}\) with \(u\le v\). We have
Proof
Let \(w_t: [0, 1]\rightarrow \mathcal {H}\) be the \(C^{1, \bar{1}}_B\) geodesic connecting u and v. By Theorem 3.4, we have
By Lemma 4.1, we have \(u\le w_t\) given \(u\le v\). Since \(w_t\) is convex in t, it follows that
It then follows that, by (4.10) and (4.11),
Next we use \(\omega _u^n\wedge \eta \le 2^n \omega _{\frac{u+v}{2}}^n\wedge \eta \) to obtain that
We write the right-hand side above as follows and apply (4.12) for \(u\le (u+v)/2\) to obtain
The lemma below implies that \(\mathrm{{d}}_p(u, (u+v)/2)\le \mathrm{{d}}_p(u, v)\), completing the proof. \(\square \)
Lemma 4.5
Suppose \(u, v, w\in \mathcal {H}\) and \(u\le v\le w\). Then we have
Proof
Let \(\alpha _t, \beta _t\) be the \(C^{1, \bar{1}}_B\) geodesic segments connecting u, v and u, w, respectively. Since \(u\le v\le w\), by Lemma 4.1 we have \(u\le \alpha _t\le v\) and \(u\le \beta _t\le w\); moreover, \(\alpha _t\le \beta _t\). Since \(\alpha _0=\beta _0\), this gives that \(0\le \dot{\alpha }_0\le \dot{\beta }_0\). Theorem 3.4 then implies that \(\mathrm{{d}}_p(u, v)\le \mathrm{{d}}_p(u, w)\). Similarly we can prove \(\mathrm{{d}}_p(v, w)\le \mathrm{{d}}_p(u, w)\). \(\square \)
Next we prove that the distance formula (4.1) is well defined and extends the original definition (3.51).
Lemma 4.6
Given \(u_0, u_1\in \mathcal {E}_p(M, \xi , \omega ^T)\), the limit (4.1) is finite and independent of the approximating sequences \(u_0^k, u_1^k\in \mathcal {H}\).
Proof
First we show that given \(u\in \mathcal {E}_p(M, \xi , \omega ^T)\) and a sequence \(\{u_k\}_{k\in \mathbb {N}}\subset \mathcal {H}\) decreasing to u. We have \(\mathrm{{d}}_p(u_l, u_k)\rightarrow 0\) as \(l, k\rightarrow \infty \). We can assume that \(l\le k\) and hence \(u_k\le u_l\). Then Lemma 4.4 implies that
Clearly, we have \(u-u_l\le u_k-u_l\le 0\) and \(u-u_l, u_k-u_l\in \mathcal {E}_p(M, \xi , \omega _{u_l})\). Hence applying Proposition 3.11 for the class \(\mathcal {E}_p(M, \xi , \omega _{u_l})\), we obtain that
As \(u_l\) decreases to \(u\in \mathcal {E}_p(M, \xi , \omega ^T)\), the monotone convergence theorem implies that the right-hand side above converges to zero as \(l\rightarrow \infty \), hence \(\mathrm{{d}}_p(u_l, u_k)\rightarrow 0\) as \(l, k\rightarrow \infty \).
Now by Lemma 3.15, we know that
Hence this proved that the limit (4.1) is convergent and finite.
Next we show that the limit is independent of the choice of approximating sequences. Let \(v_0^l, v_1^l\) be other approximating sequences. Certainly we can assume the sequences are strictly decreasing, by adding small constants if necessary. Fix k and consider the sequence \(\{\max \{u_0^{k+1}, v_0^j\}_{j\in \mathbb {N}}\}\) decreases pointwise to \(u_0^{k+1}\). By Dini’s lemma, the convergence is uniform (for fixed k) and hence we can choose \(j_k\) sufficiently large such that \(v^j_0<u^k_0\), \(j\ge j_k\). Repeating the argument we can assume \(v_1^{j}<u^k_1\), for \(j\ge j_k\). By triangle inequality again, we have
By (4.13) we know that if k is sufficiently large, \(\mathrm{{d}}_p(v_0^j, u_0^k)+\mathrm{{d}}_p(v_1^j, u_1^k)\) is sufficiently small. Hence the distance \(\mathrm{{d}}_p(u_0, u_1)\) is independent of the choice of approximating sequence. \(\square \)
For \(u_0,u_1 \in \mathcal {H}\), we can approximate \(u_0, u_1\) by constant sequences. The previous lemma indicates that the distance (4.1) on \(\mathcal {E}_p(M,\xi ,\omega ^T)\) is an extension of the distance (3.51) on \(\mathcal {H}\) for weight \(\chi (l)=\frac{|l|^p}{p}\).
For \(u_0,u_1 \in \mathcal {E}_p(M,\xi ,\omega ^T)\), we choose a decreasing sequence \(\{u_0^k\}_{k\in \mathbb {N}}, \{u_1^k\}_{k\in \mathbb {N}}\subset \mathcal {H}\) such that \(u_0^k\searrow u_0, u^k_1\searrow u_1\). We connect \(u_0^k, u_1^k\) by the unique \(C_B^{1, \bar{1}}\) geodesic segment \(u^k_t\). By Lemma 4.1, it follows that \(u^k_t\) decreases in k. Hence the limit \(\lim _{k\rightarrow \infty } u^k_t \) exists. Using Dini’s lemma as above, one can show that the limit does not depend on the choice of approximating sequence. By Proposition 4.3 and the remark before it, the limit indeed coincides with the weak geodesic \(u_t\) connecting \(u_0,u_1\):
Lemma 4.7
For \(u_0,u_1 \in \mathcal {E}_p(M,\xi ,\omega ^T)\), the weak geodesic \(u_t\) connecting them is a \(\mathrm{{d}}_p\)-geodesic in the sense that
for \(t_1,t_2 \in [0,1]\).
Proof
Let \(\{u_0^k\}_{k\in \mathbb {N}},\{u_1^k\}_{k\in \mathbb {N}} \subset \mathcal {H}\) be sequences strictly decreasing to \(u_0,u_1\), respectively, and \(u_t^k \in \mathcal {H}_{\triangle }\) the unique \(C_B^{1,\bar{1}}\) geodesic connecting \(u_0^k,u_1^k\). By Theorem 3.4, we have
For \(l \in (0,1)\), Lemma 4.1 implies that \(u_l^k\) strictly decreases to \(u_l\). Then one can choose a sequence \(\{w_l^k\}_{k\in \mathbb {N}} \subset \mathcal {H}\) such that
-
(1)
\(u_l^{k+1} \le w_l^k \le u_l^k\);
-
(2)
For the \(C_B^{1,\bar{1}}\) geodesic \(v_t^k\) connecting \(u_0^k\) and \(w_l^k\) with \(v_0^k=u_0^k, v_1^k=w_l^k\), we have
$$\begin{aligned} \left| \int _M|\dot{v}_0^k|^p \omega _{u_0^k}^n\wedge \eta -l^p\int _M|\dot{u}_0^k|^p\omega _{u_0^k}\wedge \eta \right| <\frac{1}{k}. \end{aligned}$$
In fact there exists a sequence \(\{\varphi ^j\}_{j \in \mathbb {N}} \subset \mathcal {H}\) decreasing to \(u_l^k\). By Dini’s lemma, \(\varphi ^j\) converges to \(u_k^l\) uniformly. It follows from Lemma 4.8 and Proposition 4.3 that for j big enough, \(w_l^k=\varphi ^j\) will satisfy our requirements. By Theorems 3.4 and (4.14),
Hence \(\mathrm{{d}}_p(u_0,u_l)=l\mathrm{{d}}_p(u_0,u_1)\) for \(l \in [0,1]\).
Without loss of generality, we assume that \(0 \le t_1\le t_2\le 1\). By Proposition 4.3, \(h_t=u_{(1-t)t_2}\) is the weak geodesic connecting \(u_{t_2}\) and \(u_0\). By following from the results above, we have
This completes the proof. \(\square \)
Lemma 4.8
Suppose \(u_0,u_1\in \text {PSH}(M,\xi ,\omega ^T)\cap L^{\infty }\). Let \(\{u_1^k\}_{k\in \mathbb {N}} \subset \text {PSH}(M,\xi ,\omega ^T)\cap L^{\infty }\) be a sequence decreasing to \(u_1\) and \(u_t, u_t^k \in \text {PSH}(M,\xi ,\omega ^T)\cap L^{\infty }\) the weak geodesic connecting \(u_0,u_1\) and \(u_0, u_1^k\), respectively. Then
Proof
Denote by \(C=\max (||u_1^1-u_0||_{L^{\infty }}, ||u_1-u_0||_{L^{\infty }})\). It follows Proposition 4.2 that \(||\dot{u}_0||_{L^{\infty }} \le C, ||\dot{u}_0^k||_{L^{\infty }} \le C\). By Proposition 4.3 the sequence \(\{u_t^k\}_{k \in \mathbb {N}}\) decreases to \(u_t\) hence the sequence \(\{\dot{u}_0^k\}_{k\in \mathbb {N}}\) is decreasing with \(\dot{u}_0^k \ge \dot{u}_0\).
Moreover, we have \(\dot{u}_0^k\) decreases to \(\dot{u}_0\). If this is not true, we can find \(x_0 \in M, a \in \mathbb {R}\) such that \(\dot{u}_0^k> a > \dot{u}_0\). Then there exists \(0<t_0<1\) such that \(u_t^k(x_0)> u_0+at > u_t(x_0)\) for \(t \in [0, t_0]\). It contradicts with the fact that \(u_t^k\) decreases to \(u_t\).
Then the lemma follows from Lebesgue’s dominated convergence theorem. \(\square \)
Pythagorean formula about \(\mathrm{{d}}_p\) distance involves that rooftop envelope plays an essential role in Darvas’s results [28, 29] and we have a similar formula in Sasaki setting:
Theorem 4.2
(Pythagorean formula) Given \(u_0, u_1\in \mathcal {E}_p(M, \xi , \omega ^T)\), we have \(P(u_0, u_1)\in \mathcal {E}_p(M, \xi , \omega ^T)\) and
Proof
First we prove the formula for \(u_0,u_1\in \mathcal {H}\). It follows from Theorem 3.1 that \(P(u_0,u_1) \in \mathcal {H}_{\triangle }\). Let \(u_t\) be the \(C_B^{1,\bar{1}}\) geodesic connecting \(u_0,u_1\). Let \(v_t\) be the weak geodesic connecting \(P(u_0,u_1),u_1\). It follows from Lemma 4.1 that \(P(u_0,u_1) \le v_t\) for \(t \in [0,1]\). Hence we have \(\dot{v}_0 \ge 0\). By Lemmas 4.9, Lemma 4.3, the definition of rooftop envelope, and Lemma 3.3, we have
By a similar argument we also have
Now using Theorem 3.4 we have
and the Pythagorean formula holds for smooth potentials \(u_0,u_1\in \mathcal {H}\).
For the general case we can choose sequences \(\{u_0^k\}_{k\in \mathbb {N}}, \{u_1^k\}_{k\in \mathbb {N}} \subset \mathcal {H}\) decreases to \(u_0,u_1\), respectively. Then the sequence \(P(u_0^k, u_1^k) \in \mathcal {H}_{\triangle }\) decreases to \(P(u_0,u_1)\) and the Pythagorean formula follows from Lemma 4.11. \(\square \)
Lemma 4.9
Let \(u_t\) be the weak geodesic connecting \(u_0,u_1 \in \mathcal {H}_{\triangle }\). Then the following holds:
Proof
\(v_t=u_{1-t}\) is the weak geodesic connecting \(u_1,u_0\). By Lemma 4.3, we have
Recall that \(\omega _{u_1}^n\wedge \eta \) has total finite measure \(\text {Vol}(M)\), hence except for a countably many \(s \in \mathbb {R}\) we have \(\omega _{u_1}^n\wedge \eta (\{u_0=u_1-s\})=0\) and \(\omega _{u_1}^n\wedge \eta (\{\dot{u}_1 \ge s\})=\omega _{u_1}^n\wedge \eta (\{\dot{u}_1>s\})\). For such real number s, it follows from Lemma 3.3 that
and
It follows from Lemma 4.3, the definition of rooftop envelope, that
A similar argument gives that
It follows that
Now choose sequence \(\{u_0^k\}_{k\in \mathbb {N}}, \{u_1^k\}_{k\in \mathbb {N}} \subset \mathcal {H}\) decreasing to \(u_0,u_1\), respectively. Let \(u_t^{kl},u_t\) be the \(C_B^{1,\bar{1}}\) geodesic connecting \(u_0^k, u_1^l\) and \(u_0,u_1\), respectively. Let \(u_t^k\) be the \(C_B^{1,\bar{1}}\) geodesic connecting \(u_0^k, u_1\). It follows from Lemmas 4.11, 4.8 and the above results that
Then using Lemmas 4.11, 4.8, and Proposition 4.3, we have
This completes the proof. \(\square \)
Lemma 4.10
Assume that \(u, v\in \mathcal {E}_p(M,\xi ,\omega ^T)\) with \(u\le v\).Then we have
Proof
First we can choose \(u_k, w_k \in \mathcal {H}\) strictly decreasing to u, v, respectively. Then \(\max (u_k, w_k) \in \text {PSH}(M,\xi ,\omega ^T)\) are continuous and strictly decreases to v. By Dini’s lemma there exists \(v_k \in \mathcal {H}\) such that \(\max (u_{k-1},v_{k-1}) \ge v_k \ge \max (u_k, v_k)\). Then \(v_k\) decreases to v and \(u_k \le v_k\). It follows from Lemma 4.4 that
By Proposition 3.15, the required inequality follows as \(k \rightarrow \infty \). \(\square \)
Lemma 4.11
If the sequence \(\{u_k\}_{k\in \mathbb {N}} , \{v_k\}_{k \in \mathbb {N}}\subset \mathcal {E}_p(M,\xi ,\omega ^T)\) decreases (increases) to \(u ,v\in \mathcal {E}_p(M,\xi ,\omega ^T)\), respectively, then \(\mathrm{{d}}_p(u_k, v_k) \rightarrow \mathrm{{d}}_p(u, v)\) as \(k \rightarrow \infty \). In particular, \(\mathrm{{d}}_p(u_k,u) \rightarrow 0\).
Proof
If the sequence \(\{u_k\}_{k\in \mathbb {N}}\) is decreasing, using the triangle inequality and Lemma 4.10, we have
and the lemma follows from Lemma 3.15.
If the sequence \(\{u_k\}_{k\in \mathbb {N}}\) is increasing, using the triangle inequality and Lemma 4.10, we have
and the lemma follows from Lemma 3.15. \(\square \)
Next we proceed to prove that \((\mathcal {E}_p(M,\xi ,\omega ^T),\mathrm{{d}}_p)\) is a complete metric space.
Lemma 4.12
Suppose \(u_0, u_1\in \mathcal {E}_p(M, \xi , \omega ^T)\). Then we have
Proof
It is obvious that \( P(u_0,u_1) \le P(u_0,\frac{u_0+u_1}{2}) \le u_0\) and \(P(u_0,u_1) \le P(u_0,\frac{u_0+u_1}{2}) \le \frac{u_0+u_1}{2}\). By the Pythagorean theorem 4.2, Lemmas 4.5, and 4.10, we have
This completes the proof. \(\square \)
Theorem 4.3
For any \(u_0, u_1\in \mathcal {E}_p(M, \xi , \omega ^T)\), we have
Proof
Using the triangle inequality, arithmetic–geometric mean inequality, and Lemma 4.10, we have:
By the previous lemma, the Pythagorean formula, and Lemma 4.10, there exists a constant C such that
Recall that \(\omega _{u_0}^n\wedge \eta \le 2^n\omega _{\frac{u_0+u_1}{2}}^n\wedge \eta \). Similarly, we also have:
Hence by the Holder inequality, we have:
By symmetry of \(u_0,u_1\), we also have:
Adding the last two inequalities, we obtain:
This completes the proof. \(\square \)
Lemma 4.13
Let \(\{u_k\}_{k\in \mathbb {N}} \subset \mathcal {E}_p(M,\xi ,\omega ^T)\) be a \(\mathrm{{d}}_p\)-bounded sequence decreasing (increasing) to u. Then \(u \in \mathcal {E}(M,\xi ,\omega ^T)\) and \(\mathrm{{d}}_p(u_k,u)\rightarrow 0\).
Proof
If \(\{u_k\}_{k\in \mathbb {N}}\) is decreasing, we can assume that \(u_k <0\). It follows from Lemma 4.10 that
are uniformly bounded. \(\int _M |u_k|^p(\omega ^T)^n\wedge \eta \) is uniformly bounded; the monotone convergence theorem and the dominated convergence theorem imply that \(u_k \rightarrow u \) in \(L^1\) and \(u \in \text {PSH}(M, \xi , \omega ^T)\). \(E_p(u_k)= \int _M |u_k|^p\omega _{u_k}^n\wedge \eta \) is uniformly bounded; it follows from Proposition 3.16 and Lemma 4.11 that \(u \in \mathcal {E}_p(M, \xi , \omega ^T)\) and \(\mathrm{{d}}_p(u_k, u) \rightarrow 0\).
If \(\{u_k\}_{k\in \mathbb {N}}\) is increasing, it follows from Theorem 4.3 that there exists a constant C such that
is uniformly bounded. By Propositions 3.3 and 3.4, we have \(u_k \rightarrow u\) in \(L^1\) and \(u \in \text {PSH}(M,\xi ,\omega ^T)\). By Proposition 3.16 and Lemma 4.11, we have \(u \in \mathcal {E}_p(M, \xi , \omega ^T)\) and \(\mathrm{{d}}_p(u_k, u) \rightarrow 0\). \(\square \)
Proposition 4.4
Given \(u_0, u_1, v\in \mathcal {E}_p(M, \xi , \omega ^T)\),
Proof
By Theorem 3.1 and Lemma 4.11, we only have to prove the inequality for \(u_0, u_1, v \in \mathcal {H}_{\triangle }\). In this case, \(P(u_0, v), P(u_1, v) \in \mathcal {H}_{\triangle }\) according to Theorem 3.1.
First we assume that \(u_0 \le u_1\). Let \(u_t, v_t\) be the \(C_B^{1,\bar{1}}\) geodesic connecting \(u_0,u_1\) and \(P(u_0,v), P(u_1, v)\), respectively. Then \(P(u_0, v) \le P(u_1,v) \le v\) and Proposition 4.1 imply that \(P(u_0, v) \le v_t \le v\). Hence for \(x \in \{P(u_0,v)=v\}\), \(v_t(x)\) is independent of t and \(\dot{v}_0(x)=0\). Then we have
\(P(u_0,v) \le P(u_1,v), P(u_0,v)\le u_0, P(u_1, v) \le u_1\), and Proposition 4.1 imply that \(P(u_0, v) \le v_t \le u_t\) for \(t \in [0,1]\) and \(\dot{v}_0 \ge 0\). Moreover, for \(x \in \{P(u_0,v)=u_0\}\) we have
Then it follows from Lemmas 4.9 and 3.3 that
For the general case, using the Pythagoreans formula we have
This completes the proof. \(\square \)
Proposition 4.5
\((\mathcal {E}_p(M, \xi , \omega ^T), \mathrm{{d}}_p)\) is a complete metric space.
Proof
First we show that \((\mathcal {E}_p(M,\xi ,\omega ^T), \mathrm{{d}}_p)\) is a metric space. The symmetry of \(\mathrm{{d}}_p\) is obvious and the triangle inequality is inherited from the triangle inequality for smooth potentials. We only have to check the non-degeneracy of \(\mathrm{{d}}_p\). Suppose \(w_1,w_2 \in \mathcal {E}_p(M,\xi ,\omega ^T)\) and \(\mathrm{{d}}_p(w_1,w_2)=0\). It follows from the Pythagorean formula that \(\mathrm{{d}}_p(w_1,P(w_1,w_2))=0\) and \(\mathrm{{d}}_p(P(w_1,w_2),w_2)=0\). Then Lemma 4.10 implies that \(w_1=P(w_1,w_2)=w_2\) with respect to the measure \(\omega _{P(w_1,w_2)}^n\wedge \eta \). Then the domination principle Lemma 3.6 implies that \(w_1 \le P(w_1,w_2)\) and \(w_2 \le P(W_1,w_2)\). It follows that \(w_1=P(w_1,w_2)=w_2\). Hence \((\mathcal {E}_p(M, \xi , \omega ^T),\mathrm{{d}}_p)\) is a metric space.
Next we show that the metric space \((\mathcal {E}_p(M, \xi , \omega ^T), \mathrm{{d}}_p)\) is complete. Suppose \(\{u_k\}_{k\in \mathbb {N}} \subset \mathcal {E}_p(M, \xi , \omega ^T)\) is a \(\mathrm{{d}}_p\) Cauchy sequence. We will prove that there exists \(u \in \mathcal {E}_p(M, \xi , \omega ^T)\) such that \(\mathrm{{d}}_p(u_k, u) \rightarrow 0\).
Without loss of generality we can assume that
for \(k \in \mathbb {N}\). Denote by \(u_k^l=P(u_k, u_{k+1},\ldots , u_{k+l})\) for \(k, l \in \mathbb {N}\) and \(u_k^0=u_k\). It follows from the definition of rooftop envelope and Proposition 4.4 that
and the sequence \(\{u_k^l\}_{l \in \mathbb {N}} \subset \mathcal {E}_p(M, \xi , \omega ^T)\) is \(\mathrm{{d}}_p\) bounded and decreasing. According to Lemma 4.13, \(\tilde{u}_k =\lim \limits _{l\rightarrow \infty } u_k^l \in \mathcal {E}_p(M, \xi , \omega ^T)\) and \(\mathrm{{d}}_p(u_k^l, \tilde{u}_k) \rightarrow 0\) as \(l \rightarrow \infty \). Moreover, \(u_k^{l+1} \le u_{k+1}^l\) implies that \(\tilde{u}_k \le \tilde{u}_{k+1}\) and \(\{\tilde{u}_k\}_{k\in \mathbb {N}}\) is a increasing sequence in \(\mathcal {E}_p(M, \xi , \omega ^T)\).
It follows from Lemma 4.11, the definition of rooftop envelope, and Proposition 4.4 that
and the sequence \(\{\tilde{u}_k\}_{k\in \mathbb {N}} \subset \mathcal {E}_p(M, \xi , \omega ^T)\) is \(\mathrm{{d}}_p\) bounded and increasing. By Lemma 4.13, \(u=\lim \limits _{k \rightarrow \infty } \tilde{u}_k \in \mathcal {E}_p(M, \xi , \omega ^T)\) and \(\lim \limits _{k\rightarrow \infty }\mathrm{{d}}_p(\tilde{u}_k, u)=0\). Moreover, by Proposition 4.4 we have
and
It follows from Lemma 4.11 that
By the triangle inequality
we have \(\mathrm{{d}}_p(u_k ,u) \rightarrow 0\). This completes the proof. \(\square \)
To end this section, we remark that Theorem 4.1 follows from Lemmas 4.6, 4.7, and Proposition 4.5. Our main Theorem 2 follows from Theorem 4.1, Lemmas 4.7, 4.3(3), and 4.12, and Theorem 4.3.
5 Sasaki-Extremal Metric
We give a brief discussion of existence of Sasaki-extremal metric and properness of modified \(\mathcal {K}\)-energy. Calabi’s extremal metric was extended to Sasaki setting by Boyer–Galicki–Simanca [16]. A Sasaki metric is called Sasaki-extremal if its transverse Kähler metric is extremal in the sense of Calabi [17]. As in Kähler setting, given a priori estimates [49] and the pluripotential theory developed in the paper, we have the following:
Theorem 5.1
A compact Sasaki manifold \((M, \xi , \eta , g)\) admits a Sasaki-extremal metric in the transverse Kähler class \([\omega ^T]\) if and only if the modified \(\mathcal {K}\)-energy is reduced proper.
We recall some basic notions [16, 17, 35, 36, 52]. We use the group \(\text {Aut}_0(\xi , J)\) to denote the subgroup of diffeomorphism group of M which preserves both \(\xi \) and transverse holomorphic structure. Its Lie algebra is the Lie algebra of all Hamiltonian holomorphic vector fields in the sense of [37, Definition 4.4].
First one can define Sasaki–Futaki invariant as follows, given \(X\in \mathfrak {aut}\), the Lie algebra of \(\text {Aut}_0(\xi , J)\),
where f is the potential of transverse scalar curvature,
The first step is certainly to verify that (5.1) does not depend on a particular choice of transverse Kähler form in \([\omega ^T]\) (see [16, Proposition 5.1]). We are interested in the reduced part \(\mathfrak {h}_0\) of \(\mathfrak {aut}\), which consists of Hamiltonian holomorphic vector fields such that \(\eta (Y)\) has non-empty zero. When \((M, \xi , \eta , g)\) is a Sasaki-extremal metric, then similar as in Calabi’s decomposition, we have [16, Theorem 4.8] the decomposition
where \(\mathfrak {a}\) consists of parallel vector fields of the transverse Kähler metric \(g^T\). Moreover, the reduced part \(\mathfrak {h}_0\) has the decomposition
where \(\mathfrak {z}_0=\text {aut}(\xi , \eta , g)/\{\xi \}\) and
where \(X:=(\bar{\partial } R)^{\#}\) is the dual vector and it is the extremal vector field in \(\mathfrak {h}_0\). In general, we can define Futaki–Mabuchi bilinear form [36] on \(\mathfrak {h}_0\) as in Kähler setting (in Sasaki setting this is well defined on \(\mathfrak {aut}\) since every Hamiltonian vector field has a potential, simply given by \(\eta (Y)\); for example, \(\xi \) has potential 1). Given \(Y, Z\in \mathfrak {aut}\), define
It is straightforward to check that (5.2) remains unchanged if \(\eta \rightarrow \eta +\mathrm{{d}}^c_B\phi \) for \(\phi \in \mathcal {H}\). If we restrict us on the real Hamiltonian holomorphic vector fields such that \(\eta (Y)\) is real, then there exists a unique vector field V such that
We call such V and its corresponding \(X=V-\sqrt{-1}JV\) the extremal vector field. As in Kähler setting, for JV-invariant metrics in \(\mathcal {H}\), we define the modified \(\mathcal {K}\)-energy [41, 56] as
Let \(\text {Aut}_0(\xi , J, V)\) be the subgroup of \(\text {Aut}_0(\xi , J)\) which commutes with the flow of JV.
Proposition 5.1
The \(\mathcal {K}_V\) energy is invariant under the action of \(\text {Aut}_0(\xi , J, V)\)
Proof
The proof is similar to Kähler setting [48, Lemma 2.1] and it follows in a tautologic way from Futaki invariant and definition of extremal vector field through Futaki–Mabuchi bilinear form. We fix a background transverse Kähler structure \(\omega ^T\) such that it is JV invariant. For \(\sigma \in \text {Aut}_0(\xi , J, V)\), let \(\sigma _t\) be one parameter subgroup generated by the flow of \(Y_\mathbb {R}:=\text {Re}(Y)\) for some \(Y\in \mathfrak {aut}\). Since Y commutes with V, hence \(\sigma _t^* \omega _0\) is invariant with respect to JV if \(\omega _0\in [\omega ^T]\) is invariant. We compute
The right-hand side is zero by (5.3). \(\square \)
We define the distance \(d_1\) modulo the group action \(G_0:=\text {Aut}_0(\xi , J, V)\). Fix a compact subgroup K of \(G_0\) such that K contains the flow of JV (and \(\xi \) of course). Denote
Note that \(G_0\) acts on \(\mathcal {H}_0\) through \(\omega _\phi \rightarrow \sigma ^*\omega _\phi =\omega ^T+\sqrt{-1}\partial _B\bar{\partial }_B \sigma [\phi ]\). Given any \(\phi , \psi \in \mathcal {H}_0\), we can consider the distance modulo \(G_0\) as follows [26]
Definition 5.1
We say \(\mathcal {K}_V\) is reduced proper for K-invariant metrics with respect to \(d_{1, G_0}\), if the following conditions hold
-
(1)
\(\mathcal {K}_V\) is bounded below over \(\mathcal {H}^K\).
-
(2)
There exists constant \(C, D>0\) such that for \(\phi \in \mathcal {H}^K\)
$$\begin{aligned} \mathcal {K}_V(\phi )\ge C d_{1, G_0}(0, \phi )-D. \end{aligned}$$
To prove Theorem 5.1, we proceed exactly as in [48], to consider the modified Chen’s continuity path [21], for a K-invariant transverse Kähler metric \(\omega ^T\),
Given a priori estimates as in [49] and the pluripotential theory on Sasaki manifolds developed in this paper, we can then follow [48, 49] to prove Theorem 5.1. Since the argument is almost identical, we only sketch the process and skip the details.
-
(1)
The openness of (5.5) is proved similarly [48, Theorem 3.4]; note that we assume transverse Kähler metrics and potentials are K-invariant.
-
(2)
For \(0<t<1\), \(\mathcal {K}_V\) bounded below over \(\mathcal {H}^K\) implies that the distance \(d(0, \phi _t)\) is uniformly bounded by a constant in the order \(C((1-t)^{-1}+1)\), where \(\phi _t\) is the solution of (5.5) at t. This together with the fact that \(\phi _t\) minimizes \(t\mathcal {K}_V+(1-t)\mathbb {J}\), gives the uniform upper bound of entropy of \(H(\phi _t)\) (depending on \((1-t)^{-1}\)). Hence estimates in [49, Theorem 2] apply to get the solution for any \(t<1\).
-
(3)
Choose an increasing sequence \(t_i\rightarrow 1\); first using the properness assumption, we can assume that there are \(\sigma _i\in G\) such that \(\psi _i:=\sigma _i[\phi _{t_i}]\) (\(\omega _{\psi _i}=\sigma _i^{*}\omega _{\phi _{t_i}}\)) satisfies that \(d(0, \psi _i)\) is uniformly bounded above. Then \(\psi _i\) satisfies a scalar curvature-type equation
$$\begin{aligned}&\omega _{\psi _i}^n=e^{F_i}(\omega ^T)^n\\&\Delta _{\psi _i} F_i=h_i+\text {tr}_{\psi _i}\left( Ric(\omega ^T)-\frac{1-t_i}{t_i}\omega _i\right) , \end{aligned}$$where \(h_i\) is uniformly bounded and \(\omega _i=\sigma _i^{*}(\omega ^T)\). One can use [49, Theorem 3] and arguments as in [48, Theorem 3.5] to conclude the convergence of \(\psi _i, F_i\) to a smooth Sasaki-extremal structure.
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Acknowledgements
The first named author thanks Prof. Xiuxiong Chen for his encouragement. The first named author is also grateful for Darvas, and states his lecture notes [30] helped them significantly in writing the current paper. The first named author is supported in part by an NSF Grant, Award No. 1611797. The second named author thanks Prof. Xiangyu Zhou and Prof. Yueping Jiang for their help and encouragement. He is partially supported by NSFC 11701164.
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Appendix
Appendix
1.1 Approximation Through Type-I Deformation and Regularity of Rooftop Envelop
Using Type-I deformation, we can obtain the following approximation of irregular Sasaki structure \((M, \xi , \eta , g)\), which would be important for us; see [55] and in particular [13, Theorem 7.1.10] for the approximation. Suppose \(\xi \) is irregular, then the Reeb flow generates an isometry in \(\text {Aut}(M, \xi , \eta , g)\). Let \(T^k\subset \text {Aut}(M, \xi , \eta , g)\) (\(k\ge 2\)) be the torus generated by \(\xi \) and denote \(\mathfrak {t}\) to be its Lie algebra. We can then choose \(\rho _i\rightarrow 0, \rho _i\in \mathfrak {t}\) such that \(\xi _i=\xi +\rho _i\) is quasiregular. Define
where \(\Phi \) is the (1, 1) tensor field defined on the contact bundle \(\mathcal {D}=\text {Ker}(\eta )\). We recall the following:
Theorem 6.1
(Approximation of irregular Sasaki structure) Let \((M, \xi , \eta , g)\) be an irregular Sasaki structure on a compact manifold M. Then we can choose \(\rho _i\rightarrow 0\) such that \(\xi _i\) is quasiregular and (6.1) define a quasiregular Sasaki structure which is invariant under the action of \(T^k\), the torus generated by \(\xi \) in \(\text {Aut}(M, \xi , \eta , g)\).
Lemma 6.1
Let \((M, \xi , \eta , g)\) be a Sasaki structure on a compact manifold M. Consider a torus \(T\subset \text {Aut}(M, \xi , \eta , g)\) and \(\xi _i\in \mathfrak {t}\). Choose \(\xi _i=\xi +\rho _i\) for \(\rho _i\) sufficiently small. Consider two Sasaki structures \((\xi , \eta , \Phi , g)\leftrightarrow (\xi _i, \eta _i, \Phi _i, g_j)\) via Type-I deformation. Then we have the following. Suppose u is T invariant and \(u\in \text {PSH}(M, \xi , \omega ^T)\) with \(|d \Phi \mathrm{{d}}u|\le C_0\). Then for \(\rho _i\) sufficiently small, there exists positive constant \(\epsilon _i\rightarrow 0\) (as \(\rho _i\rightarrow 0\)) such that,
Similarly, suppose \(|d \Phi \mathrm{{d}}u|\le C_0\) and \(u\in \text {PSH}(M, \xi _i, \omega _i^T)\), then there exists positive constant \(\epsilon _i \rightarrow 0\) as \(i\rightarrow \infty \), such that
Proof
Since u is \(T^k\)-invariant, hence u is a basic function with respect to both \(\xi \) and \(\xi _i\). We write
Using (6.1), we compute
If \(|\mathrm{{d}}\Phi d u|\le C_0\), then (6.4) implies that \(|\mathrm{{d}}\Phi _i d u|\le C_1\) (vice versa). Moreover, when \(\rho _i\rightarrow 0\),
We can then choose \(\epsilon _i\rightarrow 0\) as \(\rho _i\rightarrow 0\), such that
This proves (6.2). Note that given the relation of \(\Phi \) and \(\Phi _i\), then \(|d \Phi \mathrm{{d}}u|\le C_0\) implies that \(|\mathrm{{d}}\Phi _i d u|\) is uniformly bounded (we suppose \(\rho _i\) is uniformly small in smooth topology). Interchanging \(\xi \) and \(\xi _i\), this proves (6.3). \(\square \)
Remark 6.1
Note that the complex structure on the cone remains unchanged under Type-I deformation [50, Lemma 2.2]. The transverse holomorphic structure is changed since the foliation is changed, due to the change of Reeb vector foliation; on the other hand, the contact bundle \(\mathcal {D}\) remains unchanged. Note that \((\mathcal {D}, \Phi )\) and \((\mathcal {D}, \Phi _i)\) can be identified to transverse holomorphic tangent bundle \(T^{1, 0}(\mathcal {F}_\xi )\) and \(T^{1, 0}(\mathcal {F}_{\xi _i})\) (the foliations are different). Since the term \(\eta \wedge d\left( \frac{1-\mathrm{{d}}u(\Phi \rho _i)}{1+\eta (\rho _i)}\right) \) vanishes on \(\mathcal {D}\) and \(\left( \frac{1+2\mathrm{{d}}u(\Phi \rho _i)}{1+\eta (\rho _i)}-1\right) \omega ^T\) involves with only \(\mathrm{{d}}u\), hence the above statement holds if we only assume that \(|\mathrm{{d}}u|\) is uniformly bounded. Since we shall not need this, we skip the argument. However, it seems that assumption like \(|\mathrm{{d}}u|\le C\) is necessary and we are not able to extend this to \(\text {PSH}(M, \xi , \omega ^T)\).
As mentioned above, we fix a torus \(T\subset \text {Aut}(N, \xi , \eta , g)\) and consider \(\rho _i\in \mathfrak {t}\) sufficiently small. Let \(\xi _i=\xi +\rho _i\) and let \((\xi _i, \eta _i, g_i, \Phi _i)\) be the Type-I deformation of \((\xi , \eta , g, \Phi )\).
Lemma 6.2
Let \(\rho _i\rightarrow 0\). Suppose a sequence of T-invariant functions \(u_i\in \text {PSH}(M, \xi _i, \omega _i^T)\) with \(|\mathrm{{d}}\Phi d u_i|_{\omega ^T}\le C_0\) converges to \(u\in \text {PSH}(M, \xi , \omega ^T)\). Then \(|\mathrm{{d}}\Phi \mathrm{{d}}u|_{\omega ^T}\le C_0\) and we have the following weak convergence of the measure
Proof
By (6.4) and \(|\mathrm{{d}}\Phi d u_i|_{\omega ^T}\le C_0\), \(\omega _i^T+\frac{1}{2}\mathrm{{d}}\Phi _i d u_i\) and \(\omega ^T+\frac{1}{2}\mathrm{{d}}\Phi d u_i\) differ by a term with small \(L^\infty \) norm, hence we only need to prove that
Note that \(\eta _i=\eta /(1+\eta (\rho _i))\) converges smoothly to \(\eta \), then the above follows from the weak convergence of \((\omega ^T+\frac{1}{2}\mathrm{{d}}\Phi d u_i)^n\wedge \eta \). \(\square \)
Next we give a proof of Theorem 3.1 in Sasaki setting, regarding the regularity of envelop construction.
Lemma 6.3
Assume \(\beta >0\) and \(u,v \in \text {PSH}(M,\xi ,\omega ^T) \cap L^{\infty }\). If
then \(v \le u\).
Proof
By the comparison principle (3.6)
Then we have
It follows that \(\{u<v \}\) has zero Lebesgue measure and \(v \le u\) almost everywhere with respect to Lebesgue measure. Moreover, we have \(v \le u\) everywhere on M since they are \(\omega ^T\)-plurisubharmonic. \(\square \)
Using the same computation to the transverse complex Monge–Ampere equation (as in the complex Monge–Ampere equation [57, p. 99]), we can obtain the following Laplacian estimate.
Lemma 6.4
Suppose \(u \in \mathcal {H}\) is a solution for the equation
Then
where \(B>0\) is a constant which depends on \(\omega \).
Theorem 6.2
Given \(f\in C^\infty _B(M)\), then we have the following estimate
Moreover, if \(u_1, \ldots , u_k\in \mathcal {H}_\Delta \), where we use the notation
then \(P(u_1, \ldots , u_k)\in \mathcal {H}_\Delta \).
Proof
The first result was proved by Berman–Demailly [8] in Kähler setting. Since all quantities are basic and only transverse Kähler structure is involved, the argument as in Kähler setting has a direct adaption; see [30, Theorem A.7] for details in Kähler setting.
For each \(\beta >0\), consider the equation
This reads locally
The transverse version of Aubin–Yau theorem implies that there exists a unique solution \(u_\beta \) for any \(\beta >0\) and a smooth function f. The unique solution \(u_\beta \) satisfies the following:
and there exists \(\beta _0>0\) and a uniform constant C such that \(\beta \ge \beta _0\),
To prove (6.6), we choose \(x_0 \in M\) such that \(u_{\beta }-f\) obtains its maximum at \(x_0\). Combining with Eq. (6.5), we have
and
at \(x_0\). It follows that
on M where \(C=\sup _{M}\log \frac{(\omega _f^T)^n \wedge \eta }{(\omega ^T)^n \wedge \eta } \). By the definition (3.24) we have
On the other hand, we choose \(v \in \mathcal {H}\) and \(L>0\) such that
One can choose \(\beta _1>2\) such that \(\epsilon =\frac{2n\log \beta }{\beta }<1\) for all \(\beta \ge \beta _1\). Take \(\beta _2=\max \{\frac{1}{L},\beta _1\}\), then for any \(\beta \ge \beta _2\), we have
where \(\delta =\frac{1}{\beta }\). It follows that
and
By Eq. ((6.5)) and Lemma (6.3), we have
and
Combined with Eq. ((6.8)) we can derive that
for \(\beta \ge \beta _2\). Then (6.6) follows immediately.
It is standard to deduce the lower bound in (6.7) from the fact \(\omega ^T+\sqrt{-1}\partial _B\overline{\partial }_B u_{\beta } \ge 0\). By Eq. ((6.5)) and Lemma 6.4, we have
It follows that
and
Assume that \(\text {Tr}_{\omega ^T}\omega _{u_{\beta }}^Te^{-Bu_{\beta }}\) obtains its maximum s at \(x_1 \in M\) and \(C_1=\sup _M (2n+\triangle ^{\omega ^T}f)\), then we have
By the inequality (6.9) and \(P(f) \le f\), \(u_{\beta }\) is uniformly bounded. Hence we obtain an upper bound for \(\text {Tr}_{\omega ^T}\omega _{u_{\beta }}^Te^{-Bu_{\beta }}\) if \(\beta \ge \beta _0=\max \{3nB,\beta _2\}\). We conclude that \(\triangle ^{\omega ^T}u_{\beta } \le C\) for \(\beta \ge \beta _0\).
The first statement follows from (6.6) and (6.7).
For the second statement, first note that we only need to show that if \(u_0, u_1\in \mathcal {H}_\Delta \), then \(P(u_0, u_1)\in \mathcal {H}_\Delta \). Let \(u_t\) be the geodesic segment connecting \(u_0, u_1\), then by Lemma 3.9, we know that \(u_t\in \mathcal {H}_\Delta \) (see [8] and [47] for Kähler setting). Now we have already known \(P(u_0, u_1)=\inf _{t\in [0, 1]} u_t\), then by [31, Proposition 4.4] (applied to each foliation chart), \(\Delta u_t\) is uniformly bounded. This shows that \(P(u_0, u_1)\in \mathcal {H}_\Delta \). \(\square \)
More generally, one can obtain results as in [31] that \(P(f_1, \ldots , f_n)\in C^{1, \bar{1}}_B\) given \(f_1, \ldots , f_n\in C^{1, \bar{1}}_B\). The point is that given two functions \(f_1, f_2\), \(h=\min \{f_1, f_2\}\) satisfy \(\Delta h\le \max \{\Delta f_1, \Delta f_2\}\) in viscosity sense, writing \(h=\frac{f_1+f_2}{2}-\frac{|f_1-f_2|}{2}\). The argument as in [30, Theorem A.7] applies using the maximum principle in viscosity sense. Since we do not need this, we shall skip the details.
1.2 Complex Monge–Ampere Operator and Intrinsic Capacity on Compact Sasaki Manifolds
We discuss briefly the Bedford–Taylor theory on Sasaki manifolds. For details for complex Monge–Ampere operator, see Bedford–Taylor [2]. We also extend intrinsic Monge–Ampere capacity to Sasaki setting, see [43] for Kähler setting.
Given a Sasaki structure, there is a splitting of tangent bundle \(TM=L\xi \otimes \mathcal {D}\), where \(\mathcal {D}=\text {Ker}(\eta )\), with \(\Phi : \mathcal {D}\rightarrow \mathcal {D}\) inducing a splitting \(\mathcal {D}\otimes \mathbb {C}=\mathcal {D}^{1, 0}\oplus \mathcal {D}^{0, 1}\). Hence the subbundle \(\Lambda ^{2p}(\mathcal {D}^*)\) of \(\Lambda ^{2p}M\) is well defined and \(\Phi \) induces a splitting to give bidegree of forms in \(\Lambda ^{2p}(\mathcal {D}^*)\). Note that we have the following,
We do not assume that \(\theta \in \Lambda ^{2p}(\mathcal {D}^*)\) is basic. That is, the coefficients of \(\theta \) might not be invariant under the Reeb flow. A simple observation shows that if \(\theta \in \Lambda ^{2p}(\mathcal {D}^*)\), then \(\theta \) is basic if it is closed, \(d\theta =0\) (since \(\iota _\xi \theta =0\)). Hence a closed 2p-form in \(\Lambda ^{2p}(\mathcal {D}^*)\) is basic and can be regarded as a transverse closed 2p-form, defined as in [58]. In general, \(d\Lambda ^{2p}(\mathcal {D}^*)\) is not in \(\Lambda ^{2p+1}(\mathcal {D}^*)\).
Next, we give a very brief discussion of transverse positive closed currents of bidegree of (p, p) on M, \(0\le p\le n\); see [58] for similar treatment. We simply treat them as closed differential forms of bidegree (p, p) in \(\Lambda ^{2p}(\mathcal {D}^*)\) with measurable coefficients which are invariant under the Reeb flow. Its total variation is controlled by
Given \(\phi \in \text {PSH}(M, \xi , \omega ^T)\), we write \(\phi \in L^1(T)\) if \(\phi \) is integrable with respect to the measure \(T\wedge (\omega ^T)^{n-p}\wedge \eta \). In this case, the current \(\phi T\) is well defined and we write
The positivity is a local notion and we simply think T as a positive closed (p, p)-form on each foliation chart. Hence \(\omega _\phi \wedge T\) is also a transverse closed positive \((p+1, p+1)\) form. Note that we think transverse positive closed currents of bidegree of (p, p)-type as a linear functional on \(\Lambda ^{n-p, n-p}(\mathcal {D}^*)\), hence the test forms are of bidegree \((n-p, n-p)\). A main point is that test forms are not restricted to basic forms. In other words, given such a current T and \(\gamma \in \Lambda ^{n-p, n-p}(\mathcal {D}^*)\), we have the following pairing:
When \(\phi \in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty \), it follows that \(\phi \in L^1(T)\) for any transverse positive closed current T of bidegree (p, p) and hence one can define inductively \(\omega _\phi ^k\wedge (\omega ^T)^{n-k}\); in particular, this leads to the definition of transverse complex Monge–Ampere operator \(\omega _\phi ^n\) of bidegree (n, n). Moreover, the cocycle condition on transverse holomorphic structure ensures that \(\omega _\phi ^k\wedge (\omega ^T)^{n-k}\) is well defined on M. In particular, \(\omega _\phi ^n\wedge \eta \) defines a positive Borel measure on M.
It is more convenient to consider this construction locally in foliations charts \(W_\alpha =(-\delta , \delta )\times V_\alpha \). By taking test forms \(\gamma \in \Lambda ^{n-p, n-p}(\mathcal {D}^*)\) with compact support, we can consider \(T\wedge \eta \) on a foliation chart for a transverse positive closed (p, p) current T. In particular, this give a local description of the complex Monge–Ampere measures \(\omega _\phi ^k\wedge (\omega ^T)^{n-k}\wedge \eta \). By taking test functions f supported in a foliation chart, the measure \(\omega _\phi ^k\wedge (\omega ^T)^{n-k}\wedge \eta \) for each k is regarded as the product measure \(\omega _\phi ^k\wedge (\omega ^T)^{n-k}\wedge \mathrm{{d}}x\) on \(W_\alpha \), where \(\xi =\partial _x\) is the Reeb direction. Note that \(\omega _\phi ^k\wedge (\omega ^T)^{n-k}\) is defined on \(V_\alpha \) as the usual way in Kähler setting, and the cocycle condition on transverse holomorphic structure ensures that \(\omega _\phi ^k\wedge (\omega ^T)^{n-k}\) is well defined as a transverse positive closed current of bidegree (n, n). On each foliation chart, we have \(\omega _\phi ^k\wedge (\omega ^T)^{n-k}\wedge \eta =\omega _\phi ^k\wedge (\omega ^T)^{n-k}\wedge \mathrm{{d}}x\) as a product measure. This coincides with the local description given by van Coevering [58, Section 2].
Moreover, when \(u, v\in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty \), \(\mathrm{{d}}u\wedge \mathrm{{d}}^c_B v\wedge T\) can also be defined, where T is a transverse closed positive current of bidegree \((n-1, n-1)\). By the polarization formula we only need to define \(\mathrm{{d}}u\wedge \mathrm{{d}}^c_B u\wedge T\). By adding a positive constant if necessary, we assume \(u\ge 0\). Then we define
In particular, \(\mathrm{{d}}u\wedge \mathrm{{d}}^c_B u\wedge T\) is positive if T is a transverse closed positive current of bidegree \((n-1, n-1)\). We can then define \(\mathrm{{d}}u\wedge \mathrm{{d}}^c_B u\wedge T\wedge \eta \) as a positive Borel measure. Using the polarization formula, we have the following Cauchy–Schwarz inequality, for \(u, v\in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty \),
We also record the following Stokes’ theorem in Sasaki setting, and its proof follows the Bedford–Taylor theory as in Kähler setting via approximation (Lemma 3.1); see [58, Theorem 2.3.1, Proposition 2.3.2].
Lemma 6.5
Let \(u, v, \phi \in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty \), then for each \(0\le k\le n-1\), we have
We record a basic inequality in Sasaki setting, usually referred to Chern–Levine–Nirenberg inequality.
Proposition 6.1
(Chern–Levine–Nirenberg inequalities) Let T be a positive closed current of bidegree (p, p) on M and \(\phi \in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty \). Then \(\Vert \omega _\phi \wedge T\Vert =\Vert T\Vert \). Moreover, if \(\psi \in \text {PSH}(M, \xi , \omega ^T)\cap L^1(T)\), then \(\psi \in L^1(\omega _\phi \wedge T)\) and
Proof
By Stokes’ theorem, we have \(\int _M \mathrm{{dd}}^c_B (\phi T)\wedge (\omega ^T)^{n-p-1}\wedge \eta =0\), hence
To prove (6.13), we first assume \(\psi \le 0, \phi \ge 0\). By assumption, \(\psi \in L^1(T)\), then
By Stokes’ theorem, we compute
Now suppose \(\sup \psi >0\). Replacing \(\phi \) by \(\phi -\inf \phi \), we compute
The same argument as above leads to (6.13) for the general case. \(\square \)
For a Borel subset E on a Sasaki manifold \((M,\xi ,\omega ^T)\), we define the capacity as
It is obvious that \(\text {cap}_{\omega ^T}(\cup _{k=1}^{\infty }E_k)\le \sum \nolimits _{k=1}^{\infty }\text {cap}_{\omega ^T}(E_k)\) for a sequence of Borel sets \(E_k\). We have the following:
Proposition 6.2
Let \(\phi \in \text {PSH}(M, \xi , \omega ^T)\) with \(0\le \phi \le 1\) and \(\psi \in \text {PSH}(M, \xi , \omega ^T)\) such that \(\psi \le 0\). Then
Proof
We only need to prove (6.14) for canonical cutoffs \(\psi _k=\max \{\psi , -k\}\) (\(-\psi _k\) increases to \(-\psi \) and we can apply monotone convergence theorem). We have the following:
We can then proceed inductively to obtain (6.14). Note that the argument above is a special case of (6.13). \(\square \)
Proposition 6.3
Suppose that \(u \in \text {PSH}(M,\xi ,\omega ^T)\) and \(u\le 0\). Then for \(t>0\) we have
Proof
This is a direct consequence of Proposition 6.2. Denote \(K_t=\{u<-t\}\), then
\(\square \)
Proposition 6.4
Suppose that \(u_k, u \in \text {PSH}(M,\xi ,\omega ^T) \cap L^{\infty }\) and \(u_k\) decreases to u. Then for \(\delta >0\) we have
Proof
This proceeds exactly the same as in [43, Proposition 3.7]. We sketch the argument briefly. We assume \(\text {Vol}(M)=1\) for simplicity. Fix \(\delta >0\) and \(\phi \in \text {PSH}(M, \xi , \omega ^T)\) such that \(0\le \phi \le 1\). We have
By Stokes’ theorem, we write
By the Cauchy–Schwartz inequality, setting \(f_k=u_k-u\ge 0\),
We compute
Similarly, we compute
Combining all these together gives
Suppose \(u_k-u\le c_0\) for a fixed positive constant \(c_0\ge 1\). Then we have
Hence we have
We can proceed inductively by replacing \(\omega _\phi \) by \(\omega ^T+\omega _u\) to obtain
The dominated convergence theorem implies the right-hand side goes to zero, independent of \(\phi \). This completes the proof. \(\square \)
As a consequence, we have the following:
Theorem 6.3
Let \(\varphi \in \text {PSH}(M,\xi ,\omega ^T)\), then for any \(\epsilon >0\) there exists an open subset \(O_{\epsilon } \subset M\) such that \(\text {cap}_{\omega ^T}(O_{\epsilon }) < \epsilon \) and \(\varphi \) is continuous on \(M-O_{\epsilon }\).
Proof
By Proposition 6.3 there exists \(t_0>0\) such that \(\text {cap}_{\omega ^t}(O_0) <\frac{\epsilon }{2}\) for the open subset \(O_0=\{u<-t_0\}\). Take the cutoff \(u_{t_0}=\max \{u,-t_0\} \in \text {PSH}(M,\xi ,\omega ^T)\), then there exists a sequence \(u_k \in \mathcal {H}\) decreasing to u. By Proposition 6.4, we can choose a subsequence \(u_{k_j}\) such that \(\text {cap}_{\omega ^T}(O_j) < \frac{\epsilon }{2^{j+1}}\) for the open subset \(O_j=\{u_{k_j}>u+\frac{1}{j}\}\). Then for the open subset \(O_{\epsilon }=\cup _{j=0}^{\infty }O_j\) we have \(\text {cap}_{\omega ^T}(O)<\epsilon \). Moreover \(u_{K_j}\) converges uniformly to u on \(M-O_{\epsilon }\), hence u is continuous on \(M-O_{\epsilon }\). \(\square \)
Remark 6.2
The discussions above are taken from Kähler setting [43, Section 3]. Note that in (6.13) it is necessary to replace \(\sup \psi \) by \(\max \{\sup \psi , 0\}\) (similarly one needs to replace \(\sup _X\psi \) by \(\max \{\sup _X \psi , 0\}\) in [43, Proposition 3.1]).
We also need the following uniqueness in Sasaki setting, see [44, Theorem 3.3].
Theorem 6.4
Suppose \(u, v\in \mathcal {E}_1(M, \xi , \omega ^T)\) such that
then \(u-v=\text {const}\).
Proof
This follows exactly as in [44, Theorem 3.3] and we sketch the argument. The first step is that for \(u\in \mathcal {E}_1(M, \xi , \omega ^T)\) and its canonical cutoffs \(u_j=\max \{u, -j\}\), then \(\nabla u_j\in L^2(\mathrm{{d}}\mu _g)\) and has uniformly bounded \(L^2\) norm (see [44, Proposition 3.2]). We can assume that \(u\le 0\) and hence \(u_j\le 0\). Then for \(\phi \in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty \) such that \(\phi \le 0\), we know that, for any basic positive closed of \((n-1, n-1)\) type.
An inductive argument applies to \(T=\omega _\phi ^k\wedge (\omega ^T)^{n-k-1}\), we get that
Taking \(\phi =u_j\) in (6.15) and noting that the right-hand side is uniformly bounded, we get \(\nabla u_j\) is uniformly bounded in \(L^2(\mathrm{{d}}\mu _g)\), hence \(\nabla u\in L^2(\mathrm{{d}}\mu _g)\).
We assume that \(u, v\le -1\) and \(\text {Vol}(M)=1\). Set \(f=(u-v)/2\) and \(h=(u+v)/2\). We need to establish that \(\nabla f=0\) by showing that \(\int _M \mathrm{{d}}f\wedge \mathrm{{d}}^c_B f\wedge (\omega ^T)^{n-1}\wedge \eta =0\). If we assume u, v are bounded, then we have
where we use the fact that \(\mathrm{{dd}}^c_B f=(\omega _u-\omega _v)/2\). We shall also establish the following a priori bound, when u, v are bounded,
We apply (6.16) and (6.17) to the canonical cutoffs \(u_j, v_j\) (writing \(f_j, h_j\) correspondingly and using Proposition 3.15),
We can then conclude that
This implies that \(u-v\) is a constant. To establish (6.17), we need several observations as follows. First observe that for \(l=n-2, \ldots , 0\),
where the last inequality follows from \(-h\le 1\) and the normalization of the volume. We can then apply the following inequality inductively for \(T=\omega _h^l\wedge (\omega ^T)^{n-l-1}\) such that
which proves (6.17). Now we establish (6.18). We write
hence we obtain, integrating by parts,
By Cauchy–Schwartz inequality, we have
We can get a similar control
Clearly, we have the following (\(h\le 0, S=\omega _h^l\wedge (\omega ^T)^{n-l-2}\))
Combining these estimate altogether we conclude that,
The last observation is that
This completes the proof of (6.18) by combining two inequalities above. \(\square \)
1.3 Functionals in Finite-Energy Class \(\mathcal {E}_1\) and Compactness
We discuss briefly well-known functionals in Kähler geometry and their properties over finite-energy class \(\mathcal {E}_1\), see [30, Section 3.8]. The energy functionals include Monge–Ampere energy \(\mathbb {I}\) and Aubin’s I-functional on \(\mathcal {E}_1\), see [1, 4, 5, 30] for Kähler setting. These results have a direct adaption in Sasaki setting. Recall Aubin’s I-functional in Sasaki setting, for \(u, v\in \mathcal {H}\)
We also recall the J-functional
where the \(\mathbb {I}_{\omega _u}(v)\)-functional is given by
We define the \(\mathbb {I}\)-functional (with the base \(\omega ^T\)) on \(\mathcal {H}\),
The \(\mathbb {I}\)-functional is also called the Monge–Ampère energy, since if \(t\rightarrow v_t\in \mathcal {H}\) is smooth, then we have (as in Kähler setting),
We mention that I is symmetric with respect to u, v but J is not. I, J are both defined on the metric level, independent of the choice of normalization of potentials u, v; while \(\mathbb {I}_{\omega _u}(v)\) depends on the normalization of u, v. When u, v are bounded, then Bedford–Taylor theory allows to integrate by parts and the I-functional takes the formula
Hence it is non-negative.
We need more information about \(\mathbb {I}\)-functional, see [30, Section 3.7] for Kähler setting. These properties in Sasaki setting follow in a rather straightforward way given pluripotential theory extended to Sasaki setting. We include these facts here for completeness.
Proposition 6.5
Given \(u, v\in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty \), the following cocycle condition holds
Moreover, we have \(\mathbb {I}(u)\) is concave in u in the sense that,
As a direct consequence, if \(u, v\in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty \) such that \(u\ge v\). Then \(\mathbb {I}(u)\ge \mathbb {I}(v)\).
Proof
This follows almost identical as in [30, Proposition 3.8], given the pluripotential theory established in Sasaki setting in the paper. We sketch the argument. When \(u, v\in \mathcal {H}\), this follows exactly the same as in Kähler setting, by taking \(h_t=(1-t)u+tv\) and then use (6.23) to compute directly. When \(u, v\in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty \), we then use \(u_k, v_k\in \mathcal {H}\) decreasing to u, v (Lemma 3.1), respectively. Using Bedford–Taylor’s theorem in Sasaki setting [58, Theorem 2.3.1], we proceed exactly as in Kähler setting to conclude that \(\mathbb {I}(u_k)\rightarrow \mathbb {I}(u)\), etc. For the estimate (6.26), we compute
Using the estimate inductively for the terms in (6.25) leads to (6.26). Clearly, \(\mathbb {I}(u)\) is concave in u given (6.26). \(\square \)
The monotonicity property allows to define \(\mathbb {I}(u)\) for \(u\in \text {PSH}(M, \xi , \omega ^T)\) through the limit process, using the canonical cutoffs \(u_k=\max \{u, -k\}\)
Though the above limit is well defined, it may equal \(-\infty \). It turns out \(\mathbb {I}(u)\) is finite exactly on \(\mathcal {E}_1(M, \xi , \omega ^T)\). We record some further properties of \(\mathbb {I}(u)\) for \(u\in \mathcal {E}_1(M, \xi , \omega ^T)\). The proofs are almost identical and we shall skip the details, see [30, Propositions 3.40, 3.42, 3.43; Lemma 3.41].
Proposition 6.6
Let \(u\in \text {PSH}(M, \xi , \omega ^T)\). Then \(-\infty <\mathbb {I}(u)\) if and only if \(u\in \mathcal {E}_1(M, \xi , \omega ^T)\). Moreover,
Proposition 6.7
Suppose \(u_0, u_1\in \mathcal {E}_1(M, \xi , \omega ^T)\) and \(t\rightarrow u_t\) is the finite-energy geodesic connecting \(u_0, u_1\). Then \(t\rightarrow \mathbb {I}(u_t)\) is linear in t. We also have the following distance formula:
In particular, \(d_1(u_0, u_1)=\mathbb {I}(u_0)-\mathbb {I}(u_1)\) if \(u_0\ge u_1\).
We have the following (see [30, Lemma 3.47])
Lemma 6.6
Suppose \(u, u^j, v, v^j\in \mathcal {E}_1(M, \xi , \omega ^T)\) and \(u^j\searrow u\) and \(v^j\searrow v\). Then the following hold:
Moreover, \(\lim _{j\rightarrow \infty } I(u^j, v^j)=I(u, v)\).
Proof
By Proposition 3.8, we have
Hence it follows that
Interchange \(u\leftrightarrow v\), we get \(I(v, \max {\{u, v\}})=\int _{\{u>v\}} (u-v)(\omega _v^n-\omega _u^n)\wedge \eta \). This proves (6.28). We write
Since \(u^j, v^j\le \max \{u^j, v^j\}\), we can apply Proposition 3.15 to conclude \(I(u^j, \max {\{u^j, v^j\}})\rightarrow I(u, \max {\{u, v\}})\) and \(I(v^j, \max {\{u^j, v^j\}})\rightarrow I(v, \max {\{u, v\}})\), using the formula (6.19). This completes the proof. \(\square \)
We have the following well-known inequalities:
Proposition 6.8
For \(u, v\in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty \), we have
Moreover, J(u, v) is convex in v since \(\mathbb {I}_{\omega ^T}(v)\) is concave in v.
Proof
This is well known, by direct computation [38, Proposition 4.2.1] for \(u, v\in \mathcal {H}\). A direct approximation argument using Lemma 3.1 shows that this can be generalized for \(u, v\in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty \). \(\square \)
The functionals (\(I, J, \mathbb {I}\)) are well defined for \(u, v\in \mathcal {E}_1(M, \xi , \omega ^T)\) [see Proposition (3.16)]. Note that (6.26) and Proposition 6.8 both hold in \(\mathcal {E}_1(M, \xi , \omega ^T)\) (see [4] for Kähler setting). This follows by an approximation argument applying Proposition 3.15. Next we prove the following, as a direct adaption of [5, Theorem 1.8],
Lemma 6.7
There exists a positive \(C=C(n)\) such that for \(u, v, w\in \mathcal {E}_1(M, \xi , \omega ^T)\), then
Proof
With Lemma 6.6, we only need to argue (6.29) holds for bounded potentials, with u, v, w replaced by canonical cutoffs \(u_k, v_k, w_k\). The proof follows exactly as in [5, Theorem 1.8, Lemma 1.9]. and we include the proof for completeness. For \(u, v, \psi \in \text {PSH}(M, \xi , \omega ^T)\cap L^\infty \), set
Using (6.24), it is straightforward to see that
We need the following, there exists a constant \(C=C(n)\) for \(u, v, \psi \in \text {PSH}{M, \xi , \omega ^T}\cap L^\infty \), we have the following (see [5, Lemma 1.9]),
With (6.31) we prove (6.29). Taking \(\phi =\frac{u+v}{2}\), the triangle inequality gives,
Using (6.30) and (6.31), we have
By Proposition 6.8, we have
It follows that
We assume \(I(u, v)\ge \max \{I(u, w), I(v, w)\}\) (otherwise we are done). Hence it follows
This is sufficient to prove that
Now we establish (6.31) (see [5, Lemma 1.9]). First observe that
Hence we have
Hence if \(I(u, v)\ge I(u, \psi )+I(v, \psi )\), clearly we have
Now we suppose \(I(u, v)\le I(u, \psi )+I(v, \psi )\). Taking \(\phi =\frac{u+v}{2}\), we consider
By (6.30), \(b_0\le I(u, v)\) and \(b_{n-1}=\Vert \mathrm{{d}}(u-v)\Vert _{\psi }^2\). We claim that, \(p=0, \cdot , n-2\),
We compute
Using Cauchy–Schwarz inequality, we compute
where we have used that \(\omega _u\le 2\omega _\phi \) and (6.24). We can get the same estimate for
This establishes (6.33). By Proposition 6.8, we know that
Denote \(a=(I(\psi , u)+I(\psi , v))\). We write (6.33) as
Note that \(b_0=I(u, v)\le a\), hence it is evident that \(b_p\le C a\). Hence it follows that, for \(p=0, \ldots , n-2\),
A direct computation gives that,
This completes the proof. \(\square \)
More generally, we have the following [30, Proposition 3.48]
Proposition 6.9
Suppose \(C>0\) and \(\phi , \psi , u, v\in \mathcal {E}_1(M, \xi , \omega ^T)\) satisfies
Then there exists a continuous function \(f_C:\mathbb {R}^{+}\rightarrow \mathbb {R}^{+}\) depending only on C with \(f_C(0)=0\) such that
Proof
The proof is similar in philosophy as Lemma 6.7 and follows almost identically as in Kähler setting, see [30, Proposition 3.48]. Hence we skip the details. \(\square \)
As a consequence, we have the following [30, Theorem 3.46]:
Theorem 6.5
Suppose \(u_k, u\in \mathcal {E}_1(M, \xi , \omega ^T)\). Then the following holds:
-
(1)
\(d_1(u_k, u)\rightarrow 0\) if and only if \(\int _M |u_k-u|\omega ^n_T\wedge \eta \rightarrow 0\) and \(\mathbb {I}(u_k)\rightarrow \mathbb {I}(u)\).
-
(2)
If \(d_1(u_k, u)\rightarrow 0\), then \(\omega ^n_{u_k}\wedge \eta \rightarrow \omega _u^n\wedge \eta \) weakly and \(\int _M |u_k-u|\omega ^n_v\wedge \eta \rightarrow 0\) for \(v\in \mathcal {E}_1(M, \xi , \omega ^T)\).
Proof
If \(d_1(u_k, u)\rightarrow 0\), then Propositions 6.6 and 6.9 imply (1) and (2). For the reverse direction in (1), it follows almost identically as in Kähler setting, see [30, Proposition 3.52], using Proposition 6.9 and approximation argument. We sketch the process. First we have
And then one argues that
Hence this shows that \(I(u, u_k)\rightarrow 0\). Using Proposition 6.9 and Lemma 6.6, one can then show
This gives the desired convergence \(d_1(u_k, u)\rightarrow 0\). \(\square \)
As an application of results established above, we have the following compactness result in Sasaki setting, following [30, Theorem 4.45].
Theorem 6.6
Let \(u_j\in \mathcal {E}_1(M, \xi , \omega ^T)\) be a \(d_1\)-bounded sequence for which the entropy
Then \(\{u_j\}\) contains a \(d_1\)-convergence sequence.
Proof
We sketch the proof for completeness; for details see [30, Theorem 4.45]. First \(d_1\) bounded implies that \(\mathbb {I}\) and \(\sup u\) are both bounded. Together with Proposition 3.4, this implies that \(d_1\) bounded set is precompact in \(L^1\). That is, there exists \(u\in \mathcal {E}_1(M, \xi , \omega ^T)\) such that after passing by subsequence,
Moreover, we have (see [30, Proposition 4.14, Corollary 4.15])
Since all elements in \(\mathcal {E}_1(M, \xi , \omega ^T)\) have zero Lelong number, we apply Zeriahi’s uniform version of the famous Skoda integrability theorem [59] (we apply Zeriahi’s theorem in each foliation chart) to obtain: for any \(p\ge 1\), there exists \(C=C(p)\) such that
Since \(\sup u_j\le C\), we have
Now we need to use the assumption that \(H(u_j)\) is uniformly bounded above. We proceed as in the proof of [30, Theorem 4.45] to conclude
By Proposition 6.26 (which holds for \(\mathcal {E}_1\)), we can then conclude that \(\liminf \mathbb {I}(u_j)\ge \mathbb {I}(u)\). This gives \(\lim \mathbb {I}(u_j)=\mathbb {I}(u)\). Hence \(d_1(u_j, u)\rightarrow 0\), as a consequence of Theorem 6.5. \(\square \)
Finally we have the extension of \(\mathcal {K}\)-energy, see [7, Theorem 1.2] for Kähler setting.
Theorem 6.7
The \(\mathcal {K}\)-energy can be extended to a functional \(\mathcal {K}: \mathcal {E}_1\rightarrow \mathbb {R}\cup \{+\infty \}\). Such a \(\mathcal {K}\)-energy in \(\mathcal {E}^1\) is the greatest \(d_1\)-lsc extension of \(\mathcal {K}\)-energy on \(\mathcal {H}\). Moreover, \(\mathcal {K}\)-energy is convex along the finite-energy geodesics of \(\mathcal {E}^1\).
Proof
As in Kähler setting [19], we can write the \(\mathcal {K}\)-energy as the following:
where \(H(\phi )\) is the entropy part and \(\mathbb {J}\) is the entropy part, taking the formula, respectively,
As a direct consequence of this formula, \(\mathcal {K}(\phi )\) is well defined for \(\phi \in \mathcal {H}_\Delta \). More importantly, for \(\phi _0, \phi _1\in \mathcal {H}\), and \(\phi _t\in \mathcal {H}_\Delta \) being the geodesic connecting \(\phi _0, \phi _1\), \(\mathcal {K}(\phi _t)\) is convex with respect to \(t\in [0, 1]\).
Now we extend \(H(\phi )\) and \(\mathbb {J}_{-Ric}\) to \(\mathcal {E}_1\) separately. As in [7], the extension of \(\mathbb {J}_{-Ric}\) to \(\mathcal {E}_1\) is \(d_1\)-continuous, while since \(d_1(u_k, u)\rightarrow 0\) implies that \(\omega _{u_k}^n\wedge \eta \rightarrow \omega _u^n\wedge \eta \) weakly (Theorem 6.5), this implies that the extension of \(\phi \rightarrow H(\phi )\) to \(\mathcal {E}_1\) is \(d_1\) lsc. Moreover, by [49, Lemma 5.4], the extension of \(\mathcal {K}\) is the greatest lsc extension. In the end, the convexity of the extended \(\mathcal {K}\)-energy along the finite-energy geodesic segments follows exactly as in [7, Theorem 4.7]. \(\square \)
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He, W., Li, J. Geometric Pluripotential Theory on Sasaki Manifolds. J Geom Anal 31, 1093–1179 (2021). https://doi.org/10.1007/s12220-019-00257-5
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DOI: https://doi.org/10.1007/s12220-019-00257-5