On the Hamilton's isoperimetric ratio in complete Riemannian manifolds of finite volume

doi:10.1016/j.jfa.2020.108843

Journal of Functional Analysis

Volume 280, Issue 4, 15 February 2021, 108843

https://doi.org/10.1016/j.jfa.2020.108843 Get rights and content

Abstract

We study a family of geometric variational functionals introduced by Hamilton, and considered later by Daskalopulos, Sesum, Del Pino and Hsu, in order to understand the behavior of maximal solutions of the Ricci flow both in compact and noncompact complete Riemannian manifolds of finite volume. The case of dimension two has some peculiarities, which force us to use different ideas from the corresponding higher-dimensional case. Under some natural restrictions, we investigate sufficient and necessary conditions which allow us to show the existence of connected regions with a connected complementary set (the so-called “separating regions”). In dimension higher than two, the associated problem of minimization is reduced to an auxiliary problem for the isoperimetric profile (with the corresponding investigation of the minimizers). This is possible via an argument of compactness in geometric measure theory valid for the case of complete finite volume manifolds. Moreover, we show that the minimum of the separating variational problem is achieved by an isoperimetric region. The dimension two requires different techniques of proof. The present results develop a definitive theory, which allows us to circumvent the shortening curve flow approach of the above mentioned authors at the cost of some applications of the geometric measure theory and of the Ascoli-Arzela's Theorem.

Introduction

The papers [15], [16] have historically influenced the study of the Ricci flow on smooth Riemannian manifolds in the last 25 years. Recent advances can be found in [11], [12], where Daskalopoulos and Hamilton investigate the behavior of the maximal solutions of the Ricci flow over planes of finite volumes. Recent contributions in the same direction can also be found in [18], [35]. The original idea of Daskalopoulos and Hamilton was to introduce a series of isoperimetric ratios, which present some properties of monotonicity. These allow to avoid singularities, which may appear at the extinction time of the Ricci flow. Again in [11], [12], the authors assume the existence of minimizers for certain isoperimetric ratios, which correspond to the maximal solution of the 2-dimensional Ricci flow on a plane of finite volume. Our results deal with a proof of the existence of such minimizers in any dimension (possibly, higher than 2) under two sharp quantitative assumptions, which involve the isoperimetric profile function, allowing us to generalize [18, Theorem 1.1] in our Theorem 4.20 (note that for the reader convenience [18, Theorem 1.1] is reported integrally below as Theorem 4.3). The main contributions of this work are:

(1)
In dimension three and higher, the isoperimetric problem with the separability constraint is equivalent to the one without the separability constraint.
(2)
In dimension two and higher, we discuss necessary and sufficient conditions, in order to show the existence of nontrivial minimizers. (See Theorem 4.5, Theorem 4.13, Theorem 4.17, Theorem 4.18, Theorem 4.19, Theorem 4.20 and Remark 4.6).

The first general idea of our paper, treating the case of a Riemannian manifold M of dimension

n + 1 \geq 3

, is that the isoperimetric problem with the separability constraint is equivalent to the isoperimetric problem without the separability constraint. This equivalence holds only in dimension higher than 2. Proposition 3.3 shows the details. In dimension 2 the equivalence fails to be true (compare Remark 4.4) and different tools must be used, in order to generalize the results in literature.

We have Theorem 3.4, Theorem 3.5 4.5, 4.13, 4.17, 4.18, involving completely different techniques, which are proper of the geometric measure theory. Indeed, their proofs rely on an argument of compactness for finite perimeter sets in noncompact Riemannian manifolds of finite volume whose consequence is the continuity of the isoperimetric profile function. Corollary 2.2, Corollary 2.3 are among the new contributions that we offer on the topic of continuity and compactness in the present context of study. To understand why to prove continuity of the isoperimetric profile is an interesting result for itself, the reader can see also [27], [29] in which it is shown by sophisticated examples that there exist complete Riemannian manifolds with discontinuous isoperimetric profile. Compactness for isoperimetric regions and continuity of the isoperimetric profile combined with the superadditivity property of the isoperimetric ratios (compare Lemma 4.1, Lemma 4.2, Lemma 4.9, Lemma 4.10, Lemma 4.11) provide the proofs of Theorem 3.4, Theorem 3.5 4.5, 4.13, 4.17, 4.18. Similar arguments of compactness can be found in [14], [17], [21], [23], [25], [26], [32], [33]; these contributions contain several theorems about compactness and regularity for the classical isoperimetric problem and turn out to be very powerful tools, once applied to the context of [11], [12].

The study of the variational problems associated to the functionals of Daskalopulos and Hamilton (see Definition 3.1) is more difficult in dimension 2 than in dimension 3 or higher when separability constraints are involved and we described the details in Theorem 4.13, Theorem 4.17, Theorem 4.18, Theorem 4.19, Theorem 4.20. We use in dimension 2 a soft regularizing theorem; roughly speaking, we show that “the limit of simple curves is simple” in the variational problem that we consider. We do it by showing that (under our assumptions) a minimizing sequence of separating simple curves lies inside a compact set; we show that we loose perimeter in the limit, if and only if, there is more than one connected component. Again the superadditivity of the isoperimetric ratios profiles play a crucial role in the arguments of the proofs.

Our approach is completely different from the one used in the proof of [18, Theorem 1.1], and should push the theory, of the Ricci flow in dimension 2, to wider generalizations than the original framework of Daskalopous, Hamilton, Sesum and Del Pino [9], [10], [11], [12], [15], [16]. Indeed Theorem 4.19 provides new proofs and new arguments even in the compact case, generalizing [15] to the noncompact case with different techniques via curve shortening flow. To conclude this part of the introduction we highlight that our approach permits to distillate the necessary and sufficient conditions to guarantee the existence of nontrivial minimizers. This is among our main contributions.

Section 2 is devoted to illustrate some preliminaries, which are fundamental for the proofs of the main theorems of Sections 3 and 4. We offer a new proof of the continuity of the isoperimetric profile function (see Section 2), by means of an argument contained in [32]. This result has independent interest and has an important role in the structure of our proofs in Section 4. The main results are in fact here and we solve a problem of minimization for the isoperimetric ratio in the sense of Hamilton (see [11], [12]). In dimension two we use a classical Ascoli–Arzela Theorem to get uniform convergence after reducing the problem to the compact case. This allows us to deduce the existence of simple continuous connected separating curves in the limit, that furthermore is the boundary of an isoperimetric region see Theorem 4.19, Theorem 4.20. Section 4 contains the proofs of the main results of the present paper. Finally, Section 5 contains examples in which the assumptions of the main theorems are satisfied. These examples show the usefulness of replacing the original problem with our formulation.

Section snippets

Preliminaries

We introduce some terminology and notation which will be used in the rest of the paper. The symbol $M^{n + 1}$ denotes an open connected set of a smooth complete ( $n + 1$ )-dimensional Riemannian manifold. In the rest of the paper, we will write briefly M, in order to denote $M^{n + 1}$ . For any measurable set $Ω \subseteq M$ and any open set $U \subseteq M$ (here $n \geq 0$ ), $vol (Ω)$ is the ( $n + 1$ )-dimensional Hausdorff measure of Ω, $H^{k} (Ω)$ is the k-dimensional Hausdorff measure of Ω (here $k \geq 0$ ), and $P (Ω, U) = \sup {\int_{Ω} div Y d H^{n + 1} | {‖ Y ‖}_{\infty} = 1}$ is the perimeter

Isoperimetric ratio in the sense of Hamilton

In the present section we consider only complete manifolds of finite volume. We introduce some terminology, which can be found in [8], but also some new functionals for the proofs of our main results.

Definition 3.1

Let M be a complete Riemannian manifold with $vol (M) < \infty$ . If $N \subseteq M$ is a smooth embedded closed hypersurface which separates M, we define the isoperimetric ratio $I (N) = P {(N)}^{n + 1} \cdot {(\frac{1}{vol (M_{1})} + \frac{1}{vol (M_{2})})}^{n}$ and $C (N) = P (N) \cdot (\frac{1}{vol (M_{1})} + \frac{1}{vol (M_{2})}) .$ If $H \subset M$ is a smooth embedded closed (possibly disconnected) hypersurface

Minimization problems

An interesting question is to know whether the inequalities in Theorem 3.4, Theorem 3.5 become equalities or not. An answer to this question may depend on the topology of the ambient manifold M and on the dimension of M. We will investigate such aspects in the present section, beginning with two useful lemmas which provide information on the number of connected components of the regions whose boundary minimize C (in the sense of Definition 3.1). We apply an argument of algebraic nature, which

Some examples

The difficulty of applying the argument of Theorem 4.5 is due to the fact that ${\tilde{I}}_{M} (V)$ may be or not continuous. In the present section, we provide some examples in order to show different behaviors, when we test the condition (i) of Theorem 4.5 on complete Riemannian manifolds of finite volume. Of course, these behaviors depend on the metric which we are considering.

Example 5.1

The manifolds considered in [11], [12], [15], [16], [18].

Example 5.2

The present example illustrates Theorem 4.5. We take a rotationally

Acknowledgements

The authors thank Pierre Pansu for his valuable comments which improved the original results. Special thanks go to Luis Eduardo Osorio Acevedo, who helped us with the figures, and to the anonymous reviewer for relevant comments on the previous version. The first author has been partially sponsored by FAPESP (2018/22938-4), which allowed his Visiting Fellow at the Department of Mathematics of Princeton in 2020, where a substantial part of this work has been written. He also thanks CNPq (

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On the Hamilton's isoperimetric ratio in complete Riemannian manifolds of finite volume

Abstract

Introduction

Section snippets

Preliminaries

Isoperimetric ratio in the sense of Hamilton

Minimization problems

Some examples

Acknowledgements

Ann. Inst. Henri Poincaré, Anal. Non Linéaire

J. Math. Anal. Appl.

Functions of Bounded Variation and Free Discontinuity Problems

A proof of a general isoperimetric inequality for surfaces

Math. Z.

Sur le volume minimal de R2

Ann. Sci. Éc. Norm. Supér. (4)

A new isoperimetric comparison theorem for surfaces of variable curvature

Duke Math. J.

Inegalités isopérimétrique et applications

Ann. Sci. Éc. Norm. Supér. (4)

Misura e integrazione sugli insiemi dimensionalmente orientati, Note I e II

Atti Accad. Naz. Lincei, VIII, Ser. Rend. Cl. Sci. Fis. Mat. Nat.

Riemannian Geometry: A Modern Introduction

The Ricci Flow: An Introduction

On a singular diffusion equation

Commun. Anal. Geom.

Geometric estimates for the logarithmic fast diffusion equation

Commun. Anal. Geom.

Classification of compact ancient solutions to the Ricci flow on surfaces

J. Differ. Geom.

Definizione ed espressione analitica del perimetro di un insieme

Atti Accad. Naz. Lincei, Rend. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat.

Isoperimetric estimates for the curve shrinking flow in the plane

An isoperimetric estimate for the Ricci flows on the two–sphere

The isoperimetric problem on surfaces of revolution of decreasing Gauss curvature

Trans. Am. Math. Soc.

Sur le volume minimal de $R^{2}$