Abstract
We prove existence of instantaneously complete Yamabe flows on hyperbolic space of arbitrary dimension \(m\ge 3\). The initial metric is assumed to be conformally hyperbolic with conformal factor and scalar curvature bounded from above. We do not require initial completeness or bounds on the Ricci curvature. If the initial data are rotationally symmetric, the solution is proven to be unique in the class of instantaneously complete, rotationally symmetric Yamabe flows.
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The Yamabe flow was introduced by Richard Hamilton [11]. It describes a family of Riemannian metrics g(t) subject to the equation \(\partial _{t}g=-\mathrm {R}g\) and tends to evolve a given initial metric towards a metric of vanishing scalar curvature. Hamilton showed that global solutions always exist on compact manifolds without boundary. Their asymptotic behaviour was subsequently analysed by Chow [5], Ye [19], Schwetlick and Struwe [15] and Brendle [3, 4]. Less is known about the Yamabe flow on noncompact manifolds. Daskalopoulos and Sesum [6] analysed the profiles of self-similar solutions (Yamabe solitons). Ma and An [13] proved short-time existence of Yamabe flows on noncompact, locally conformally flat manifolds \(M\) under the assumption that the initial manifold \((M,g_0)\) is complete with Ricci tensor bounded from below. More recently, Bahuaud and Vertman [1, 2] constructed Yamabe flows starting from spaces with incomplete edge singularities such that the singular structure is preserved along the flow.
In dimension \(m=2\) the Yamabe flow coincides with the Ricci flow. Peter Topping and Gregor Giesen [7, 16, 17] introduced the notion of instantaneous completeness and obtained existence and uniqueness of instantaneously complete Ricci/Yamabe flows on arbitrary surfaces. The analysis of the flow on the hyperbolic disc plays an important role in their work. It relies on results which exploit the fact that the Ricci tensor is bounded by the scalar curvature in dimension 2.
The goal of this paper is to find techniques which allow a generalisation of Giesen and Topping’s results to the Yamabe flow on hyperbolic space \((\mathbb {H},g_{\mathbb {H}})\) of dimension \(m\ge 3\). \((\mathbb {H},g_{\mathbb {H}})\) is a complete, noncompact, simply connected manifold of constant sectional curvature \(-1\) and it is conformally equivalent to the Euclidean unit ball \((B_1,g_{\mathbb {E}})\).
A family \((g(t))_{t\in [0,T]}\) of Riemannian metrics on a manifold \(M\) with scalar curvature \(\mathrm {R}=\mathrm {R}_{g(t)}\) is called a Yamabe flow, if \(\tfrac{\partial }{\partial t}g=-\mathrm {R}\, g\). The family \((g(t))_{t\in [0,T]}\) is called instantaneously complete, if the Riemannian manifold \((M,g(t))\) is geodesically complete for every \(0<t\le T\).
Since the Yamabe flow preserves the conformal class of the metric, any conformally hyperbolic Yamabe flow \((g(t))_{t\in [0,T]}\) on \(\mathbb {H}\) is given by \(g(t)=u(\cdot ,t)\,g_{\mathbb {H}}\), where the conformal factor \(u:\mathbb {H}\times [0,T]\rightarrow \mathbb {R}\) is a positive function evolving by the equation
where \(m=\dim \mathbb {H}\), where \(\Delta _{g_{\mathbb {H}}}\) denotes the Laplace-Beltrami operator with respect to the hyperbolic background metric \(g_\mathbb {H}\) and where \(\left|\nabla u\right|_{g_{\mathbb {H}}}^2=g_{\mathbb {H}}(\nabla u,\nabla u)\). Introducing the exponent \(\eta :=\frac{m-2}{4}\) to define \(U=u^{\eta }\), Eq. (1) is equivalent to
which follows by virtue of \((\eta -1)=\frac{(m-6)}{4}\) and \(\tfrac{1}{\eta }\Delta _{g_{\mathbb {H}}} u^\eta =(\eta -1)u^{\eta -2}\left|\nabla u\right|_{g_{\mathbb {H}}}^2+u^{\eta -1}\Delta _{g_{\mathbb {H}}} u\). While Eq. (2) has a simpler structure, pointwise bounds on u follow easier from Eq. (1). We prove the following statements.
(Existence) Let \(g_0=u_0g_{\mathbb {H}}\) be any (possibly incomplete) conformal metric on \((\mathbb {H},g_{\mathbb {H}})\) with bounded conformal factor \(0<u_0\in \mathrm {C}^{4,\alpha }(\mathbb {H})\) and scalar curvature \(\mathrm {R}_{g_0}\le K_0\) bounded from above. Then, for any \(T>0\) there exists an instantaneously complete family of metrics \((g(t))_{t\in [0,T]}\) satisfying the Yamabe flow equation
Moreover, \(g(t)\ge m(m-1)\,t\,g_{\mathbb {H}}\) for any \(t\in {]}0,T{]}\). As \(t\searrow 0\), the metric g(t) converges locally in class \(\mathrm {C}^2\) to \(g_0\).
FormalPara RemarkOn noncompact, locally conformally flat manifolds \(M\), Ma and An [13] require bounded scalar curvature, a lower bound on the Ricci tensor and completeness of \((M,g_0)\) for short-time existence and additionally non-positive scalar curvature for global existence.
FormalPara Theorem 2(Uniqueness) Let \((g(t))_{t\in [0,T]}\) and \((\tilde{g}(t))_{t\in [0,T]}\) be two conformally hyperbolic Yamabe flows on \((\mathbb {H},g_{\mathbb {H}})\) satisfying
- (i)
\(\exists b\in \mathbb {R}: \quad \tilde{g}(0)\le b\,g_{\mathbb {E}}\),
- (ii)
\(\forall t\in [0,T]:\quad g(t)\ge m(m-1)t\,g_{\mathbb {H}}\).
Then, if \(\tilde{g}(0)\le g(0)\), we have \(\tilde{g}(t)\le g(t)\) for all \(t\in [0,T]\).
In particular, if g(t) and \(\tilde{g}(t)\) both satisfy (ii) and if \(g(0)=\tilde{g}(0)\le b\,g_{\mathbb {E}}\), then \(g\equiv \tilde{g}\).
FormalPara RemarkIn the Poincaré ball model for \(\mathbb {H}\), the conformally hyperbolic initial metric \(\tilde{g}(0)\) can be compared to the Euclidean metric, whose pullback we also denote as \(g_{\mathbb {E}}\). Assumption (i) means, that the initial manifold \((\mathbb {H},\tilde{g}(0))\) is incomplete and has finite diameter.
Assumption (ii) implies instantaneous completeness of g(t). We conjecture that instantaneously complete, conformally hyperbolic Yamabe flows always satisfy (ii). For rotationally symmetric flows, this is proved in Proposition 2.2.
The instantaneously complete flow Topping [16] constructs on 2-dimensional manifolds has a certain maximality property which we also observe in higher dimensions: Theorem 2 implies, that if \(g_0\le b\,g_{\mathbb {E}}\), then the Yamabe flow constructed in Theorem 1 is maximally stretched in the sense that any other Yamabe flow with the same or lower initial data stays below it.
Moreover, Theorem 2 implies, that if \(g_0\le b\,g_{\mathbb {E}}\), then any two solutions \((g(t))_{t\in [0,T]}\) and \((\tilde{g}(t))_{t\in [0,\tilde{T}]}\) constructed in Theorem 1 agree on \([0,T]\cap [0,\tilde{T}]\). Since \(T>0\) is arbirtary in Theorem 1, we then obtain global existence, i. e. an instantaneously complete Yamabe flow \((g(t))_{t\in {[}0,\infty {[}}\) on \(\mathbb {H}\) with \(g(0)=g_0\).
FormalPara Theorem 3Let \(g_0=u_0g_{\mathbb {R}^m}\) be a conformally Euclidean metric on \((\mathbb {R}^m,g_{\mathbb {R}^m})\) with \(m\ge 3\). If \(u_0(x)\le b\left|x\right|^{-4}\) for some finite constant b and all \(x\in \mathbb {R}^m\), then any Yamabe flow \((g(t))_{t\in [0,T]}\) on \(\mathbb {R}^m\) with \(g(0)=g_0\) is geodesically incomplete for all \(t\in [0,T]\).
FormalPara RemarkTheorem 3 shows that the results about instantaneously complete Yamabe flow on hyperbolic space do not equally hold on arbitrary manifolds of dimension \(m\ge 3\). It contrasts with the 2-dimensional case, where instantaneously complete Yamabe flows always exist [7].
For example, there does not exist an instantaneously complete Yamabe flow starting from the punctured unit sphere \((\dot{\mathbb {S}}^{m},g_{\mathbb {S}^{m}})\) in dimension \(m\ge 3\). Indeed, if \(\pi :\dot{\mathbb {S}}^{m}\rightarrow \mathbb {R}^m\) is stereographic projection, then \(\pi _*g_{\mathbb {S}^{m}}=4{(1+\left|x\right|^2)}^{-2}g_{\mathbb {R}^m}\) and Theorem 3 applies.
1 Existence
In this section, we prove Theorem 1. As a first step, short-time existence of a solution u to Eq. (1) for given \(u(\cdot ,0)=u_0>0\) on convex, bounded domains \(\Omega \subset \mathbb {H}\) with suitable boundary data is proven by applying the inverse function theorem on Banach spaces. Richard Hamilton [9, § IV.11] uses the same technique to prove existence of solutions to the heat equation for manifolds. Local Hölder estimates then lead to a uniform existence time for all domains.
In a second step we derive uniform gradient estimates, which do not depend on the domain. By considering an exhaustion of \(\mathbb {H}\) with convex, bounded domains, we obtain a locally uniformly bounded sequence of solutions which allows a subsequence converging to a solution of (1) on all of \(\mathbb {H}\).
1.1 Existence on bounded domains
We denote the non-linear terms in Eq. (1) by
Given a smooth, bounded domain \(\Omega \subset \mathbb {H}\) and \(T>0\), we consider the problem
for given \(0<u_0\in \mathrm {C}^{2,\alpha }(\overline{\Omega })\) and \(\phi \in \mathrm {C}^{2,\alpha ;1,\frac{\alpha }{2}}(\partial \Omega \times [0,T])\) satisfying \(\phi (\cdot ,t)\ge m(m-1)t\) and the first order compatibility conditions
Such boundary data \(\phi \) exist since \(u_0\) and \(\mathrm {R}_{g_0}\) are bounded on the compact set \(\partial \Omega \) and \(u_0>0\). In Sect. 1.2 we choose \(\phi \) explicitly.
For small times \(t>0\), we expect the solution u to (3) to be close to the solution \(\tilde{u}\) of the linear problem
Since \(\Omega \) is bounded and since \(u_0>0\) in \(\mathbb {H}\), there exists some \(\delta >0\) depending on \(\Omega \) and \(u_0\) such that \(u_0\ge \delta \) in \(\Omega \). Therefore, Eq. (5) is uniformly parabolic with regular coefficients and the compatibility conditions given in (4) are satisfied. According to linear parabolic theory [12, § IV.5, Theorem 5.2], problem (5) has a unique solution \(\tilde{u}\in \mathrm {C}^{2,\alpha ;1,\frac{\alpha }{2}}(\overline{\Omega }\times [0,T])\). Since \(u_0>0\) and \(\phi (\cdot ,t)\ge m(m-1)t\) for all \(t\in [0,T]\), the parabolic maximum principle (Prop. A.2) applied to \(m(m-1)t-\tilde{u}(\cdot ,t)\) implies
In particular, \(\tilde{u}\ge \varepsilon \) on \(\Omega \times [0,T]\) for some \(\varepsilon >0\) depending on \(\Omega \) and \(\tilde{u}\).
Lemma 1.1
(Short-time existence on bounded domains) Let \(\Omega \subset \mathbb {H}\) be a smooth, convex, bounded domain. Then, there exists \(T>0\) such that problem (3) is solvable.
Proof
A solution u to (3) is of the form \(u=\tilde{u}+v\), where \(\tilde{u}\) solves (5) and
Given the Hölder exponent \(0<\alpha <1\), let
The map
then is well-defined because the compatibility conditions (4) imply that at every \(p\in \partial \Omega \) for every \(v\in X\), we have
The linearisation of \(Q[\tilde{u}]\) around \(\tilde{u}\in \mathrm {C}^{2,\alpha ;1,\frac{\alpha }{2}}(\overline{\Omega }\times [0,T])\) defines the linear operator
The map S is Gateaux differentiable at \(0\in X\) with derivative
The mapping \(u\mapsto L(u)\) is continuous near \(\tilde{u}\) because \(\tilde{u}\) is bounded away from zero. Hence, DS(0) is in fact the Fréchet-derivative of S at \(0\in X\). Moreover, the linear operator \(\frac{\partial }{\partial t}-L(\tilde{u})\) is uniformly parabolic.
Let \(f\in Y\) be arbitrary. By definition, \(0=f(\cdot ,0)\) is satisfied on \(\partial \Omega \) which is the first order compatibility condition for the linear parabolic problem
As before, linear parabolic theory states that (7) has a unique solution \(w\in X\). Therefore, the linear map \(DS(0):X\rightarrow Y\) is invertible.
By the Inverse Function Theorem (Proposition A.1), S is invertible in some neighbourhood \(V\subset Y\) of S(0). We claim that V contains an element e such that \(e(\cdot ,t)=0\) for \(0\le t\le \varepsilon \) and sufficiently small \(\varepsilon >0\). Let \(f:=S(0)=\frac{\partial }{\partial t}\tilde{u}-Q[\tilde{u}]\) be fixed. Let \(\theta :[0,T]\rightarrow [0,1]\) be a smooth cutoff function such that
We claim \(\theta f\in V\) for sufficiently small \(\varepsilon >0\). Since \(\tilde{u}\) is smooth in \(\overline{\Omega }\times [0,T]\), we have \(f\in \mathrm {C}^{1}(\overline{\Omega }\times [0,T])\). Since at \(t=0\), we have
we can estimate
Let \(t,s\in [0,T]\) such that \(t>s\). If \(s>2\varepsilon \), then \((f-\theta f)(\cdot ,s)=(f-\theta f)(\cdot ,t)=0\). Therefore, we may assume \(s\le 2\varepsilon \). In this case we estimate
Due to (8), the special case \(s=0\) reduces to
Since the left-hand side of (11) vanishes for \(t>2\varepsilon \), we have in fact
If \(\left|t-s\right|<\varepsilon \), estimate (10) directly implies
If \(\left|t-s\right|\ge \varepsilon \), we replace the estimate by
Therefore, \([f-\theta f]_{\frac{\alpha }{2},t}\le 32\varepsilon ^{1-\frac{\alpha }{2}}\left||f\right||_{\mathrm {C}^1}\).
For the spatial Hölder seminorm, we obtain a similar estimate from (9) and
where d(x, y) denotes the Riemannian distance between x and y in \((\mathbb {H},g_{\mathbb {H}})\) and where convexity of \(\Omega \) is used. To conclude, \(\left||f-\theta f\right||_{Y}\le C\varepsilon ^{\beta -\alpha }\left||f\right||_{\mathrm {C}^1}\). Thus, \(\theta f\) belongs to the neighbourhood V of f if \(\varepsilon >0\) is sufficiently small. By construction, \(S^{-1}(\theta f)\) is a solution to (6) in \(\Omega \times [0,\varepsilon ]\). Redefining \(T=\varepsilon >0\), we obtain the claim. \(\square \)
1.2 Local estimates
Let \(\Omega \subset \mathbb {H}\) be a smooth, convex, bounded domain. Let \(u\in \mathrm {C}^{2,\alpha ;1,\frac{\alpha }{2}}(\overline{\Omega }\times [0,T])\) be a solution to the nonlinear problem (3) as determined in Lemma 1.1. Restricting the hyperbolic background metric \(g_{\mathbb {H}}\) to \(\Omega \), we obtain the Yamabe flow \(g(t)=u(\cdot ,t)g_{\mathbb {H}}\) on \(\Omega \) with initial metric \(g_0=u_0g_{\mathbb {H}}\). In order to estimate the scalar curvature \(\mathrm {R}=\mathrm {R}_{g(t)}\) of \((\Omega ,g(t))\) by means of the maximum principle, we will assume \(u_0\in \mathrm {C}^{4,\alpha }(\overline{\Omega })\) such that \(\mathrm {R}_{g_0}\in \mathrm {C}^{2,\alpha }(\overline{\Omega })\) and specify the parabolic boundary data \(\phi \) explicitly. We define the function \({{\,\mathrm{v}\,}}\in \mathrm {C}^{2,\alpha }(\overline{\Omega })\) by
which is the relative initial velocity of the Yamabe flow in question compared to the “big bang”-Yamabe flow \(m(m-1)tg_{\mathbb {H}}\). Defining the constant
we have \(\left|{{\,\mathrm{v}\,}}\right|\le 2u_0\kappa \). For \(s\ge 0\), let
where \(\chi _{[0,1]}\) denotes the characteristic function of the interval [0, 1] (Fig. 1).
As parabolic boundary data for problem (3) we choose
which satisfies the desired inequalities
and the first order compatibility conditions (4) by construction, i.e. \(\phi (\cdot ,0)=u_0\) and
Moreover, we have \(\phi \in \mathrm {C}^{2,\alpha ;1,\frac{\alpha }{2}}(\overline{\Omega }\times [0,T])\) because \(u_0\in \mathrm {C}^{4,\alpha }(\overline{\Omega })\) and \(\mathrm {R}_{g_0}\in \mathrm {C}^{2,\alpha }(\overline{\Omega })\) and since the derivatives
are continuous at \(s=1\) and bounded in \({[}0,\infty {[}\). We observe that for any \(s\in {[}0,\infty {[}\)
Given \(\varepsilon >0\) to be chosen, the estimates (15), \(\left|\psi '\right|\le 1\) and \(\left|{{\,\mathrm{v}\,}}\right|\le 2u_0\kappa \) imply
if \(\varepsilon >0\) is chosen sufficiently small depending only on \(u_0\), \(\Omega \) and m. Hence,
Let \(0<K_0\le \kappa \) be a constant such that \(\mathrm {R}_{g_0}\le K_0\) in \(\overline{\Omega }\). For \(0\le t<\tfrac{1}{K_0}\) we have
To estimate (17) we set \(a:=\frac{K_0}{\kappa }\in [0,1]\) and \(s=\kappa t\) and observe that the expression
is decreasing in \(s\in [0,1]\) as long as \(a\le 1\) and therefore bounded from above by \(a\psi (0)+\psi '(0)=1\) and from below by \(\frac{a}{3}\ge 0\). Substituting the term \(0\le \Xi \le 1\) in (17), we conclude
For every \(t\in {[}0,\tfrac{1}{K_0}{[}\) we obtain
Finally, for all \(t\ge \frac{1}{\kappa }\) we have \(\psi '(\kappa t)=0\) and thus \(\frac{\partial }{\partial t}\phi (\cdot ,t)=m(m-1)\). Some of the previous estimates are illustrated in Fig. 2.
Lemma 1.2
(Scalar curvature bound) Let \(0\le K_0\in \mathbb {R}\) be a constant such that \(\mathrm {R}_{g_0}\le K_0\) in \({\Omega }\). Let \(T_0:=(\max \{K_0,\frac{1}{T}\})^{-1}\). Then, there exists \(\varepsilon >0\) depending only on \(u_0\), \(\Omega \) and m such that for all \((x,t)\in \Omega \times {[}0,T_0{[}\)
Proof
In \(\Omega \times [0,T]\) we can express the scalar curvature in the form \(\mathrm {R}=-\frac{1}{u}\frac{\partial u}{\partial t}\). In particular, on \(\partial \Omega \times [0,T]\) we have
where the right hand side satisfies the lower bound (16) and the upper bound (18). Scalar curvature evolves by the equation (see [5, Lemma 2.2])
Let \(w(t)=\frac{K_0}{1-K_0 t}\) for \(t\in {[}0,T_0{[}\) with \(\frac{\mathrm {d}w}{\mathrm {d}t}=w^2\). Then \(\mathrm {R}\le w\) on \((\Omega \times \{0\})\cup (\partial \Omega \times {[}0,T_0{[})\) by (18). Moreover, we have
Let \(T_1<T_0\le T\) be fixed. Then, \((\mathrm {R}+w)\) is bounded from above in \(\Omega \times [0,T_1]\) and the inequality \(\mathrm {R}\le w\) in \(\Omega \times [0,T_1]\) follows from the parabolic maximum principle (Proposition A.2). Since \(T_1<T_0\) is arbitrary, we have \(\mathrm {R}\le w\) in \(\Omega \times {[}0,T_0{[}\).
Let \(\varepsilon >0\) be sufficiently small depending only on \(u_0\), \(\Omega \) and m, such that (16) holds and such that additionally, \(\mathrm {R}_{g_0}\ge -\frac{1}{\varepsilon }\) in \(\Omega \). In the argument above we replace w(t) by \(-\frac{1}{t+\varepsilon }\) and conclude \(\mathrm {R}\ge -\frac{1}{t+\varepsilon }\) analogously. \(\square \)
Lemma 1.3
(Upper and lower bound) Let \(0<u\in \mathrm {C}^{2,\alpha ;1,\frac{\alpha }{2}}(\overline{\Omega }\times [0,T])\) be a solution to problem (3) with boundary data (13) and bounded initial data \(u_0>0\). Then, for every \(0\le t\le T\),
Proof
From the equation for u, we deduce that given any constant \(c\in \mathbb {R}\) the function \(w(\cdot ,t)=u(\cdot ,t)-m(m-1)t-c\) satisfies
Since \(u>0\), equation (20) is uniformly parabolic. For \(c=\frac{1}{3}\min _{\overline{\Omega }}u_0\) (respectively \(c=\frac{5}{3}\max _{\overline{\Omega }}u_0\) ) we have \(w\ge 0\) (respectively \(w\le 0\)) on \((\partial \Omega \times [0,T])\cup (\Omega \times \{0\})\) by (14) and the parabolic maximum principle (Proposition A.2) implies \(w\ge 0\) (respectively \(w\le 0\)) in \(\Omega \times [0,T]\). \(\square \)
Lemma 1.4
(Uniqueness on bounded domains) Let \(u,v\in \mathrm {C}^{2,\alpha ;1,\frac{\alpha }{2}}(\overline{\Omega }\times [0,T])\) be two positive solutions of problem (3) with equal initial and boundary data. Then \(u=v\).
Proof
With derivatives and inner products taken with respect to \(g_{\mathbb {H}}\), we have by (1)
which can be considered as linear parabolic equation for \(u-v\) with bounded coefficients because \(u,v\in \mathrm {C}^{2,\alpha ;1,\frac{\alpha }{2}}(\overline{\Omega }\times [0,T])\) are uniformly bounded away from zero and from above and \(\left|\nabla u\right|\), \(\left|\nabla v\right|\), \(\Delta v\) are bounded functions in \(\overline{\Omega }\). Since \((u-v)\) vanishes along \((\Omega \times \{0\})\cup (\overline{\Omega }\times [0,T])\), the parabolic maximum principle (Proposition A.2) implies \(u-v=0\) in \(\overline{\Omega }\times [0,T]\) as claimed. \(\square \)
Lemma 1.5
(Local Hölder estimate) Let \(u\in \mathrm {C}^{2,\alpha ;1,\frac{\alpha }{2}}(\overline{\Omega }\times [0,T])\) be a solution to problem (3) with boundary data (13) and \(T<\frac{1}{K_0}\), where \(K_0\ge 0\) is an upper bound for \(\mathrm {R}_{g_0}\) in \(\Omega \). Then, there exists a constant C, depending only on the dimension m, the initial data \(u_0\), the constant \(K_0\) and the domain \(\Omega \) such that for every \(t\in [0,T]\)
Proof
Let \(U=u^{\eta }\) be the corresponding solution to equation (2), i.e.
Lemmata 1.2 and 1.3 yield uniform bounds on the function U and the scalar curvature \(\mathrm {R}\) in \(\Omega \times [0,T]\). Therefore, equation (21) implies
for every \(t\in [0,T]\), where C is a finite constant depending only on m, \(u_0\), \(K_0\) and \(\Omega \). Elliptic \(L^p\)-Theory [8, § 9.5] implies \(\left||U(\cdot ,t)\right||_{W^{2,p}(\overline{\Omega })}\le C\) for every \(1<p<\infty \). With \(p>m\), Sobolev’s embedding theorem implies \(\left||U(\cdot ,t)\right||_{\mathrm {C}^{1,\alpha }(\overline{\Omega })}\le C\). In particular, since \(U=u^{\eta }\) is bounded away from zero by Lemma 1.3, we have
Hence, the equation
has sufficiently regular coefficients for linear parabolic theory [12, § IV.5, Theorem 5.2] to apply: It follows that \(V=U\) is the unique solution to (22) with the given initial and boundary data. Moreover, U satisfies
Since U is bounded away from zero in \(\overline{\Omega }\times [0,T]\), the claim follows. \(\square \)
Corollary 1.6
(Extension in time) If the initial scalar curvature \(\mathrm {R}_{g_0}\) is bounded from above by \(K_0\ge 0\) in \(\mathbb {H}\), then problem (3) with boundary data (13) is solvable in \(\Omega \times {[}0,\frac{1}{K_0}{[}\) for every smooth, bounded domain \(\Omega \subset \mathbb {H}\).
Proof
According to Lemma 1.1, problem (3) is solvable in \(\Omega \times [0,T]\) for some \(T>0\). Let \(T_{*}>0\) be the maximal existence time, i. e. the supremum over all \(T>0\) such that problem (3) has a solution defined in \(\Omega \times [0,T]\). By Lemma 1.4, two such solutions agree on their common domain, therefore there exists a solution u defined on \(\overline{\Omega }\times {[}0,T_{*}{[}\). Suppose, that for some \(\Omega \) the maximal existence time is \(T_{*}<\frac{1}{K_0}\). Then, Lemma 1.5 implies that u can be extended to \(u\in \mathrm {C}^{2,\alpha ;1,\frac{\alpha }{2}}(\overline{\Omega }\times [0,T_{*}])\) and that \(u(\cdot ,T_{*})\in \mathrm {C}^{2,\alpha }(\overline{\Omega })\) is suitable initial data for problem (3). The boundary data (13) are defined also for \(t\ge T_{*}\) and they are compatible with \(u(\cdot ,T_{*})\) at time \(T_{*}\). Therefore, we may apply Lemma 1.1 to extend the solution regularly in time in contradiction to the maximality of \(T_{*}\). \(\square \)
1.3 Uniform estimates
We assume that the initial metric \(g_0=u_0g_{\mathbb {H}}\) and its scalar curvature satisfy the upper bounds \(u_0\le C_0\) and \(\mathrm {R}_{g_0}\le K_0\) in \(\mathbb {H}\) with some constant \(K_0\ge 0\). Let \(0<T<\frac{1}{K_0}\) be fixed. From the previous section we recall that for any smooth, bounded domain \(\Omega \subset \mathbb {H}\), there exists a uniformly bounded solution u of (3) on \(\Omega \times [0,T]\). However, the previous Hölder estimates on u may depend on the domain \(\Omega \). In the following, we derive independent bounds. As before, spatial derivatives and inner products are taken with respect to the hyperbolic background metric but in the following we will suppress the index \(g_{\mathbb {H}}\) to ease notation.
Lemma 1.7
(Uniform gradient estimate) Let \(B_\ell \) be the metric ball of radius \(\ell >1\) around the origin in \((\mathbb {H},g_{\mathbb {H}})\). Let \(u\in \mathrm {C}^{2,\alpha ;1,\frac{\alpha }{2}}(B_\ell \times [0,T])\) be a solution to problem (3) with boundary data (13) as constructed in Corollary rm 1.6. Then, \(U=u^{\eta }\) with \(\eta =\frac{m-2}{4}\) satisfies
where the constant C depends on the dimension m and the constants \(C_0\), \(K_0\), T but not on \(\ell \). Similar bounds hold for higher derivatives of U.
Proof
Let \(p\in \mathbb {R}\) be an exponent. As in [13], we consider the function \(w=U^p\left|\nabla U\right|^2\) and compute
We recall \(\mathrm {R}=-\frac{1}{\eta U}\frac{\partial U}{\partial t}\) and \(\frac{1}{m-1}U^{\frac{1}{\eta }}\frac{\partial U}{\partial t}=m\eta U+\Delta U\) from equation (2). Since
we have
Bochner’s identity implies
Together with \({{\,\mathrm{Ric}\,}}_{g_{\mathbb {H}}}=-(m-1)g_{\mathbb {H}}\), we apply (24) in the following computation.
Hence,
We insert (25) into (23) and resubstitute \(w=U^p\left|\nabla U\right|^2\) to obtain
Choosing \(p=-\frac{1}{2}\) and deducing \(\frac{2}{m-1}\mathrm {R}\le K_1\) from Lemma 1.2 for some constant \(K_1\ge 0\) depending on \(K_0\) and T, we obtain
Let \(\varphi :\mathbb {R}\rightarrow [0,1]\) be a smooth, non-increasing cutoff function satisfying \(\varphi (x)=1\) for \(x\le \ell -1\) and \(\varphi (x)=0\) for \(x\ge \ell \). Let \(r:\mathbb {H}\rightarrow {[}0,\infty {[}\) be the Riemannian distance function from the origin in \((\mathbb {H},g_{\mathbb {H}})\) and let \(\chi =\varphi \circ r\). Before we multiply both sides of Eq. (26) by \(\chi \), we compute
Lemma A.3 stating \(\Delta r\le 2(m-1)\) on \(\mathbb {H}\setminus B_1\) and Lemma A.4 about cutoff functions (both given in the “Appendix”) provide an estimate of the last term: There exists a constant \(c_m\) depending only on the dimension m such that
and such that \(\chi ^{-3}\left|\nabla \chi \right|^4\le c_m^2\) which will be used later. (27) and (28) lead to
Young’s inequality for \(a,b\in \mathbb {R}\), \(\delta >0\) and \(p,q>1\) with \(\frac{1}{p}+\frac{1}{q}=1\) states that
We apply it with \(p=\frac{4}{3}\) and \(q=4\) and recall \(w=U^{-\frac{1}{2}}\left|\nabla U\right|^2\), i.e. \(\left|\nabla U\right|=w^{\frac{1}{2}}U^{\frac{1}{4}}\) to obtain
Young’s inequality with \(p=2=q\) also yields
Let the sum of all terms in (29) containing \(\Delta (\chi w)\) or \(\nabla (\chi w)\) be denoted by
and let the largest of the occurring factors which depend only on the dimension m be denoted by \(C_m\). Then
Choosing \(\delta =\frac{1}{16}\), we have \(-(\tfrac{1}{4}-2\delta )=-\frac{1}{8}<0\). Since \(-\chi w^2\le -(\chi w)^2\) and since
for any \(c_1,c_2>0\), we obtain
with a different constant \(\tilde{C}_m\). By Lemma 1.3, we have
in \(B_\ell \times [0,T]\). Since \((\frac{3}{2}-\frac{1}{\eta })\eta >-1\), the right hand side of (30) is bounded from above by a spatially constant, positive function \(f\in L^1([0,T])\). Let \(F'(t)=f(t)\) with \(F(0)=\max (\chi w)(0)\) define a primitive function F for \(t\mapsto f(t)\). Then,
Since \(\chi w-F=-F\le 0\) on \(\partial B_\ell \times [0,T]\) and \(\chi w\le F\) on \(B_\ell \times \{0\}\), the parabolic maximum principle (Proposition A.2) implies \(\chi w\le F\) in \(B_\ell \times [0,T]\). The map \(t\mapsto F(t)\) is increasing and F(T) depends only on \(m,C_0,K_0,T\). Therefore, in \(B_{\ell -1}\times [0,T]\), we finally have
Since the Yamabe flow equation is only of second order, similar estimates on higher derivatives of U follow analogously. \(\square \)
Proof of Theorem 1
Let \(\eta =\frac{m-2}{4}\) and let the initial metric \(g_0=u_0g_{\mathbb {H}}\) be given by \(0<U_0=u_0^{\eta }\in \mathrm {C}^{2,\alpha }(\mathbb {H})\). Let \(B_r\) be the metric ball of radius \(r>0\) around the origin in \((\mathbb {H},g_{\mathbb {H}})\). Then, \(B_1\subset B_2\subset \ldots \subset \mathbb {H}\) is an exhaustion of \(\mathbb {H}\) with smooth, bounded domains. We fix \(0<T<\frac{1}{K_0}\) and choose \(\phi \) as in (13). By Corollary 1.6, the problem
is solvable for every \(k>0\). According to Lemma 1.7, the sequence \(\{{U_k}|_{B_1\times [0,T]}\}_{2\le k\in \mathbb {N}}\) is uniformly bounded in \(\mathrm {C}^{2,\alpha ;1,\frac{\alpha }{2}}(B_1\times [0,T])\). Since \(B_1\) is a bounded domain, the embedding \(\mathrm {C}^{2,\alpha ;1,\frac{\alpha }{2}}(B_1\times [0,T])\hookrightarrow \mathrm {C}^{2;1}(B_1\times [0,T])\) is compact and we obtain a subsequence \(\Lambda _1\subset \mathbb {N}\) such that
converges in \(\mathrm {C}^{2;1}(B_1\times [0,T])\) to a solution of the Yamabe flow equation (2) on \(B_1\). We repeat this argument to obtain a subsequence \(\Lambda _2\subset \Lambda _1\) such that
converges to a solution of (2) on \(B_2\). Iterating this procedure leads to a diagonal subsequence of \(\{U_k\}_{2\le k}\) which converges to a limit \(U\in \mathrm {C}^{2;1}(\mathbb {H}\times [0,T])\) satisfying the Yamabe flow equation (2). Since the bounds from Lemma 1.3 are preserved in the limit, we have \(m(m-1)t\le U^{\frac{1}{\eta }}\), i.e. \((g(t))_{t\in [0,T]}\) given by \(g(t)=U(\cdot ,t)^{\frac{1}{\eta }}g_{\mathbb {H}}\) is an instantaneously complete Yamabe flow.
It remains to show that the Yamabe flow constructed above can be extended in time. Let \(T_*\) be the supremum over all \(T>0\) such that there exists a Yamabe flow \(g(t)=u(\cdot ,t)g_{\mathbb {H}}\) on \(\mathbb {H}\) which is defined for \(t\in [0,T]\) and satisfies \(u(\cdot ,0)=u_0\) as well as
We have already shown \(T_{*}\ge \frac{1}{K_0}\). Suppose, \(T_{*}<\infty \) and let \(0<\varepsilon <\frac{1}{5}T_{*}\) be arbitrary. For \(T=T_{*}-\varepsilon \), there exists \(u:\mathbb {H}\times [0,T]\rightarrow \mathbb {R}\) satisfying \(u(\cdot ,0)=u_0\) together with estimate (31) and Eq. (2) which can be written in divergence form:
where we recall \(\eta =\frac{m-2}{4}\). Around an arbitrary point \(p\in \mathbb {H}\), we choose geodesic normal coordinates x and given \(0<r<\tfrac{1}{3}\sqrt{T}\) we consider the parabolic cylinder
According to (31) and the choices of r, \(\varepsilon \) and T, we have
because \(T-(2r)^2>T-\frac{4}{9}T=\frac{5}{9}(T_{*}-\varepsilon )>\frac{4}{9}T_{*}\). Hence, (32) can be interpreted as a linear equation with uniformly bounded coefficient \(\frac{1}{u}\). Therefore we may apply parabolic DeGiorgi–Nash–Moser Theory [12, Theorem III.10.1] (see also [18]) to Eq. (32) in order to obtain
for some \(0<\alpha <1\) and some constants C, \(C'\) depending only on the indicated quantities. In particular, the Hölder estimate (33) holds uniformly in \(p\in \mathbb {H}\). As in the proof of Lemma 1.5 we obtain
Consequently, the scalar curvature \(\mathrm {R}=-(m-1)u^{-\eta -1}(\frac{1}{\eta }\Delta u^{\eta }+m u^{\eta })\) stays uniformly bounded up to time T by some constant \(K_1(m,T_{*},\sup u_0)\). With the same methods as before, we can extend the solution, first locally in bounded domains and then via exhaustion in all of \(\mathbb {H}\). As initial data, we choose \(u_1=u(\cdot ,T-\varepsilon )\). This allows us build compatible boundary data from a suitable extension of \(u|_{\mathbb {H}\times [T-\varepsilon ,T]}\). It also ensures the extended solution to be regular by Lemma 1.4. The extended solution is defined in \(\mathbb {H}\times [0,T-\varepsilon +T_1]\) with any given \(T_1<\frac{1}{K_1}\). If \(\varepsilon >0\) is chosen small enough such that
we obtain a contradiction to the maximality of \(T_{*}\). \(\square \)
2 Uniqueness
This section contains the proofs of Theorems 2 and 3. As before, \((\mathbb {H},g_{\mathbb {H}})\) denotes hyperbolic space of dimension \(m\ge 3\) and \(g_{\mathbb {E}}=h^{-2}g_{\mathbb {H}}\) the pullback of the Euclidean metric to \(\mathbb {H}\), where \(h>0\) is a smooth function provided by the Poincaré ball model.
2.1 Upper and lower bounds
The Yamabe flow constructed in the previous section satisfies the upper and lower bounds given in (31). The aim of this section is to find conditions under which any Yamabe flow necessarily satisfies such bounds.
Proposition 2.1
(Upper bound) Let \(\eta =\tfrac{m-2}{4}\). If \(g(t)=u(\cdot ,t)g_{\mathbb {H}}\) for \(t\in [0,T]\) is any Yamabe flow on \(\mathbb {H}\) with initial data \(g(0)\le b\, g_{\mathbb {E}}\) for some constant \(b\in \mathbb {R}\), then
in \(\mathbb {H}\) for every \(t\in [0,T]\).
Proof
Let \(\varphi :\mathbb {R}\rightarrow [0,1]\) be a smooth, non-increasing cutoff function satisfying \(\varphi (x)=1\) for \(x\le 1\) and \(\varphi (x)=0\) for \(x\ge 2\). Introducing geodesic normal coordinates \((r,\vartheta )\in {]}0,\infty {[}\times \mathbb {S}^{m-1}\) on \((\mathbb {H},g_{\mathbb {H}})\) and a parameter \(A>1\), we define the rotationally symmetric function \(\phi =\varphi \circ \frac{r}{A}\) on \(\mathbb {H}\). As before, spatial derivatives and inner products are taken with respect to the hyperbolic background metric \(g_{\mathbb {H}}\). With any regular function \(v:\mathbb {H}\rightarrow \mathbb {R}\), we have
Setting \(f:=b h^{-2}\) and \(v:=(u^{\eta }-f^{\eta })\), we denote
Since \(f g_{\mathbb {H}}=b\,g_{\mathbb {E}}\) is a flat metric, it is a static solution to the Yamabe flow equation \(\partial _tg=-\mathrm {R}g\) which by (2) implies \(-\Delta f^{\eta }=m\eta f^{\eta }\). Thus,
Since \(\nabla \phi =\frac{1}{A}(\varphi '\circ \frac{r}{A})\nabla r\), we have
As \(\varphi '\circ \frac{r}{A}\) is non-positive in \(\mathbb {H}\) and identically zero in the unit ball around the origin, we may apply Lemma A.3 stating \(\Delta r\le 2(m-1)\) and Lemma A.4 about cutoff functions (both given in the “Appendix”) to estimate
at points where \(\phi \ne 0\) with a constant C depending only on the dimension and the choice of \(\varphi \). Let time \(t_0\in {]}0,T{]}\) be fixed. The function \([(u^{\eta }-f^{\eta })\phi ](\cdot ,t_0)\) has a global, non-negative maximum at some point \(q_{0}\in \mathbb {H}\) which depends on \(t_0\) and the parameters A and b. If \([(u^{\eta }-f^{\eta })\phi ](q_0,t_0)>0\), we have
Equation (34) and estimate (35) then yield
Denoting \(v=u^{\eta }-f^{\eta }\) as before, we obtain
The assumption \(g(0)\le b\, g_{\mathbb {E}}\) implies \(\max (u^{\eta }-f^{\eta })(0)\le 0\). Since \(t_0>0\) is arbitrary, we may apply Lemma A.6 stated in the “Appendix” to conclude
Letting \(A\rightarrow \infty \) such that \(\phi \rightarrow 1\) pointwise on \(\mathbb {H}\), we obtain
in \(\mathbb {H}\) for all \(t\in [0,T]\). \(\square \)
A similar approach as for Proposition 2.1 leads to a proof of Theorem 3.
Proof of Theorem 3
Suppose, \(g(t)=u(\cdot ,t)g_{\mathbb {R}^m}\) is a Yamabe flow on \(\mathbb {R}^m\) for \(t\in [0,T]\) with \(g(0)=g_0\). Then
where \(\eta =\frac{m-2}{4}\) and \(\Delta \) denotes the Euclidean Laplacian. In the proof of Proposition 2.1, we replace the equation for \(u^{\eta }\) by (37) and f by \(f(x)=b\left|x\right|^{-4}\). Then, \(f^{\eta }(x)=b^{\eta }\left|x\right|^{2-m}\) is harmonic on \(\mathbb {R}^m\setminus \{0\}\) which implies
With a cutoff function \(\phi \) as in (35), we gain
from the assumption \(u(\cdot ,0)\le f\) as in (36). Letting \(A\rightarrow \infty \), we conclude \(u(x,t)\le f(x)\) for every \((x,t)\in \mathbb {R}^m\times {[}0,T{[}\). In particular, the g(t)-length of radial curves \(\gamma (r)=r\sigma \) emitting from \(\sigma \in \mathbb {S}^{m-1}\subset \mathbb {R}^m\) is estimated by
which means that \((\mathbb {R}^m,g(t))\) is geodesically incomplete. \(\square \)
Proposition 2.2
(Lower barrier, rotationally symmetric case) Let \((g(t))_{t\in [0,T]}\) be a conformally hyperbolic, instantaneously complete Yamabe flow on \(\mathbb {H}\). Under the assumption, that g(t) is rotationally symmetric around some point \(x_0\in \mathbb {H}\), we have
The proof of Proposition 2.2 is based on the following Lemma.
Lemma 2.3
Let \((g(t))_{t\in [0,T]}\) be an instantaneously complete Yamabe flow on \(\mathbb {H}\) given by \(g(t)=u(\cdot ,t)g_{\mathbb {H}}\) such that \(u(\cdot ,t)\) depends only on the hyperbolic distance r from some point \(x_0\in \mathbb {H}\) for every \(t\in \mathbb {H}\). Let \(\varrho :\mathbb {H}\times {]}0,T{]}\rightarrow {[}0,\infty {[}\) be given by
and let \(A:=\max \limits _{t\in [0,T]}\left|\frac{\partial }{\partial r}u^{-\frac{1}{2}}(a,t)\right|\). Then, whenever \(r>a\),
Proof
Given a conformal metric \(g=u g_{\mathbb {H}}\) on \(\mathbb {H}\) and any smooth function \(f:\mathbb {H}\rightarrow \mathbb {R}\),
In the case \(f=\varrho (\cdot ,t)\) and since \(\Delta _{g_{\mathbb {H}}}r=\frac{m-1}{\tanh r}\) by Lemma A.3, we have
where we used the assumption of rotational symmetry. If \(r>a\), then
since \((\tanh r)\ge \frac{m-1}{m}\) for \(r>a\) by definition of a. From Eq. (1) for u, we conclude
Since \(\Delta _{g_{\mathbb {H}}} u^{-\frac{1}{2}} =-\tfrac{1}{2}u^{-\frac{3}{2}}\Delta _{g_{\mathbb {H}}} u +\tfrac{3}{4}u^{-\frac{5}{2}}\left|\nabla u\right|_{g_{\mathbb {H}}}^2\) we have in fact
Moreover, with
we arrive at
For any \(b,c>0\) and all \(X\in \mathbb {R}\) the inequality
follows by completing the square. We apply (40) with \(X=\frac{\partial }{\partial r}u^{-\frac{1}{2}}\), \(b=\frac{m-1}{\tanh r}\) and \(c=\frac{1}{2}m\sqrt{u}\) to estimate
where the last inequality requires \((\tanh r)\ge \frac{m-1}{m}\) which holds by construction whenever \(r\ge a\). Hence,
The claim follows by subtracting (39) from (41). \(\square \)
Proof of Proposition 2.2
Let \((g(t))_{t\in {[}0,T{[}}\) be an instantaneously complete Yamabe flow on \(\mathbb {H}\) given by \(g(t)=u(\cdot ,t)g_{\mathbb {H}}\) and let \(v=\frac{1}{u}\). The goal is to prove the uniform estimate \(v(\cdot ,t)\le \frac{1}{m(m-1)t}\) for every \(t\in {]}0,T{[}\). Equation (1) implies that v satisfies
and hence evolves by the equation
Let \(\eta =\frac{m-2}{4}\) provided that \(m\ge 3\). Applying Eq. (38) to \(g_{\mathbb {H}}=v g\), we have
Hence, Eq. (42) implies
where we stress the fact, that (43) involves the Laplace–Beltrami operator \(\Delta _g\) with respect to the time-dependent metric g(t).
By assumption, there exists a point \(x_0\in \mathbb {H}\) such that \(u(\cdot ,t)\) and \(v(\cdot ,t)\) depend only on the hyperbolic distance r from \(x_0\) for all \(t\in [0,T]\). Let \(\varphi :\mathbb {R}\rightarrow [0,1]\) be a smooth, nonincreasing cutoff function as in Lemma A.4 satisfying \(\varphi (s)=1\) if \(s\le 1\) and \(\varphi (s)=0\) if \(s\ge 2\). Let \(\varrho :\mathbb {H}\times {]}0,T{]}\rightarrow {[}0,\infty {[}\) be defined as in Lemma 2.3. Given \(0<\varepsilon <\frac{1}{4}\), we introduce the functions
which are smooth in \(\mathbb {H}\times {]}0,T{]}\) because \(\varphi \) is constant around zero. We remark that while \(\varphi '\) denotes the actual first derivative of the function \(\varphi \) of one variable, \(\chi '\) is just a convenient shorthand notation. In fact, \(\left|\nabla \chi \right|_{g}^2=\left|\chi '\nabla \varrho \right|_{g}^2=\left|\chi '\right|^2\) and \(\Delta _{g}\chi =\chi ''+\chi '\Delta _g\varrho \). Hence,
Recall that \(\chi '\le 0\). With the estimate from Lemma 2.3, we have
Surprisingly, since \(\frac{\partial }{\partial r}\chi =\chi '\sqrt{u}\), the term
cancels the last term in (45). Thus,
By Lemma A.4, we may choose \(\varphi \) such that there exists a constant C depending only on m such that
With this choice,
By Young’s inequality, \(ab\le \frac{a^p}{p}+\frac{b^q}{q}\) for any \(a,b\ge 0\) and \(p,q>1\) with \(\frac{1}{p}+\frac{1}{q}=0\). We apply it with \(p=2\eta +2\) to estimate
and with \(p=\eta +1\) to obtain
Consequently, introducing the constant \(\tilde{C}=C^{2\eta +2}+\bigl ((A+1)C\bigr )^{\eta +1}\),
Provided that \(0<\varepsilon <\frac{1}{4}\), the term involving \(\chi v^{\eta +1}\) in (48) has a negative sign. In this case, since \(0\le \chi \le 1\), we may replace (48) by
The assumption of instantaneous completeness of g(t) implies that \(\varrho (r,t)\rightarrow \infty \) as \(r\rightarrow \infty \) for every \(t\in {]}0,T{]}\). Therefore, \(\chi =\varphi \circ (\varepsilon \varrho )\) is compactly supported in \(\mathbb {H}\) for every \(t\in {]}0,T{]}\) and \(w:{]}0,T{]}\rightarrow {]}0,\infty {[}\) given by
is well-defined. Let \(t_0\in {]}0,T{]}\) be arbitrary. Let \(q_0\in \mathbb {H}\) such that \(w(t_0)=\chi v^{\eta }(q_0,t_0)\). We compute
where \(-\Delta _{g}(\chi v^{\eta })(q_0,t_0)\ge 0\) since \(q_0\) is a maximum. We conclude that either
which is equivalent to \(\bigl (w(t_0)\bigr )^{\frac{\eta +1}{\eta }}<\tilde{C}\sqrt{\varepsilon }\), or
which shows that w is decreasing as long as \(w^{\frac{\eta +1}{\eta }}\ge \tilde{C}\sqrt{\varepsilon }\). Hence,
for some \(\beta \in {]}0,T{[}\) because the map \(t\mapsto w(t)\) is continuous. By Lemma A.5, we have
for every \(t\in {]}0,\beta {[}\). For all \(t\in {]}0,T{[}\) we may conclude
Letting \(\varepsilon \searrow 0\) such that \(\chi \rightarrow 1\) pointwise in \(\mathbb {H}\) proves the claim. \(\square \)
2.2 Generalisation of Topping’s interior area estimate
Topping [17] proves uniqueness of instantaneously complete Ricci flows on surfaces by estimating differences of area. In the following, we adapt his method to the Yamabe flow in dimension \(m\ge 3\).
The Poincaré ball model realises hyperbolic space \((\mathbb {H},g_{\mathbb {H}})\) as the unit ball in \(\mathbb {R}^m\) equipped with polar coordinates \(P:{]}0,1{[}\times \mathbb {S}^{m-1}\rightarrow \mathbb {H}\) mapping \((\rho ,\vartheta )\mapsto \rho \vartheta \) and Riemannian metric
In this section however, logarithmic polar coordinates \(\tilde{P}:{]}0,\infty {[}\times \mathbb {S}^{m-1}\rightarrow \mathbb {H}\) given by \(\tilde{P}(s,\vartheta )=P(\mathrm {e}^{-s},\vartheta )\) are more suitable. We record
and note that the Riemannian manifold \((Z,\zeta ):=\bigl ({]}0,\infty {[}\times \mathbb {S}^{m-1},\mathrm {d}s^2+g_{\mathbb {S}^{m-1}}\bigr )\) has constant scalar curvature \(\mathrm {R}_\zeta \equiv (m-1)(m-2)\).
Proof of Theorem 2
Let \(g(t)=u(\cdot ,t) g_{\mathbb {H}}\) and \(\tilde{g}(t)=v(\cdot ,t) g_{\mathbb {H}}\) be two Yamabe flows on \(\mathbb {H}\) for \(t\in [0,T]\). Let \(U,V:Z\times [0,T]\rightarrow {]}0,\infty {[}\) such that \(\tilde{P}^*g(t)=U(\cdot ,t)\,\zeta \) and \(\tilde{P}^*\tilde{g}(t)=V(\cdot ,t)\,\zeta \). From Eq. (2) follows that U and V both solve
where \(4\eta =(m-2)=\frac{1}{m-1}\mathrm {R}_{\zeta }\) and \(\Delta _{\zeta }=\tfrac{\partial ^2}{\partial s^2}+\Delta _{g_{\mathbb {S}^{m-1}}}\) is the Laplace-Beltrami operator with respect to the metric \(\zeta =\mathrm {d}s^2+g_{\mathbb {S}^{m-1}}\) on \(Z\). Note that U, V and their derivatives with respect to s have exponential decay for \(s\rightarrow \infty \). In fact,
Since u is positive and regular at the origin, \([u^{\eta -1} \frac{\partial u}{\partial \rho }](\mathrm {e}^{-s}\theta ,t)\) stays bounded as \(s\rightarrow \infty \).
By assumption (i) and Eq. (50), applying Proposition 2.1 to \(\tilde{g}(t)\), we have
Assumption (ii) is equivalent to
Combining (54) and (55), we obtain
Let \(S,s_0\in {]}0,\infty {[}\) be such that \(0<S\le \frac{1}{3}s_0<s_0\le \log 2\). Let \(\varphi :{]}0,\infty {[}\rightarrow [0,1]\) be a cutoff function which is identically equal to 1 in the interval \({]}s_0,\infty {[}\), vanishes in \({]}0,S{[}\) and satisfies \(\varphi ''(s)\le 0\) for \(s>\frac{1}{2}s_0\). Let \(Z_S:={]}S,\infty {[}\times \mathbb {S}^{m-1}\). We analyse the evolution of the quantity \(J:[0,T]\rightarrow \mathbb {R}\) given by
where \(x_{+}:=\max \{x,0\}\). Abbreviating \(w:=V^{\eta +1}-U^{\eta +1}\), we have for every fixed \(t\in {]}0,T{]}\) and every \(0<\tau <t\)
We obtain
where we may interchange limit and integral because with \(f:=V^{\eta }-U^{\eta }\) and (51) we have
which is bounded in \(Z_S\times [0,T]\) with exponential decay for \(s\rightarrow \infty \). We claim that
Indeed, let \((m_k)_{k\in \mathbb {N}}\) be a sequence of regular values for \(f(\cdot ,t)\) such that \(m_k\searrow 0\) as \(k\rightarrow \infty \). Then, \(\{f(\cdot ,t)>m_k\}\subset Z\) is a regular, open set with outer unit normal \(\nu \) in the direction of \(-\nabla f\). Moreover, since f and \(\nabla f\) have exponential decay for \(s\rightarrow \infty \) according to (52) and (53), since \(\varphi (S)=0\) and since \(\nabla \varphi \) is supported in \([S,s_0]\), we have by Green’s formula
Passing to the limit \(k\rightarrow \infty \) proves (58) since \(Z_S\setminus Z_{s_0}\) is a bounded domain. Hence,
Introducing the exponent \(\lambda \in {]}0,\frac{1}{3}{[}\) we modify estimate (56) as follows.
We apply estimate (60) and Hölder’s inequality with exponents \(\frac{1}{\lambda }\) and \(\frac{1}{1-\lambda }\) to gain
with constant \(C_1=\frac{\eta +1}{\eta }b^{(1-\lambda )\eta }\). The restriction of the integration domain to \([S,\frac{1}{2}s_0]\) is justified since \(\varphi ''\le 0\) in \({]}\frac{1}{2}s_0,\infty {[}\). Substituting \(\lambda =\tfrac{\gamma }{1+\gamma }\) we also have \(\tfrac{\lambda }{1-\lambda }=\gamma \) and \(\tfrac{1}{1-\lambda }=1+\gamma \) with \(\gamma \in {]}0,\frac{1}{2}{[}\), and we obtain
Since \(s\mapsto \sinh (s)\) is convex for positive arguments, we have
The lower barrier (55) then implies
with constant \(C_2=\frac{1}{m(m-1)}\bigl (\frac{3}{4(\log 2)}\bigr )^2\). Substituting (62) into (61), we obtain
with constant \(C=(m-1)C_1C_2^{\frac{\gamma }{1+\gamma }}\left|\mathbb {S}^{m-1}\right|^{\frac{1}{1+\gamma }}\), i. e.
which resembles Topping’s [17, (3-10)] “main differential inequality”
in the 2-dimensional case. By Lemma A.7 stated in the “Appendix”, we conclude
The assumption \(\tilde{g}(0)\le g(0)\) implies \(J(0)=0\). Therefore, we have
for every \(t\in [0,T]\). With a clever choice of \(\varphi \), Topping [17, Prop. 3.2] proves
In the limit \(S\searrow 0\) we have \(Q\rightarrow 0\) by (63) which yields \(V\le U\) in \({]}s_0,\infty {[}\times \mathbb {S}^{m-1}\). Since \(s_0>0\) is arbitrary, we obtain \(V\le U\) globally and hence \(\tilde{g}(t)\le g(t)\) as claimed.
In the case that g(t) and \(\tilde{g}(t)\) both satisfy (ii) and \(g(0)=\tilde{g}(0)\le b\,g_{\mathbb {E}}\), the reverse inequality \(\tilde{g}(t)\ge g(t)\) follows similarly by switching the roles of U and V. \(\square \)
References
Bahuaud, E., Vertman, B.: Yamabe flow on manifolds with edges. Math. Nachr. 287(2–3), 127–159 (2014)
Bahuaud, E., Vertman, B.: Long-time existence of the edge Yamabe flow. ArXiv e-prints, May (2016)
Brendle, S.: Convergence of the Yamabe flow for arbitrary initial energy. J. Differ. Geom. 69(2), 217–278 (2005)
Brendle, S.: Convergence of the Yamabe flow in dimension 6 and higher. Invent. Math. 170(3), 541–576 (2007)
Chow, B.: The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature. Commun. Pure Appl. Math. 45(8), 1003–1014 (1992)
Daskalopoulos, P., Sesum, N.: The classification of locally conformally flat Yamabe solitons. Adv. Math. 240, 346–369 (2013)
Giesen, G., Topping, P.M.: Existence of Ricci flows of incomplete surfaces. Commun. Partial Differ. Equ. 36(10), 1860–1880 (2011)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001). (Reprint of the 1998 edition)
Hamilton, R.S.: Harmonic Maps of Manifolds with Boundary. Lecture Notes in Mathematics, vol. 471. Springer, Berlin (1975)
Hamilton, R.S.: Four-manifolds with positive curvature operator. J. Differ. Geom. 24(2), 153–179 (1986)
Hamilton, Richard S.: Lectures on geometric flows. Unpublished (1989)
Ladyženskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasi-linear Equations of Parabolic Type. American Mathematical Society, Translations of Mathematical Monographs. American Mathematical Society, Providence (1988)
Ma, L., An, Y.: The maximum principle and the Yamabe flow. In: Chen, H., Rodino, L. (eds.) Partial Differential Equations and their Applications (Wuhan, 1999), pp. 211–224. World Scientific Publishing, River Edge (1999)
Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Springer, New York (1984). (Corrected reprint of the 1967 original)
Schwetlick, H., Struwe, M.: Convergence of the Yamabe flow for “large” energies. J. Reine Angew. Math. 562, 59–100 (2003)
Topping, P.M.: Ricci flow compactness via pseudolocality, and flows with incomplete initial metrics. J. Eur. Math. Soc. 12(6), 1429–1451 (2010)
Topping, P.M.: Uniqueness of instantaneously complete Ricci flows. Geom. Topol. 19(3), 1477–1492 (2015)
Trudinger, N.S.: Pointwise estimates and quasilinear parabolic equations. Commun. Pure Appl. Math. 21, 205–226 (1968)
Ye, Rugang: Global existence and convergence of Yamabe flow. J. Differ. Geom. 39(1), 35–50 (1994)
Acknowledgements
The author is very grateful to the anonymous referee for his work and care including valuable comments and suggestions.
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Appendix A: Auxiliary results
Appendix A: Auxiliary results
The previous sections depend on some standard results and computations which we collect in this appendix for convenience of the reader.
Proposition A.1
(Inverse function theorem for Banach spaces) Let X, Y be Banach spaces and \(\tilde{X}\subset X\) be open. Let \(S:\tilde{X}\rightarrow Y\) be Fréchet differentiable at \(x_0\) with \(S(x_0)=0\) and invertible derivative \(D S(x_0)\). Then there exists a neighbourhood \(U\subset \tilde{X}\) of \(x_0\) such that
- (i)
\(V=SU\) is open,
- (ii)
\(S|_U:U\rightarrow V\) is a homeomorphism,
- (iii)
\((S|_U)^{-1}\) is Fréchet differentiable.
In the following proposition, we denote partial derivatives by subscripts and understand a sum \(\sum ^{m}_{i=1}\) whenever an index i appears twice in an expression.
Proposition A.2
(Linear parabolic maximum principle [14, § 3.3]) Let \(\Omega \subset \mathbb {R}^m\) be an open, bounded domain. Suppose, u satisfies
where the function \(c<\lambda \in \mathbb {R}\) is bounded from above and ellipticity \(a_{ij}\,\xi _i\,\xi _j\ge 0\) for all \(\xi \in \mathbb {R}^m\) holds uniformly. Then, \(u\le 0\) in \(\Omega \times [0,T]\).
Proof
The function \(v(x,t)=u(x,t)\mathrm {e}^{-\lambda t}\) satisfies the equation
Assume that \(v(x_0,t_0)=\max _{\overline{\Omega }\times [0,T]} v\). If \(t_0=0\) or if \(x_0\in \partial \Omega \), then \(v\le 0\) follows. If \((x_0,t_0)\in \Omega \times {]}0,T{]}\), then \(v_t(x_0,t_0)\ge 0\), \(v_{x_k}(x_0,t_0)=0\) and \(-a_{ij}v_{x_i x_j}(x_0,t_0)\ge 0\). Thus, (64) implies \(-(c-\lambda )v(x_0,t_0)\le 0\). From \((c-\lambda )<0\) follows \(v(x_0,t_0)\le 0\). Therefore, \(v\le 0\) in \(\overline{\Omega }\times [0,T]\). Since u and v share the same sign, the claim follows. \(\square \)
Lemma A.3
(Laplacian of hyperbolic distance) Let \(r:\mathbb {H}\rightarrow {[}0,\infty {[}\) be the Riemannian distance function with respect to some origin in hyperbolic space \((\mathbb {H},g_{\mathbb {H}})\) of dimension \(m\ge 2\). Then,
In particular, \((\Delta _{g_\mathbb {H}}r)(x)\le 2(m-1)\) for every \(x\in \mathbb {H}\) with \(r(x)\ge 1\).
Proof
The Poincaré ball model realises hyperbolic space \((\mathbb {H},g_{\mathbb {H}})\) as the unit ball equipped with polar coordinates \(P:{]}0,1{[}\times \mathbb {S}^{m-1}\rightarrow \mathbb {H}\) mapping \((\rho ,\vartheta )\mapsto \rho \vartheta \) and conformal Riemannian metric \(P^*g_{\mathbb {H}}=h^2g_{\mathbb {E}}\), where
Here, \(g_{\mathbb {S}^{m-1}}\) is the standard metric on the unit sphere \(\mathbb {S}^{m-1}\subset \mathbb {R}^m\) and \(g_{\mathbb {E}}\) is the Euclidean metric on the unit ball in \(\mathbb {R}^m\). We denote the radial coordinate on the unit ball by \(\rho \) and reserve r for the hyperbolic distance which is given by
By (66), we have
On the one hand, Eq. (38) implies
On the other hand, Eq. (66) implies
Combined with (67), the claim follows. \(\square \)
Lemma A.4
(cutoff function) Let \(\varepsilon >0\) and \(a,b>0\) be real parameters. Then there exists a non-increasing cutoff function \(\varphi \in \mathrm {C}^{2}(\mathbb {R})\) given by
which satisfies the inequality
in \(\{x\in \mathbb {R}\mid \varphi (x)\ne 0\}\) with a constant C depending only on a, b and \(\varepsilon \).
Proof
There exists a non-increasing function \(\psi \in \mathrm {C}^{2}(\mathbb {R})\) satisfying \(\psi (x)=1\) for \(x\le 1\) and \(\psi (x)=0\) for \(x\ge 2\) as well as \(\left|\psi \right|,\left|\psi '\right|,\left|\psi ''\right|\le C_0\). Let \(p=\frac{2}{\varepsilon }\) and \(\varphi =\psi ^p\). We may assume \(0<\delta \le 2\) and \(p\ge 1\). Then,
which implies
\(\square \)
Lemma A.5
Let \(Q\in \mathbb {R}\) and let \(f:[0,T]\rightarrow \mathbb {R}\) be continuous satisfying
for every \(0<\xi \le T\). Then \(f(t)-f(0)\ge Qt\) for every \(t\in [0,T]\).
Proof
We follow an argument by Richard Hamilton [10, Lemma 3.1]. The function
is continuous and satisfies
Let \(0<t\le T\) and \(\varepsilon >0\) be arbitrary but fixed. Then there exists \(\delta >0\) such that
because otherwise, we would obtain a contradiction to (68). We may assume that \(\delta \in {]}0,t{]}\) is the largest value such that (69) holds. By continuity of w, (69) implies
If \(t-\delta >0\), we may repeat the argument to find some \(\delta '>0\) such that
In particular, (70) and (71) can be combined to
for all \(\tau \in {[}0,\delta '{[}\) in contradiction to the maximality of \(\delta \). Hence, \(\delta =t\) and we obtain
Since \(\varepsilon >0\) is arbitrary and \(w(0)=0\) we have \(w(t)\ge 0\) which proves the claim. \(\square \)
Lemma A.6
Let \(\eta >0\) and \(Q\ge 0\). Let \(v:[0,T]\rightarrow {[}0,\infty {[}\) be continuous satisfying
for every \(0<\xi \le T\) where \(v(\xi )\ne 0\). Then \(v(t)^{\frac{1}{\eta }}-v(0)^{\frac{1}{\eta }}\le Q t\) for every \(t\in [0,T]\).
Proof
We consider the function \(f=v^{\frac{1}{\eta }}\). Let \(0<\xi \le T\) be arbitrary. If \(v(\xi )\ne 0\), then by assumption
Dividing by \(\eta \) yields
If \(v(\xi )=0\) we have
since \(f(\xi )=v(\xi )^{\frac{1}{\eta }}=0\) and \(f\ge 0\). We conclude that (73) holds not only where \(v(\eta )\ne 0\) but in fact for every \(0<\xi \le T\) because \(Q\ge 0\) is assumed. The claim follows by multiplying (73) with \(-1\) and applying Lemma A.5. \(\square \)
Lemma A.7
Let \(\alpha \in {]}0,1{[}\) and \(C\ge 0\). Let \(J:[0,T]\rightarrow {[}0,\infty {[}\) be continuous satisfying
for every \(0<t\le T\). Then \(J(t)^{\alpha }-J(0)^{\alpha }\le C t^{\alpha }\) for every \(t\in [0,T]\).
Proof
As in the proof of Lemma A.6 (with \(\alpha =\frac{1}{\eta }\)) we obtain
for every fixed \(t\in {]}0,T{]}\). In particular, \(f(\xi )=J(\xi )^{\alpha }-C\xi ^{\alpha }\) satisfies
and we may conclude as before by multiplying with \(-1\) and applying Lemma A.5. \(\square \)
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