L^p Metric Geometry of Big and Nef Cohomology Classes

Abstract

Let (X,ω) be a compact Kähler manifold of dimension n, and let 𝜃 be a closed smooth real (1,1)-form representing a big and nef cohomology class. We introduce a metric d_p,p ≥ 1, on the finite energy space $\mathcal {E}^{p}(X,\theta )$, making it a complete geodesic metric space.

1 Introduction

Finding canonical (Kähler-Einstein, cscK, extremal) metrics on compact Kähler manifolds is one of the central questions in differential geometry (see [13, 41, 42] and the references therein). Given a Kähler metric ω on a compact Kähler manifold X, one looks for a Kähler potential φ such that ω_φ := ω + dd^cφ is “canonical”. Mabuchi introduced a Riemannian structure on the space of Kähler potentials $\mathcal {H}_{\omega }$. As shown by Chen [15] $\mathcal {H}_{\omega }$ endowed with the Mabuchi d₂ distance is a metric space. Darvas [21] showed that its metric completion coincides with a finite energy class of plurisubharmonic functions introduced by Guedj and Zeriahi [36]. Other Finsler geometries d_p, p ≥ 1, on $\mathcal {H}_{\omega }$ were studied by Darvas [20] and they lead to several spectacular results related to a longstanding conjecture on existence of cscK metrics and properness of K-energy (see [6, 16,17,18, 29]). Employing the same technique as in [29] and extending the L¹-Finsler structure of [20] to big and semipositive classes via a formula relating the Monge-Ampère energy and the d₁ distance, Darvas [22] established analogous results for singular normal Kähler varieties. Motivated by the same geometric applications, the L^p (p ≥ 1) Finsler geometry in big and semipositive cohomology classes was constructed in [32] via an approximation method.

In this note we extend the main results of [20, 32] to the context of big and nef cohomology classes. Assume that X is a compact Kähler manifold of complex dimension n and let 𝜃 be a smooth closed real (1,1) form representing a big & nef cohomology class. Fix p ≥ 1.

Main Theorem

The space $\mathcal E^{p}(X,\theta )$ endowed with d _p is a complete geodesic metric space.

For the definition of $\mathcal E^{p}(X,\theta )$, d_p and relevant notions we refer to Section 2. When p = 1 Main Theorem was established in [26] in the more general case of big cohomology classes using the approach of [22]. Here, we use an approximation argument as in [32] with an important modification due to the fact that generally potentials in big cohomology classes are unbounded. Interestingly, this modification greatly simplifies the proof of [32, Theorem A].

Organization of the Note

We recall relevant notions in pluripotential theory in big cohomology classes in Section 2. The metric space $(\mathcal E^{p},d_{p})$ is introduced in Section 3 where we prove Main Theorem. In case p = 1 we show in Proposition 3.18 that the distance d₁ defined in this note and the one defined in [26] do coincide.

2 Preliminaries

Let (X,ω) be a compact Kähler manifold of dimension n. We use the following real differential operators $d= \partial +\bar {\partial }$, $d^{c} = i(\bar {\partial }-\partial )$, so that $dd^{c} =2i \partial \bar {\partial }$. We briefly recall known results in pluripotential theory in big cohomology classes, and refer the reader to [5, 12, 24,25,26,27] for more details.

2.1 Quasi-plurisubharmonic Functions

A function $u: X \rightarrow \mathbb {R} \cup \{-\infty \}$ is quasi-plurisubharmonic (or quasi-psh) if it is locally the sum of a psh function and a smooth function. Given a smooth closed real (1,1)-form 𝜃, we let PSH(X,𝜃) denote the set of all integrable quasi-psh functions u such that 𝜃_u := 𝜃 + dd^cu ≥ 0, where the inequality is understood in the sense of currents. A function u is said to have analytic singularities if locally $u=\log {\sum }_{j=1}^{N} |f_{j}|^{2} + h$, where the fj′s are holomorphic and h is smooth.

The De Rham cohomology class {𝜃} is Kähler if it contains a Kähler potential, i.e., a function $u\in \text {PSH}(X,\theta )\cap \mathcal {C}^{\infty }(X,\mathbb {R})$ such that 𝜃 + dd^cu > 0. The class {𝜃} is nef if {𝜃 + εω} is Kähler for all ε > 0. It is pseudo-effective if the set PSH(X,𝜃) is non-empty, and big if {𝜃 − εω} is pseudo-effective for some ε > 0. The ample locus of {𝜃}, which will be denoted by Amp(𝜃), is the set of all points x ∈ X such that there exists ψ ∈PSH(X,𝜃 − εω) with analytic singularities and smooth in a neighborhood of x. It was shown in [11, Theorem 3.17] that {𝜃} is Kähler if and only if Amp(𝜃) = X.

Throughout this note we always assume that {𝜃} is big and nef. Typically, there are no bounded functions in PSH(X,𝜃), but there are plenty of locally bounded functions as we now briefly recall. By the bigness of {𝜃} there exists ψ ∈PSH(X,𝜃 − εω) for some ε > 0. Regularizing ψ (by [30, Main Theorem 1.1]) we can find a function $u\in \text {PSH}(X,\theta -\frac {\varepsilon }{2} \omega )$ smooth in a Zariski open set Ω of X. Roughly speaking, 𝜃_u locally behaves as a Kähler form on Ω. As shown in [11, Theorem 3.17], u and Ω can be constructed in such a way that Ω is the ample locus of {𝜃}.

If u and v are two 𝜃-psh functions on X, then u is said to be less singular than v if v ≤ u + C for some C ∈ ℝ, while they are said to have the same singularity type if u − C ≤ v ≤ u + C, for some $C\in \mathbb {R}$. A 𝜃-psh function u is said to have minimal singularities if it is less singular than any other 𝜃-psh function. An example of a 𝜃-psh function with minimal singularities is

$$V_{\theta}:=\sup\{ u\in \text{PSH}(X, \theta) | u\leq 0\}.$$

For a function $f: X \rightarrow \mathbb {R}$, we let f^∗ denote its upper semicontinuous regularization, i.e.,

$$ f^{*}(x) := \limsup_{X\ni y \to x} f(y). $$

Given a measurable function f on X we define

$$ P_{\theta}(f) := \left( x \mapsto \sup \{u(x) | u \in \text{PSH}(X,\theta), \ u \leq f \}\right)^{*}. $$

Essential Supremum

For u,v quasi-psh functions, the function u − v is defined almost everywhere on X (off the set where v = −∞). By abuse of notation we let supX(u − v) denote the essential supremum of u − v. By basic properties of plurisubharmonic functions we have

$$ u- \sup_{X} (u-v) \leq v \leq u + \sup_{X} (v-u), \text{ on } X. $$

We will need the following result on regularity of quasi plurisubharmonic envelope due to Berman [4].

Theorem 2.1

Let f be a continuous function such thatdd^cf ≤ Cωon X, for someC > 0.Then Δ_ω(P_𝜃(f)) is locally bounded on Amp(𝜃),and

$$ (\theta+dd^{c} P_{\theta}(f))^{n}= \mathbf{1}_{\{P_{\theta}(f)=f\}} (\theta+dd^{c} f)^{n}. $$

(2.1)

If 𝜃 is moreover Kähler then Δ_ω(P_𝜃(f)) is globally bounded on X.

If f = min(u,v) for u,v quasi-psh then f is upper semicontinuous on X and there is no need to take the upper semicontinuous regularization in the definition of P(u,v) := P_𝜃(min(u,v)). The latter is the largest 𝜃-psh function lying below both u and v, and is called the rooftop envelope of u and v in [28].

The proof of Theorem 2.1 can be found in [4]. In the Kähler case, Theorem 2.1 was also surveyed in [23]. For convenience of the reader, and per recommendation of the referee, we briefly recall the arguments here.

Proof of Theorem 2.1

We first assume that f is smooth and fix ε ∈ (0,1]. By nefness of {𝜃}, the form η := 𝜃 + εω represents a Kähler class.

Fix β > 1 and let u_β ∈PSH(X,η) ∩ C^∞(X) be the unique smooth function such that

$$ (\eta +dd^{c} u_{\beta})^{n} =e^{\beta (u_{\beta}-f)} \omega^{n}. $$

(2.2)

The existence (and smoothness) of u_β follows from Aubin [1] and Yau [42].

By [4, Theorem 1.1], u_β converges uniformly to P_η(f) along with a uniform estimate for dd^cu_β. The proof of [4, Theorem 1.2] actually establishes a Laplacian estimate for u_β independent of ε and β.

We fix ψ ∈PSH(X,𝜃) such that supXψ = 0, ψ is smooth in Ω, the ample locus of {𝜃} and 𝜃 + dd^cψ ≥ aω, where a > 0 is a small constant. Note that ψ and a, whose existence follows from the bigness of {𝜃} as recalled in Section 2.1, are independent of ε.

Consider

$$ H := \log \text{Tr}_{\omega} (\eta +dd^{c} u_{\beta}) - A (u_{\beta} -\psi), $$

defined on Ω, where A > 0 is a constant to be specified later. Then, H is smooth on Ω and tends to −∞ on the boundary of Ω. Let x ∈Ω be a point where H attains its maximum in Ω. Setting ω^′ := η + dd^cu_β, it follows from [14, Lemma 2.2] (which is an improvement of [40]) that

$$ {\Delta}_{\omega^{\prime}} \log \text{Tr}_{\omega} (\omega^{\prime}) \geq \frac{{\Delta}_{\omega}(\beta(u_{\beta}-f))}{\text{Tr}_{\omega}(\omega^{\prime})} - B \text{Tr}_{\omega^{\prime}}(\omega), $$

where − B is a negative lower bound for the holomorphic bisectional curvature of ω. In the remainder of this paragraph we carry all computations at the point x. By the maximum principle, we have

$$ 0 \geq {\Delta}_{\omega^{\prime}} H \geq \beta -\beta \frac{\text{Tr}_{\omega}(\eta +dd^{c} f)}{\text{Tr}_{\omega}(\omega^{\prime})} -B \text{Tr}_{\omega^{\prime}}(\omega) - An + A a \text{Tr}_{\omega^{\prime}} (\omega). $$

Let C₁ ≥ 0 be a constant such that $\theta +\omega + dd^{c} f \leq e^{C_{1}}\omega$. Then, choosing A = B/a, we arrive at

$$ 0 \geq (\beta -An) - \beta \frac{ne^{C_{1}}}{\text{Tr}_{\omega}(\omega^{\prime})}. $$

Thus, for β ≥ 2An we have

$$ \text{Tr}_{\omega}(\omega^{\prime}) \leq \frac{\beta n e^{C_{1}}}{\beta -An}\leq 2n e^{C_{1}}. $$

(2.3)

Let also ρ₀ be the unique 𝜃-psh function with minimal singularities such that

$$ (\theta +dd^{c} \rho_{0})^{n} = C_{3}\omega^{n}, \ \sup_{X} \rho_{0}=0, $$

for a uniform normalization constant C₃ = C(𝜃,ω) > 0. The existence of ρ₀ follows from [5, 12]. By [12, Theorem 4.1] we obtain a lower bound for ρ₀:

$$ \rho_{0} \geq V_{\theta} -C(\theta,\omega). $$

Since ρ₀ ≤ f − infXf we have that ρ₀ + infXf + (log C₃)/β is a subsolution to the Monge-Ampère equation defining u_β, (2.2). By [24, Lemma 2.5] and the fact that V_𝜃 ≥ ψ, we have that

$$ u_{\beta} \geq \rho_{0}+\inf_{X} f + (\log C_{3})/\beta \geq \psi-C_{4}, $$

where C₄ > 0 depends on 𝜃,ω,infXf. From this and (2.3), we thus obtain

$$H(x) \leq \log (2n e^{C_{1}}) + A C_{4}.$$

We finally have, for all β ≥ 2nA,

$$ \text{Tr}_{\omega}(\eta +dd^{c} u_{\beta}) \leq C_{5} e^{-A \psi} \ \text{on} \ {\Omega}. $$

Letting β → +∞ and noting that u_β converges uniformly to P_𝜃+εω(f), we obtain

$$ {\Delta}_{\omega} (P_{\theta +\varepsilon \omega}(f)) \leq C_{6} e^{-A\psi}, $$

where C₆ depends on B,a, C₁, infXf. Letting ε → 0 we arrive at

$$ {\Delta}_{\omega} (P_{\theta}(f)) \leq C_{6} e^{-A\psi}. $$

We finally remove the smoothness assumption on f. Assume that f is a continuous function such that dd^cf ≤ Cω. We approximate f uniformly by smooth functions f_j such that dd^cf_j ≤ (C + 1)ω. This is possible thanks to Demailly [30]. Then, the previous steps yield

$$ {\Delta}_{\omega}(P_{\theta}(f_{j})) \leq C^{\prime} e^{-A \psi}, $$

where C^′ > 0 depends only on C,B,a,infXf,𝜃,ω. Letting j → +∞ we arrive at the conclusion. Having the Laplacian bound, one can then argue as in [37, Theorem 9.25] to get (2.1), completing the proof of Theorem 2.1. □

2.2 Non-pluripolar Monge-Ampère Products

Given u₁,…,u_p𝜃-psh functions with minimal singularities, $\theta _{u_{1}}\wedge {\dots } \wedge \theta _{u_{p}}$, as defined by Bedford and Taylor [2, 3] is a closed positive current in Amp(𝜃). For general u₁,…,u_p ∈PSH(X,𝜃), it was shown in [12] that the non-pluripolar product of $\theta _{u_{1}},\ldots ,\theta _{u_{p}}$, that we still denote by

$$ \theta_{u_{1}}\wedge {\ldots} \wedge \theta_{u_{p}}, $$

is well-defined as a closed positive (p,p)-current on X which does not charge pluripolar sets. For a 𝜃-psh function u, the non-pluripolar complex Monge-Ampère measure of u is simply ${\theta _{u}^{n}}:=\theta _{u}\wedge \ldots \wedge \theta _{u}$.

If u has minimal singularities then ${\int }_{X} {\theta _{u}^{n}}$, the total mass of ${\theta _{u}^{n}}$, is equal to ${\int }_{X} \theta _{V_{\theta }}^{n}$, the volume of the class {𝜃} denoted by Vol(𝜃). For a general u ∈PSH(X,𝜃), ${\int }_{X} {\theta _{u}^{n}}$ may take any value in [0,Vol(𝜃)]. Note that Vol(𝜃) is a cohomological quantity, i.e., it does not depend on the smooth representative we choose in {𝜃}.

2.3 The Energy Classes

From now on, we fix p ≥ 1. Recall that for any 𝜃-psh function u we have ${\int }_{X} {\theta _{u}^{n}} \leq \text {Vol}(\theta )$. We denote by $\mathcal {E}(X, \theta )$ the set of 𝜃-psh functions u such that ${\int }_{X} {\theta _{u}^{n}} = \text {Vol}(\theta )$. We let $\mathcal E^{p}(X,\theta )$ denote the set of $u\in \mathcal E(X,\theta )$ such that ${\int }_{X} |u-V_{\theta }|^{p} {\theta _{u}^{n}}<+\infty$. For $u,v \in \mathcal {E}^{p}(X, \theta )$ we define

$$ I_{p}(u,v):=I_{p,\theta}(u,v):={\int}_{X} |u-v|^{p} \left( {\theta_{u}^{n}}+{\theta_{v}^{n}}\right). $$

It was proved in [34, Theorem 1.6] that I_p satisfies a quasi triangle inequality:

$$ I_{p,\theta}(u,v) \leq C(n,p) (I_{p,\theta}(u,w)+ I_{p,\theta}(v,w)), \ \forall u,v,w\in \mathcal E^{p}(X,\theta). $$

In particular, applying this for w = V_𝜃 and using Theorem 2.1, we obtain I_p,𝜃(u,v) < +∞, for all $u,v\in \mathcal E^{p}(X,\theta )$. Moreover, it follows from the domination principle [24, Proposition 2.4] that I_p is non-degenerate:

$$ I_{p,\theta}(u,v)=0 \Longrightarrow u=v. $$

2.4 Weak Geodesics

Geodesic segments connecting Kähler potentials were first introduced by Mabuchi [38]. Semmes [39] and Donaldson [33] independently realized that the geodesic equation can be reformulated as a degenerate homogeneous complex Monge-Ampère equation. The best regularity of a geodesic segment connecting two Kähler potentials is known to be $\mathcal {C}^{1,1}$ (see [8, 15, 19]).

In the context of a big cohomology class, the regularity of geodesics is very delicate. To avoid this issue, we follow an idea of Berndtsson [7] considering geodesics as the upper envelope of subgeodesics (see [24]).

For a curve [0,1] ∋ t↦u_t ∈PSH(X,𝜃), we define

$$ X\times D \ni (x,z) \mapsto U(x,z) := u_{\log |z|}(x), $$

(2.4)

where $D:= \{z\in \mathbb {C} | 1< |z|<e \}$. We let π : X × D → X be the projection on X.

Definition 2.2

We say that t↦u_t is a subgeodesic if (x,z)↦U(x,z) is a π^∗𝜃-psh function on X × D.

Definition 2.3

For φ₀,φ₁ ∈PSH(X,𝜃), we let $\mathcal {S}_{[0,1]}(\varphi _{0},\varphi _{1})$ denote the set of all subgeodesics [0,1] ∋ t↦u_t such that $\limsup _{t\to 0} u_{t}\leq \varphi _{0}$ and $\limsup _{t\to 1} u_{t}\leq \varphi _{1}$.

Let φ₀,φ₁ ∈PSH(X,𝜃). We define, for (x,z) ∈ X × D,

$$ {\Phi}(x,z) := \sup \{ U(x,z) | U \in \mathcal{S}_{[0,1]}(\varphi_{0},\varphi_{1}) \}. $$

The curve t↦φ_t constructed from Φ via (2.4) is called the weak Mabuchi geodesic connecting φ₀ and φ₁.

Geodesic segments connecting two general 𝜃-psh functions may not exist. If $\varphi _{0}, \varphi _{1} \in \mathcal {E}^{p}(X,\theta )$, it was shown in [24, Theorem 2.13] that $P(\varphi _{0},\varphi _{1}) \in \mathcal {E}^{p}(X,\theta )$. Since P(φ₀,φ₁) ≤ φ_t, we obtain that t → φ_t is a curve in $\mathcal {E}^{p}(X,\theta )$. Each subgeodesic segment is in particular convex in t:

$$ \varphi_{t}\leq \left( 1-t\right)\varphi_{0} + t\varphi_{1}, \ \forall t\in [0,1]. $$

Consequently, the upper semicontinuous regularization (with respect to both variables x,z) of Φ is again in $\mathcal {S}_{[0,1]}(\varphi _{0},\varphi _{1})$, hence so is Φ. In particular, if φ₀,φ₁ have minimal singularities, then the geodesic φ_t is Lipschitz on [0,1] (see [24, Lemma 3.1]):

$$ |\varphi_{t}-\varphi_{s}| \leq |t-s| \sup_{X} |\varphi_{0}-\varphi_{1}|, \ \forall t,s \in [0,1]. $$

(2.5)

2.5 Finsler Geometry in the Kähler Case

Darvas [20] introduced a family of distances in the space of Kähler potentials

$$\mathcal{H}_{\omega}:= \{\varphi\in \mathcal{C}^{\infty} (X, \mathbb{R}) | \omega_{\varphi}>0\}. $$

Definition 2.4

Let $\varphi _{0},\varphi _{1}\in \mathcal {H}_{\omega }$. For p ≥ 1, we set

$$ d_{p}(\varphi_{0},\varphi_{1}):=\inf \{ \ell_{p}(\psi) | \psi \text{ is a smooth path joining } \varphi_{0} \text{ to } \varphi_{1} \}, $$

where $\ell _{p}(\psi ):={{\int }_{0}^{1}} \left (\frac {1}{V} {\int }_{X} | \dot {\psi }_{t}|^{p} \omega _{\psi _{t}}^{n}\right )^{1/p} dt$ and $V:= \text {Vol}(\omega )={\int }_{X} \omega ^{n}$.

It was then proved in [20, Theorem 1] (generalizing Chen’s original arguments [15]) that d_p defines a distance on $\mathcal {H}_{\omega }$, and for all $\varphi _{0}, \varphi _{1}\in \mathcal {H}_{\omega }$,

$$ d_{p}(\varphi_{0},\varphi_{1})= \left( \frac{1}{V}{\int}_ X |\dot{\varphi_{t}}|^{p} \omega_{\varphi_{t}}^{n} \right)^{1/p}, \quad \forall t\in [0,1], $$

(2.6)

where t → φ_t is the Mabuchi geodesic (defined in Section 2.4). It was shown in [20, Lemma 4.11] that (2.6) still holds for φ₀,φ₁ ∈PSH(X,ω) with dd^cφ_i ≤ Cω,i = 0,1, for some positive constant C.

By [9, 30], potentials in $\mathcal E^{p}(X,\omega )$ can be approximated from above by smooth Kähler potentials. As shown in [21], the metric d_p can be extended for potentials in $\varphi _{0}, \varphi _{1}\in \mathcal {E}^{p}(X, \omega )$: if ${\varphi _{i}^{k}}$ are smooth strictly ω-psh functions decreasing to φ_i, i = 0,1; then, the limit

$$d_{p}(\varphi_{0}, \varphi_{1}):= \lim_{k\rightarrow +\infty} d_{p}({\varphi_{0}^{k}}, {\varphi_{1}^{k}})$$

exists and it is independent of the approximants. By [20, Lemmas 4.4 and 4.5], d_p defines a metric on $\mathcal {E}^{p}(X,\omega )$ and $(\mathcal E^{p}(X,\omega ),d_{p})$ is a complete geodesic metric space.

3 The Metric Space $(\mathcal {E}^{p}(X,\theta ),d_{p})$

The goal of this section is to define a distance d_p on $\mathcal {E}^{p}(X, \theta )$ and prove that the space $(\mathcal E^{p}(X,\theta ), d_{p})$ is a complete geodesic metric space. We follow the strategy in [32], approximating the space of “Kähler potentials” $\mathcal {H}_{\theta }$ by regular spaces. Throughout this note we will use the notation

$$\omega_{\varepsilon}:= \theta+\varepsilon \omega, \ \varepsilon >0.$$

By nefness of 𝜃, ω_ε := 𝜃 + εω represents a Kähler cohomology class for any ε > 0. Note that ω_ε is not necessarily a Kähler form but there exists a smooth potential $f_{\varepsilon }\in \mathcal {C}^{\infty }(X,\ \mathbb {R})$ such that ω_ε + dd^cf_ε is a Kähler form. For notational convenience we normalize 𝜃 so that $\text {Vol}(\theta )= {\int }_{X} \theta _{V_{\theta }}^{n}=1$ and we set V_ε := Vol(ω_ε).

Typically there are no smooth potentials in PSH(X,𝜃) but the following class contains plenty of potentials sufficiently regular for our purposes:

$$ \mathcal{H}_{\theta}:=\{ \varphi\in \text{PSH}(X, \theta) | \varphi=P_{\theta}(f), f\in \mathcal{C}(X, \mathbb{R}), dd^{c} f\leq C(f) \omega \}.$$

Here C(f) denotes a positive constant which depends only on f. Note that any $u=P_{\theta }(f)\in \mathcal {H}_{\theta }$ has minimal singularities because, for some constant C > 0, V_𝜃 − C is a candidate defining P_𝜃(f). The following elementary observation will be useful in the sequel.

Lemma 3.1

If $u,v\in \mathcal {H}_{\theta }$ then $P_{\theta }(u,v)\in \mathcal {H}_{\theta }$ .

Proof

Set $h= \min (f,g)\in \mathcal {C}^{0}(X,\mathbb {R})$, where $f,g\in \mathcal {C}^{0}(X,\mathbb {R})$ are such that u = P_𝜃(f) and v = P_𝜃(g) and dd^cf ≤ Cω, dd^cg ≤ Cω. Then, − h = max(−f,−g) is a Cω-psh function on X, hence dd^c(−h) + Cω ≥ 0. □

3.1 Defining a Distance d_p on $\mathcal {H}_{\theta }$

By Darvas [20], the Mabuchi distance d_p,ω is well defined on $\mathcal {E}^{p}(X, \omega )$ when the reference form ω is a Kähler form. With the following observation, we show that such a distance behaves well when we change the Kähler representative in {ω}.

Proposition 3.2

Letω_f := ω + dd^cf ∈{ω} be another Kählerform. Then, given$\varphi _{0}, \varphi _{1}\in \mathcal {E}^{p}(X, \omega ),$wehave

$$d_{p, \omega} (\varphi_{0}, \varphi_{1})= d_{p,\omega_{f}}(\varphi_{0}-f, \varphi_{1}-f).$$

Proof

Let φ_t be the Mabuchi geodesic (with respect to ω) joining φ₀ and φ₁ and let ${\varphi _{t}^{f}}$ be the Mabuchi geodesic (with respect to ω_f) joining φ₀ − f and φ₁ − f. We claim that ${\varphi _{t}^{f}}= \varphi _{t}-f$. Indeed, φ_t − f is an ω_f-subgeodesic connecting φ₀ − f and φ₁ − f. Hence, $\varphi _{t}-f\leq {\varphi _{t}^{f}}$. On the other hand, ${\varphi _{t}^{f}} +f$ is a candidate defining φ_t, thus ${\varphi _{t}^{f}} +f \leq \varphi _{t}$, proving the claim.

Assume φ₀,φ₁ are Kähler potentials. By (2.6) we have

$$ \begin{array}{@{}rcl@{}} Vd_{p, \omega}^{p} (\varphi_{0}, \varphi_{1})&=& {\int}_{X} |\dot{\varphi_{0}}|^{p} (\omega+dd^{c} \varphi_{0})^{n}\\ &=& {\int}_{X} \left|\lim_{t\rightarrow 0^{+}} \frac{(\varphi_{t}-f) -(\varphi_{0}-f)}{t} \right|^{p} \left( \omega_{f}+dd^{c} (\varphi_{0}-f)\right)^{n}\\ &=& {\int}_{X} |\dot{{\varphi^{f}_{0}}}|^{p} (\omega_{f}+dd^{c} (\varphi_{0}-f))^{n}\\ &=& Vd^{p}_{p,\omega_{f}}(\varphi_{0}-f, \varphi_{1}-f). \end{array} $$

The identity for potentials in $\mathcal E^{p}(X,\omega )$ follows from the fact that the distance d_p,ω between potentials $\varphi _{0}, \varphi _{1}\in \mathcal {E}^{p}(X, \omega )$ is defined as the limit $\lim _{j} d_{p, \omega }(\varphi _{0,j}, \varphi _{1,j})$, where {φ_i,j} is a sequence of smooth strictly ω-psh functions decreasing to φ_i, for i = 0,1. □

Thanks to the above proposition, we can then define the Mabuchi distance with respect to any smooth (1,1)-form η in the Kähler class {ω}:

$$ d_{p,\eta}(\varphi_{0}, \varphi_{1}):= d_{p, \eta_{f}} (\varphi_{0}-f, \varphi_{1}-f),\quad \varphi_{0}, \varphi_{1}\in \mathcal{E}^{p}(X,\eta), $$

(3.1)

where η_f = η + dd^cf is a Kähler form. Proposition 3.2 reveals that the definition is independent of the choice of f.

We next extend the Pythagorean formula of [20, 21] for Kähler classes.

Lemma 3.3

If {η} is Kählerand$u,v\in \mathcal E^{p}(X,\eta )$then

$$ d_{p,\eta}^{p}(u,v) = d_{p,\eta}^{p}(u,P_{\eta}(u,v)) + d_{p,\eta}^{p}(v,P_{\eta}(u,v)). $$

Proof

By [20, Corollary 4.14] and (3.1), we have

$$d_{p,\eta}^{p}(u,v)= d_{p,\eta_{f}}^{p}(u-f,P_{\eta_{f}}(u-f,v-f)) + d_{p,\eta_{f}}^{p}(v-f,P_{\eta_{f}}(u-f,v-f)).$$

The conclusion follows observing that $P_{\eta _{f}}(u-f,v-f)= P_{\eta } (u,v)-f$. □

The following results play a crucial role in the sequel.

Lemma 3.4

Let $\varphi =P_{\theta }(f),\psi =P_{\theta }(g) \in \mathcal {H}_{\theta }$ . Set $\varphi _{\varepsilon }:= P_{\omega _{\varepsilon }}(f)$ and $\psi _{\varepsilon }= P_{\omega _{\varepsilon }}(g)$ . Then,

$$ \lim_{\varepsilon\rightarrow 0} I_{p, \omega_{\varepsilon} }(\varphi_{\varepsilon}, \psi_{\varepsilon})= I_{p,\theta}(\varphi, \psi). $$

Proof

Observe that |φ_ε − ψ_ε|→|φ − ψ| almost everywhere on X (in fact this holds off a pluripolar set) and they are uniformly bounded:

$$ |\varphi_{\varepsilon}-\psi_{\varepsilon}| \leq \sup_{X}|f-g|. $$

Indeed, take a point x ∈ X such that φ(x) > −∞ and ψ(x) > −∞. Recall that ω_ε := 𝜃 + εω ≥ 𝜃 and {ω_ε} is increasing in ε. Therefore, φ_ε decreases to a 𝜃-psh function on X as ε → 0. The latter must be φ. We thus have that φ_ε(x) → φ(x) and ψ_ε(x) → ψ(x) as ε → 0. Also, ψ_ε + supX|f − g| is a candidate defining φ_ε; hence, the claimed bound follows.

By Lemma 3.5 below and Lebesgue’s dominated convergence theorem,

$$ \lim\limits_{\varepsilon \to 0} {\int}_{X} \left| \varphi_{\varepsilon}- \psi_{\varepsilon}\right|^{p} (\omega_{\varepsilon}+dd^{c}\varphi_{\varepsilon})^{n} = {\int}_{X} \left| \varphi- \psi\right|^{p} (\theta+dd^{c} \varphi)^{n}. $$

Similarly, the other term in the definition of $I_{p, \omega _{\varepsilon }}$ also converges to the desired limit. □

Lemma 3.5

Let$\varphi = P_{\theta }(f)\in \mathcal {H}_{\theta }$.Forε > 0 weset$\varphi _{\varepsilon }= P_{\omega _{\varepsilon }}(f)$andwrite

$$ (\omega_{\varepsilon}+dd^{c} \varphi_{\varepsilon})^{n} = \rho_{\varepsilon} \omega^{n}; \ (\theta+dd^{c}\varphi)^{n} = \rho \omega^{n}. $$

Then, ε↦ρ_ε is increasing, uniformly bounded and ρ_ε → ρ pointwise on X.

Proof

Define, for ε > 0, D_ε := {x ∈ X|φ_ε(x) = f(x)}. Since {φ_ε} is increasing and φ_ε ≤ f, {D_ε} is also increasing. We set D := ∩_ε> 0D_ε. Then, D = {x ∈ X|φ(x) = f(x)}.

For ε^′ > ε > 0, it follows from Theorem 2.1 that

Here we use the fact that $0 \leq \omega _{\varepsilon } +dd^{c} f \leq \omega _{\varepsilon ^{\prime }} +dd^{c} f$ on D_ε. This proves the first statement. The second statement follows from the bound dd^cf ≤ Cω. We now prove the last statement. If x ∈ D, using (𝜃 + dd^cf) ≤ C^′ω, we can write

$$ \begin{array}{@{}rcl@{}} \rho_{\varepsilon}(x) \omega^{n} &=& (\theta + \varepsilon \omega +dd^{c} f)^{n} \leq (\theta +dd^{c} f)^{n} + O(\varepsilon) \omega^{n}\\ &=& (\rho(x)+O(\varepsilon))\omega^{n}. \end{array} $$

Hence, ρ_ε(x) → ρ(x). If x∉D then x∉D_ε for ε > 0 small enough, hence ρ_ε(x) = 0 = ρ(x). □

Lemma 3.6

Let$\varphi _{j}= P_{\theta }(f_{j}) \in \mathcal {H}_{\theta }$,forj = 0,1. Letφ_t(resp.φ_t,ε) be weak Mabuchigeodesics joiningφ₀andφ₁(resp.$\varphi _{0,\varepsilon }= P_{\omega _{\varepsilon }}(f_{0})$and$\varphi _{1,\varepsilon }= P_{\omega _{\varepsilon }}(f_{1})$).Then, we have the following pointwise convergence

Proof

Since $P_{\omega _{\varepsilon }}(f_{j})\geq P_{\theta }(f_{j})$, j = 0,1, it follows from the definition that φ_t,ε ≥ φ_t (the curve φ_t is a candidate defining φ_t,ε for any ε > 0). Set D_ε = {φ_0,ε = f₀} and D = {φ₀ = f₀}. Then, D_ε is increasing and ∩_ε> 0D_ε = D since φ₀ ≤ φ_0,ε ≤ f₀. If x ∈ D then, for all small s > 0,

$$ \dot{\varphi}_{0}(x)=\lim\limits_{t\rightarrow 0} \frac{\varphi_{t}(x)-f_{0}(x)}{t} \leq \dot{\varphi}_{0,\varepsilon}(x) \leq \frac{\varphi_{s,\varepsilon}(x)-\varphi_{0,\varepsilon}(x)}{s}, $$

where in the last inequality we use the convexity of the geodesic in t. Letting first ε → 0 and then s → 0 shows that $\dot {\varphi }_{0,\varepsilon }(x)$ converges to $\dot {\varphi }_{0}(x)$. If x∉D then x∉D_ε, for ε > 0 small enough. In this case the convergence we want to prove is trivial. □

Theorem 3.7

Let$\varphi _{0}:= P_{\theta }(f_{0}), \varphi _{1}:= P_{\theta }(f_{1})\in \mathcal {H}_{\theta }$andlet$\varphi _{i,\varepsilon }=P_{\omega _{\varepsilon }}(f_{i})$,i = 0,1. Letd_p,εbe the Mabuchi distance withrespect toω_εdefined in (3.1).Then,

$$ \lim\limits_{\varepsilon \rightarrow 0} d^{p}_{p,\varepsilon}(\varphi_{0, \varepsilon}, \varphi_{1, \varepsilon}) = {\int}_{X} |\dot{\varphi}_{0}|^{p} (\theta+dd^{c} \varphi_{0})^{n}={\int}_{X} |\dot{\varphi}_{1}|^{p} (\theta+dd^{c} \varphi_{1})^{n}, $$

where φ_t is the weak Mabuchi geodesic connecting φ₀ and φ₁.

Compared with [32] our approach is slightly different. We also emphasize that by [31, Example 4.5], there are functions in $\mathcal E^{p}(X,\theta )$ which are not in $\mathcal E^{p}(X,\omega )$.

Proof

Let φ_t,ε denote the ω_ε-geodesic joining φ_0,ε and φ_1,ε. Set D_ε = {φ_0,ε = f₀} and D = {φ₀ = f₀}. Combining (2.6) and Theorem 2.1, we obtain

$$V_{\varepsilon} d_{p, \varepsilon}^{p} (\varphi_{0, \varepsilon}, \varphi_{1,\varepsilon}) = {\int}_{X} |\dot{\varphi}_{0, \varepsilon}|^{p} (\omega_{\varepsilon}+dd^{c} \varphi_{0, \varepsilon})^{n}= {\int}_{D_{\varepsilon}} |\dot{\varphi}_{0, \varepsilon}|^{p} (\omega_{\varepsilon}+dd^{c} f_{0})^{n}.$$

Since |φ_0,ε − φ_1,ε|≤ supX|f₀ − f₁| and f₀ − f₁ is bounded, (2.5) ensures that $\dot {\varphi }_{0,\varepsilon }$ is uniformly bounded. It follows from Lemma 3.5 and Lemma 3.6 that the functions and are uniformly bounded and converges pointwise to . We also observe that V_ε decreases to Vol(𝜃) = 1. Lebesgue’s dominated convergence theorem then yields

$$ \lim\limits_{\varepsilon\to 0} d_{p, \varepsilon}^{p} (\varphi_{0, \varepsilon}, \varphi_{1,\varepsilon}) = {\int}_{D} |\dot{\varphi}_{0}|^{p} (\theta+dd^{c} f_{0})^{n}={\int}_{X} |\dot{\varphi}_{0}|^{p} (\theta+dd^{c} \varphi_{0})^{n}, $$

where in the last equality we use Theorem 2.1. This shows the first equality in the statement. The second one is obtained by reversing the role of φ₀ and φ₁. □

Definition 3.8

Assume that $\varphi _{0}:= P_{\theta }(f_{0}),\varphi _{1}:= P_{\theta }(f_{1}) \in \mathcal {H}_{\theta }$. Let d_p,ε be the Mabuchi distance with respect to ω_ε := 𝜃 + εω defined in (3.1). We define

$$ d_{p}(\varphi_{0},\varphi_{1}) := \lim\limits_{\varepsilon\to 0} d_{p,\varepsilon}(\varphi_{0,\varepsilon},\varphi_{1,\varepsilon}), $$

where $\varphi _{0,\varepsilon } := P_{\omega _{\varepsilon } } (f_{0})$ and $\varphi _{1,\varepsilon } := P_{\omega _{\varepsilon } } (f_{1})$.

The limit exists and is independent of the choice of ω as shown in Theorem 3.7.

Lemma 3.9

d _p is a distance on $\mathcal {H}_{\theta }$ .

Proof

The triangle inequality immediately follows from the fact that d_p,ε is a distance. From [20, Theorem 5.5] we know that

$$d_{p, \varepsilon}^{p} (\varphi_{0, \varepsilon}, \varphi_{1, \varepsilon})\geq \frac{1}{C} I_{p,\omega_{\varepsilon}} (\varphi_{0, \varepsilon}, \varphi_{1, \varepsilon}), \qquad C>0.$$

Also, by Lemma 3.4 we have $\lim _{\varepsilon \rightarrow 0} I_{p,\omega _{\varepsilon }} (\varphi _{0, \varepsilon }, \varphi _{1, \varepsilon }) = I_{p, \theta } (\varphi _{0}, \varphi _{1})$. It follows from the domination principe (see [10, 24]) that

$$ I_{p, \theta} (\varphi_{0}, \varphi_{1})= 0 \Leftrightarrow \varphi_{0}=\varphi_{1}. $$

Hence, d_p is non-degenerate. □

3.2 Extension of d_p to $\mathcal E^{p}(X,\theta )$

The following comparison between I_p and d_p was established in [20, Theorem 3] in the Kähler case.

Proposition 3.10

Given$\varphi _{0},\varphi _{1} \in \mathcal H_{\theta }$thereexists a constantC > 0 (depending only on n) such that

$$ \frac{1}{C}I_{p}(\varphi_{0},\varphi_{1}) \leq {d_{p}^{p}}(\varphi_{0},\varphi_{1}) \leq C I_{p}(\varphi_{0},\varphi_{1}). $$

(3.2)

Proof

By Darvas [20, Theorem 3] we know that

$$ \frac{1}{C}I_{p, \omega_{\varepsilon}}(\varphi_{0,\varepsilon}, \varphi_{1,\varepsilon}) \leq d_{p, \varepsilon}^{p}(\varphi_{0,\varepsilon}, \varphi_{1,\varepsilon}) \leq C I_{p,\omega_{\varepsilon}}(\varphi_{0,\varepsilon},\varphi_{1,\varepsilon}).$$

Letting ε to zero and using Lemma 3.4 and Definition 3.8, we get (3.2). □

Now, let $\varphi _{0}, \varphi _{1}\in \mathcal {E}^{p}(X, \theta )$. Let {f_i,j} be a sequence of smooth functions decreasing to φ_i, i = 0,1. We then clearly have that $\varphi _{i,j}:=P_{\theta }(f_{i,j})\in \mathcal {H}_{\theta }$ and P_𝜃(f_i,j) ↘ φ_i.

Lemma 3.11

The sequenced_p(φ_0,j,φ_1,j) converges and the limit is independent of the choice of the approximantsf_i,j.

Proof

Set a_j := d_p(φ_0,j,φ_1,j). By the triangle inequality and Proposition 3.10, we have

$$ \begin{array}{@{}rcl@{}} a_{j} & \leq & d_{p} (\varphi_{0,j}, \varphi_{0,k})+ d_{p} (\varphi_{0,k}, \varphi_{1,k})+ d_{p} (\varphi_{1,k}, \varphi_{1,j}) \\ &\leq & a_{k} +C\left( I_{p}^{1/p} (\varphi_{0,j}, \varphi_{0,k}) + I_{p}^{1/p} (\varphi_{1,j}, \varphi_{1,k}) \right), \end{array} $$

where C > 0 depends only on n,p. Hence,

$$|a_{j}-a_{k}| \leq C\left( I_{p}^{1/p} (\varphi_{0,j}, \varphi_{0,k}) + I_{p}^{1/p} (\varphi_{1,j}, \varphi_{1,k}) \right).$$

By [34, Theorem 1.6 and Proposition 1.9], it then follows that |a_j − a_k|→ 0 as j,k → +∞. This proves that the sequence d_p(φ_0,j,φ_1,j) is Cauchy; hence, it converges.

Let $\tilde {\varphi }_{i,j}= P_{\theta }(\tilde {f}_{i,j})$ be another sequence in $\mathcal {H}_{\theta }$ decreasing to φ_i, i = 0,1. Then, applying the triangle inequality several times, we get

$$ d_{p} (\varphi_{0,j}, \varphi_{1,j}) \leq d_{p} (\varphi_{0,j}, \tilde{\varphi}_{0,j}) + d_{p} (\tilde{\varphi}_{0,j}, \tilde{\varphi}_{1,j}) + d_{p} (\tilde{\varphi}_{1,j}, \varphi_{1,j}), $$

and thus

$$| d_{p} (\varphi_{0,j}, \varphi_{1,j}) -d_{p} (\tilde{\varphi}_{0,j}, \tilde{\varphi}_{1,j})| \leq C\left( I_{p}^{1/p} (\varphi_{0,j}, \tilde{\varphi}_{0,j}) + I_{p}^{1/p} (\varphi_{1,j}, \tilde{\varphi}_{1,j}) \right).$$

It then follows again from [34, Theorem 1.6 and Proposition 1.9] that the limit does not depend on the choice of the approximants. □

Given $\varphi _{0}, \varphi _{1} \in \mathcal {E}^{p}(X, \theta )$, we then define

$$ d_{p}( \varphi_{0},\varphi_{1}) := \lim_{j\rightarrow +\infty} d_{p} (P_{\theta}(f_{0,j}), P_{\theta}(f_{1,j})). $$

Proposition 3.12

d_pis a distance on$\mathcal {E}^{p}(X, \theta )$andthe inequalities comparingd_pandI_pon$\mathcal {H}_{\theta }$ (3.2) hold on$\mathcal {E}^{p}(X, \theta )$.Moreover, if$u_{j}\in \mathcal E^{p}(X,\theta )$decreasesto$u\in \mathcal E^{p}(X,\theta )$thend_p(u_j,u) → 0.

Proof

By the definition of d_p on $\mathcal {E}^{p}(X,\theta )$ we infer that the comparison between d_p and I_p in Proposition 3.10 holds on $\mathcal E^{p}(X,\theta )$. From this and the domination principle [24], we deduce that d_p is non-degenerate. The last statement follows from (3.2) and [34, Proposition 1.9]. □

The next result was proved in [6, Lemma 3.4] for the Kähler case.

Lemma 3.13

Let u _t be the Mabuchi geodesic joining $u_{0}\in \mathcal {H}_{\theta }$ and let $u_{1}\in \mathcal E^{p}(X,\theta )$ . Then,

$$ {d_{p}^{p}}(u_{0},u_{1}) = {\int}_{X} |\dot{u}_{0}|^{p} (\theta+dd^{c} u_{0})^{n}. $$

Proof

We first assume that u₀ ≥ u₁ + 1. We approximate u₁ from above by ${u_{1}^{j}}\in \mathcal {H}_{\theta }$ such that ${u_{1}^{j}}\leq u_{0}$, for all j. Let ${u_{t}^{j}}$ be the Mabuchi geodesic joining u₀ to ${u_{1}^{j}}$. Note that ${u_{t}^{j}}\geq u_{t}$ and that $\dot {u}_{t}^{j}\leq 0$. By Theorem 3.7,

$$ {d_{p}^{p}}(u_{0},{u_{1}^{j}}) = {\int}_{X} (-\dot{u}_{0}^{j})^{p} \theta_{u_{0}}^{n}. $$

Also, $\dot {u}_{0}^{j}$ decreases to $\dot {u}_{0}$; hence, the monotone convergence theorem and Proposition 3.12 give

$$ {d_{p}^{p}}(u_{0},u_{1}) = {\int}_{X} (- \dot{u}_{0})^{p} \theta_{u_{0}}^{n} <+\infty. $$

In particular $|\dot {u}_{0}|^{p} \in L^{1}(X, \theta _{u_{0}}^{n})$.

For the general case we can find a constant C > 0 such that u₁ ≤ u₀ + C since u₀ has minimal singularities. Let w_t be the Mabuchi geodesic joining u₀ and u₁ − C − 1. Note that $w_{t}\leq {u_{t}^{j}}$ since $w_{1}=u_{1}-C-1< u_{1}\leq {u_{1}^{j}}$ and $w_{0}=u_{0}={u_{0}^{j}}$ and $\dot {w}_{t}\leq 0$. It then follows that

$$ \dot{w}_{0} \leq \dot{u}_{0}^{j} \leq {u_{1}^{j}}-u_{0} \leq ({u_{1}^{j}}-V_{\theta}) + (V_{\theta}-u_{0})\leq \sup_{X} {u_{1}^{j}} + \sup_{X} (V_{\theta}-u_{0})\leq C_{1}, $$

for a uniform constant C₁ > 0. In the second inequality above, we use the fact that the Mabuchi geodesic ${u_{t}^{j}}$ connecting u₀ to ${u_{1}^{j}}$ is convex in t, while in the last inequality we use the fact that u₀ has minimal singularities.

The previous inequalities then yield $|\dot {u}_{0}^{j}|^{p} \leq C_{2} + 2^{p-1}|\dot {w}_{0}|^{p}$, where C₂ is a uniform constant. On the other hand by Theorem 3.7, we have

$$ {d_{p}^{p}}(u_{0},{u_{1}^{j}}) = {\int}_{X} |\dot{u}_{0}^{j}|^{p} \theta_{u_{0}}^{n}. $$

We claim that $|\dot {u}_{0}^{j}|^{p}$ converges a.e. to $|\dot {u}_{0}|^{p}$. Indeed, the convergence is pointwise at points x such that u₁(x) > −∞, but the set {u₁ = −∞} has Lebesgue measure zero. Also, the above estimate ensures that $|\dot {u}_{0}^{j}|^{p}$ are uniformly bounded by $2^{p-1}(-\dot {w}_{0})^{p}+C_{2}$ which is integrable with respect to the measure $\theta _{u_{0}}^{n}$ since ${\int }_{X} |\dot {w}_{0}|^{p} \theta _{u_{0}}^{n}= {d_{p}^{p}}(u_{0},u_{1}-C-1)<+\infty$. Proposition 3.12 and Lebesgue’s dominated convergence theorem then give the result. □

Proposition 3.14

If $u,v\in \mathcal {E}^{p}(X, \theta )$ then

• (i) ${d_{p}^{p}} (u,v) ={d_{p}^{p}}(u,P_{\theta }(u,v)) +{d_{p}^{p}} (v,P_{\theta }(u,v))$ and

• (ii) d_p(u,max(u,v)) ≥ d_p(v,P_𝜃(u,v)).

We recall that from [24, Theorem 2.13] $P_{\theta }(u,v)\in \mathcal E^{p}(X, \theta )$. The identity in the first statement is known as the Pythagorean formula and it was established in the Kähler case by Darvas [20]. The second statement was proved for p = 1 in [26] using the differentiability of the Monge-Ampère energy. As will be shown in Proposition 3.18, our definition of d₁ and the one in [26] do coincide.

Proof

To prove the Pythagorean formula, we first assume that $u=P_{\theta }(f),v=P_{\theta }(g)\in \mathcal {H}_{\theta }$. Set $u_{\varepsilon }:=P_{\omega _{\varepsilon }}(f)$, $v_{\varepsilon }:=P_{\omega _{\varepsilon }}(g)$. It follows from Lemma 3.3 that

$$ \begin{array}{@{}rcl@{}} d_{p,\varepsilon}^{p} (u_{\varepsilon}, v_{\varepsilon})& =& d_{p,\varepsilon}^{p} (u_{\varepsilon}, P_{\omega_{\varepsilon}}(u_{\varepsilon},v_{\varepsilon})) + d_{p,\varepsilon}^{p} (v_{\varepsilon}, P_{\omega_{\varepsilon}}(u_{\varepsilon},v_{\varepsilon}))\\ &=& d_{p,\varepsilon}^{p} (u_{\varepsilon}, P_{\omega_{\varepsilon}}(\min(f,g)) + d_{p,\varepsilon}^{p} (v_{\varepsilon}, P_{\omega_{\varepsilon}}(\min(f,g)), \end{array} $$

where in the last identity we use that $P_{\omega _{\varepsilon }}(u_{\varepsilon },v_{\varepsilon })=P_{\omega _{\varepsilon }}(\min (f,g))$. It follows from Lemma 3.1 that dd^c min(f,g) ≤ Cω. Applying Theorem 3.7 we obtain (i) for this case. To treat the general case, let u_j = P_𝜃(f_j),v_j = P_𝜃(g_j) be sequences in $\mathcal {H}_{\theta }$ decreasing to u,v. By Lemma 3.1, $P_{\theta }(u_{j},v_{j})=P_{\theta }(\min (f_{j},g_{j}))\in \mathcal {H}_{\theta }$ and it decreases to P_𝜃(u,v). Then, (i) follows from the first step and Proposition 3.12 since

$$|d_{p}(u_{j}, v_{j}) - d_{p}(u,v)| \leq d_{p}(u_{j}, u) +d_{p}(v, v_{j}). $$

To prove the second statement, in view of Proposition 3.12, we can assume that $u=P_{\theta }(f), v=P_{\theta }(g) \in \mathcal {H}_{\theta }$. By Lemma 3.13 we have

$$ {d_{p}^{p}}(u,\max(u,v)) = {\int}_{X} |\dot{u}_{0}|^{p} {\theta_{u}^{n}}, $$

where t↦u_t is the Mabuchi geodesic joining u₀ = u to u₁ = max(u,v).

Let φ_t be the Mabuchi geodesic joining φ₀ = P_𝜃(u,v) to φ₁ = v. We note that $0\leq \dot {\varphi }_{0}\leq v-P(u,v)$. Indeed, $\dot {\varphi }_{0}\geq 0$ since φ₀ ≤ φ₁ while the second inequality follows from the convexity in t of the geodesic. Using this observation and the fact that φ_t ≤ u_t, we obtain

Since P_𝜃(u,v) = P_𝜃(min(f,g)) with dd^c min(f,g) ≤ Cω, Theorem 2.1, Theorem 3.7, and [34, Lemma 4.1] then yield

$$ \begin{array}{@{}rcl@{}} {d_{p}^{p}}(P_{\theta}(u,v),v)& =& {\int}_{X} \dot{\varphi}_{0}^{p} (\theta+ dd^{c} \varphi_{0})^{n} \leq {\int}_{\{P(u,v)=u\}} \dot{\varphi}_{0}^{p} (\theta+ dd^{c} u)^{n}\\ & \leq& {\int}_{\{P(u,v)=u\}} \dot{u}_{0}^{p} (\theta+ dd^{c} u)^{n} \leq {d_{p}^{p}}(u,\max(u,v)). \end{array} $$

□

Remark 3.15

By Proposition 3.14 we have a “Pythagorean inequality” for max:

$$ {d_{p}^{p}}(u,\max(u,v)) + {d_{p}^{p}}(v,\max(u,v)) \geq {d_{p}^{p}}(u,v), \ \forall u,v\in \mathcal E^{p}(X,\theta). $$

3.3 Completeness of $(\mathcal {E}^{p}(X,\theta ),d_{p})$

In the sequel we fix a smooth volume form dV on X such that ${\int }_{X} dV=1$.

Lemma 3.16

Let$u\in \mathcal {E}^{p}(X, \theta )$andletϕbe a𝜃-psh function withminimal singularities, supXϕ = 0 satisfying$\theta _{\phi }^{n}=dV$. Then, thereexist uniform constantsC₁ = C₁(n,𝜃) andC₂ = C₂(n) > 0 such that

$$|\sup\limits_{X} u |\leq C_{1}+C_{2} d_{p}(u, \phi).$$

Proof

Using the Hölder inequality and [35, Proposition 2.7], we obtain

$$ \begin{array}{@{}rcl@{}} |\sup\limits_{X} u| &\leq & {\int}_{X} |u-\sup\limits_{X} u| dV +{\int}_{X} |u| dV \leq A+ \left( {\int}_{X} |u|^{p} dV \right)^{1/p}\\ &\leq & A+ \left( \|u-\phi\|_{L^{p}(dV)}+\|\phi\|_{L^{p}(dV)} \right). \end{array} $$

By Proposition 3.12,

$$ {\int}_{X} |u-\phi|^{p} dV = {\int}_{X} |u-\phi|^{p} \theta_{\phi}^{n} \leq I_{p} (u, \phi)\leq C(n) {d_{p}^{p}} (u, \phi). $$

Combining the above inequalities we get the conclusion. □

Theorem 3.17

The space $(\mathcal {E}^{p}(X, \theta ), d_{p})$ is a complete geodesic metric space which is the completion of $(\mathcal {H}_{\theta },d_{p})$ .

Proof

Let $(\varphi _{j})\in \mathcal {E}^{p}(X, \theta )^{\mathbb {N}}$ be a Cauchy sequence for d_p. Extracting and relabelling we can assume that there exists a subsequence (u_j) ⊆ (φ_j) such that

$$d_{p}(u_{j}, u_{j+1} )\leq 2^{-j}.$$

Define v_j,k := P_𝜃(u_j,…,u_j+k) and observe that it is decreasing in k. Also, by Proposition 3.14 (i) and the triangle inequality,

$$d_{p} (u_{j}, v_{j,k})= d_{p} (u_{j}, P_{\theta}(u_{j}, v_{j+1,k}))\leq d_{p} (u_{j},v_{j+1,k} ) \leq 2^{-j} + d_{p} (u_{j+1},v_{j+1,k} ).$$

Hence,

$$d_{p} (u_{j}, v_{j,k}) \leq \sum\limits_{\ell=j }^{k-1} 2^{-\ell}\leq 2^{-j+1}.$$

In particular I_p(u_j,v_j,k) is uniformly bounded from above. We then infer that v_j,k decreases to v_j ∈PSH(X,𝜃) as k → +∞ and a combination of Proposition 3.12 and [34, Proposition 1.9] gives

$$ d_{p}(u_{j},v_{j})\leq 2^{1-j}, \ \forall j. $$

(3.3)

Let ϕ be the unique 𝜃-psh function with minimal singularities such that supXϕ = 0 and $\theta _{\phi }^{n}=dV$. By Lemma 3.16,

$$ \begin{array}{@{}rcl@{}} |\sup\limits_{X} v_{j}| &\leq & C_{1}+C_{2} d_{p}(v_{j}, \phi)\leq C_{1}+C_{2} \left( d_{p}(v_{j}, u_{1})+ d_{p}(u_{1}, \phi)\right)\\ &\leq & C_{1}+C_{2} \left( d_{p}(v_{j}, u_{j})+ d_{p}(u_{j}, u_{1})+ d_{p}(u_{1}, \phi)\right)\\ &\leq &C_{1}+ C_{2} (4+d_{p}(u_{1}, \phi)). \end{array} $$

It thus follows that v_j increases a.e. to a 𝜃-psh function v. By the triangle inequality we have

$$ d_{p}(\varphi_{j}, v)\leq d_{p}(\varphi_{j}, u_{j})+ d_{p}(u_{j}, v_{j})+ d_{p}(v_{j}, v). $$

Since (φ_j) is Cauchy, d_p(φ_j,u_j) → 0. By [34, Proposition 1.9] and Proposition 3.12, we have d_p(v_j,v) → 0. These facts together with (3.3) yield d_p(φ_j,v) → 0; hence, $(\mathcal {E}^{p}(X, \theta ), d_{p})$ is a complete metric space.

Also, any $u\in \mathcal E^{p}(X,\theta )$ can be approximated from above by functions $u_{j}\in \mathcal {H}_{\theta }$ such that d_p(u_j,u) → 0 (Proposition 3.12). It thus follows that $(\mathcal E^{p}(X, \theta ),d_{p})$ is the metric completion of $\mathcal {H}_{\theta }$.

Let now u_t be the Mabuchi geodesic joining $u_{0},u_{1}\in \mathcal E^{p}(X,\theta )$. We are going to prove that, for all t ∈ [0,1],

$$ d_{p}(u_{t},u_{s}) =|t-s|d_{p}(u_{0},u_{1}). $$

We claim that for all t ∈ [0,1],

$$ d_{p}(u_{0},u_{t})= t d_{p}(u_{0},u_{1}) \ \text{and}\ d_{p}(u_{1},u_{t})= (1-t) d_{p}(u_{0},u_{1}). $$

(3.4)

We first assume that $u_{0},u_{1}\in \mathcal {H}_{\theta }$. The Mabuchi geodesic joining u₀ to u_t is given by w_ℓ = u_tℓ, ℓ ∈ [0,1]. Lemma 3.13 thus gives

$$ {d_{p}^{p}}(u_{0},u_{t}) = {\int}_{X} |\dot{w}_{0}|^{p} \theta_{u_{0}}^{n} = t^{p} {\int}_{X} |\dot{u}_{0}|^{p}\theta_{u_{0}}^{n}=t^{p} {d_{p}^{p}}(u_{0},u_{1}), $$

proving the first equality in (3.4). The second one is proved similarly.

We next prove the claim for $u_{0},u_{1}\in \mathcal {E}^{p}(X,\theta )$. Let $({u_{i}^{j}}), i=0,1, j\in \mathbb {N}$, be decreasing sequences of functions in $\mathcal {H}_{\theta }$ such that ${u_{i}^{j}} \downarrow u_{i}$, i = 0,1. Let ${u_{t}^{j}}$ be the Mabuchi geodesic joining ${u_{0}^{j}}$ and ${u_{1}^{j}}$. Then, ${u_{t}^{j}}$ decreases to u_t. By the triangle inequality we have $|d_{p}({u_{0}^{j}}, {u_{t}^{j}})-d_{p}(u_{0}, u_{t})|\leq d_{p}({u_{0}^{j}}, u_{0})+d_{p}(u_{t}, {u_{t}^{j}})$. The claim thus follows from Proposition 3.12 and the previous step.

Now, if 0 < t < s < 1 then applying twice (3.4), we get

$$ d_{p}(u_{t},u_{s})= \frac{s-t}{s} d_{p}(u_{0},u_{s}) = (s-t) d_{p}(u_{0},u_{1}). $$

□

We end this section by proving that the distance d₁ defined by approximation (see Definition 3.8) coincides with the one defined in [26] using the Monge-Ampère energy.

Proposition 3.18

Assume $u_{0},u_{1}\in \mathcal E^{1}(X,\theta )$ . Then,

$$ d_{1}(u_{0},u_{1})=E(u_{0})+E(u_{1})-2E(P(u_{0},u_{1})). $$

Here the Monge-Ampère energy E is defined as

$$ E(u) := \frac{1}{n+1} \sum\limits_{j=0}^{n} {\int}_{X} (u-V_{\theta}) {\theta_{u}^{j}} \wedge \theta_{V_{\theta}}^{n-j}. $$

Proof

We first assume that $u_{0},u_{1} \in \mathcal {H}_{\theta }$ and u₀ ≤ u₁. Let [0,1] ∋ t↦u_t be the Mabuchi geodesic joining u₀ and u₁. By [24, Theorem 3.12], t↦E(u_t) is affine, hence for all t ∈ [0,1],

$$ \frac{ E(u_{t})-E(u_{0})}{t} = E(u_{1})-E(u_{0}) = \frac{E(u_{1})-E(u_{t})}{1-t}. $$

Since E is concave along affine curves (see [5, 12], [26, Theorem 2.1]), we thus have

$$ {\int}_{X} \frac{u_{t}-u_{0}}{t} \theta_{u_{0}}^{n} \geq E(u_{1})-E(u_{0}) \geq {\int}_{X} \frac{u_{1}-u_{t}}{1-t} \theta_{u_{1}}^{n}. $$

Letting t → 0 in the first inequality and t → 1 in the second one, we obtain

$$ {\int}_{X} \dot{u}_{0} \theta_{u_{0}}^{n} \geq E(u_{1})-E(u_{0}) \geq {\int}_{X} \dot{u}_{1} \theta_{u_{1}}^{n}. $$

By Theorem 3.7 we then have

$$ d_{1}(u_{0},u_{1})= {\int}_{X} \dot{u}_{0} \theta_{u_{0}}^{n}= {\int}_{X} \dot{u}_{1} \theta_{u_{1}}^{n}= E(u_{1})-E(u_{0}).$$

We next assume that $u_{0},u_{1}\in \mathcal {H}_{\theta }$ but we remove the assumption that u₀ ≤ u₁. By Lemma 3.1, $P(u_{0},u_{1}) \in \mathcal {H}_{\theta }$. By the Pythagorean formula (Proposition 3.14) and the first step, we have

$$ \begin{array}{@{}rcl@{}} d_{1}(u_{0},u_{1}) &=&d_{1}(u_{0},P(u_{0},u_{1})) + d_{1}(u_{1},P(u_{0},u_{1})) \\ & = &E(u_{0})-E(P(u_{0},u_{1})) + E(u_{1})-E(P(u_{0},u_{1})). \end{array} $$

We now treat the general case. Let $({u_{i}^{j}}), i=0,1, j\in \mathbb {N}$ be decreasing sequences of functions in $\mathcal {H}_{\theta }$ such that ${u_{i}^{j}} \downarrow u_{i}$, i = 0,1. Then, $P({u_{0}^{j}},{u_{1}^{j}}) \downarrow P(u_{0},u_{1})$. By [26, Proposition 2.4], $E({u_{i}^{j}}) \to E(u_{i})$, for i = 0,1 and $E(P({u_{0}^{j}},{u_{1}^{j}})) \to E(P(u_{0},u_{1}))$ as j → +∞. The result thus follows from Proposition 3.12, the triangle inequality, and the previous step. □