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

In two classical works of 1995, R.S. Hamilton investigated some deep problems of differential geometry, in which it was involved the Ricci flow and a specific functional defined on hypersurfaces (called, isoperimetric ratio), whose analytic expression has interesting properties. We show some conditions of minimization for this functional, by using the approach of the geometric measure theory and a recent method developed in [M. Ritor\'e and C. Rosales, Existence and characterization of regions minimizing perimeter under a volume constraint inside euclidean cones, Trans. Amer. Math. Soc. 356 (2004), 4601--4622].


Introduction
The papers [9,10] have historically influenced the study of the Ricci flow on smooth Riemannian manifold in the last 20 years. Recent advances can be found in [5,6], where Daskalopoulos and Hamilton investigate the behaviour of the maximal solutions of the Ricci flow over planes of finite volumes. They introduce a series of isoperimetric ratios, which present some properties of monotonicity. These allow us to avoid singularities, which may appear at the extinction time of the Ricci flow. Again in [5,6], 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 existence of such minimizers in any dimension (eventually, higher than 2) under two sharp quantitive assumptions, which are naturally involved in the isoperimetric profile function. In order to do this, we apply methods of the geometric measure theory, which can be found in [8,11,13,15,16,17,19,20]. The results of such papers contain various theorems of compactness and regularity for the classical isoperimetric problem and turn out to be very powerful tools, once applied to the context of [5,6]. Section 2 is devoted to illustrate some preliminaries, which are fundamental for the purposes of the main theorems of Sections 3 and 4. In doing this, we offer a new proof of the continuity of the isoperimetric profile function (see Section 2), by means of an argument contained in [19]. 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 [5,6]). The 2-dimensional case, the analytic expressions and the connectedness of the minimizers force us to use different methods, up to the specific situation which we encounter. Roughly speaking, our strategy is based on a replacement of the original minimization of [5,6] with an auxiliary problem, which involves the isoperimetric profile. We find the minimizers of the auxialiary problem and discover that they coincide with the minimizers of the original problem. This is mainly due to two technical lemmas, which are related to the analytic expression of the isoperimetric ratio (see Lemmas 4.1 and 4.2 below). More details can be found in Section 4. Finally, we offer examples in which the assumptions of the main theorems are satisfied. These examples show the usefulness of replacing the original problem with our formulation.

Previous results in literature
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 ⊆ M and any open set U ⊆ M (here n ≥ 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 ≥ 0) and is the perimeter of Ω relative to U , where Y is a smooth vector field with compact support contained in U , and div Y denotes the divergence of Y . Briefly, we write P(Ω) = P(Ω, M ) and say that Ω has finite perimeter in U , if vol(Ω ∩ U ) < ∞ and P(Ω, U ) < ∞. As well known, these are fundamental notions in geometric measure theory, introduced by Caccioppoli [2] (via the geometric perimeter), De Giorgi [7] (via the heat semigroup) and recently adapted to the context of Riemannian manifolds in [12]. We recall from [1] that for a finite perimeter set Ω ⊆ M and an open set U ⊆ M , the reduced boundary ∂ * Ω is the boundary of Ω in the sense of [1, Definition 3.54] (in general, ∂ * Ω ⊆ ∂Ω but the converse may be false) and a classical result of De Giorgi [1,Thoerem 3.59] shows that P(Ω, U ) = H n ((∂ * Ω) ∩ U ). In particular, when ∂Ω is smooth and U = M , ∂ * Ω = ∂Ω and P(Ω) = H n (∂Ω). This precisation is important for the notions which we introduce in Definition 3.1.
We refer to [1,3,14,18] for classical aspects of geometric measure theory and differential geometry. One of these is, for instance, the following notion. The isoperimetric profile of M is the function  The behaviour of a minimizing sequence for fixed volume was investigated in various contributions in the last years, but we concentrate on [8,13,15,16,17,19,20], since we focus on a perspective of Riemannian geometry. The following result of Ritoré and Rosales [19] characterizes the existence of regions minimizing perimeter under a fixed volume constraint. The arguments overlap some techniques in [13,15]. (v) Ω is an isoperimetric region (eventually empty) for the volume it encloses.
The condition (iv) of Theorem 2.1 can be expressed by saying that {Ω c k } k∈N converges to Ω in the finite perimeter sense (see [19, pp. 4601-4603] or [1] for a rigorous definition). A priori we note that vol(Ω) may be strictly less than V in (i) of Theorem 2.1. A consequence of Theorem 2.1 is a significant result of compactness, when the ambient manifold is of finite volume. This is expressed by the following corollary. Proof. Looking at the conditions (ii) and (iv) of Theorem 2.1, there is a splitting of the volume in the following form where V 1 is the term which is at finite distance from Ω and V 2 is the term which is at infinite distance from Ω. Assume that Such Ω d k turns out to be a sequence that lies outside every fixed compact K inside M . The details of this construction can be found at [19, pp.4604-4606]. Then it must be vol and so V 2 = lim k→∞ vol(Ω d k ) = 0. This gives contradiction. Therefore V 2 = 0, hence V 1 = V and the result follows.
Another interesting corollary of Theorem 2.1 is related with the continuity of the isoperimetric profile. In order to prove this second consequence, we recall a technical lemma from [8]. Lemma 2.3 (Deformation's Lemma, see [8], Lemma 4.5). Let M be a connected unbounded open set of a complete Riemannian manifold and Ω ⊆ M a finite perimeter set. Then there exists a finite perimeter setΩ r ⊇ Ω with 0 < r < ∞ and a constant C Ω > 0, depending only on Ω, such that The continuity of the isoperimetric profile is shown below. Proof. Consider a sequence of volumes V i such that V = lim i→∞ V i . By Corollary 2.2, we have that where Ω i is an isoperimetric region of vol(Ω i ) = V i and Ω is an isoperimetric region of vol(Ω) = V such that Ω i converges to Ω in finite perimeter sense. This allows us to conclude only the lower semicontinuity of I M (V ).
In order to show the upper semicontinuity, we need to prove that where Ω is an isoperimetric region of volume V and Ω i suitable sets approximating Ω. By Lemma 2.3, there exist a finite perimeter set In particular, C Ω is fixed with respect to i. Passing through the limit The result follows.
The basic regularity properties of the boundary of isoperimetric regions are stated below.
By Theorem 2.5 (iii), the low dimensional cases shows that the isoperimetric regions have smooth boundary.

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 [4], but also some new functionals for the purposes of our main results. .
If H ⊂ M is a smooth embedded closed (eventually disconnected) hypersurface which is the boundary of an open region R, we define .
By default, we get four isoperimetric constants which lead to the isoperimetric constants In particular, if (M, g) is isometric to (R 2 , g) with a complete Riemannian metric g, we may specialize I(N ), in Definition 3.1, writing n = 1 and N = γ, which turns out to be a closed simple curve of R 2 of length L(γ), and let A 1 (γ) and A 2 (γ) denote the areas of the regions inside and outside γ respectively. In this way, we get I(γ) = L(γ) · (1/A 1 (γ) + 1/A 2 (γ)) and I = inf{I(γ) | γ separates R 2 }. This special case presents some peculiarities and was studied in [5,6]. We will focus on it in Examples 5.1 and 5. What can we say aboutĨ ♯ and C ? The previous argument of Remark 3.2 cannot be applied and we need of a new proof. Proof. From Theorem 2.1, for every volume V there always exists an isoperimetric region Ω with vol(Ω) = V . Theorem 2.5 implies that ∂Ω is smooth in low dimensions (i.e. n+ 1 ≤ 7). In higher dimensions there is a sequence {Ω i } i∈N of regions of finite perimeter with smooth boundaries {∂Ω i } i∈N converging to ∂Ω (see [12,Proposition 1.4]). Now if N ⊆ M is a smooth embedded closed (eventually disconnected) hypersurface which separates M , then Passing through the infimums, we get A priori I ♭ may be zero or not. We will give more details on this point in the next section. Now we proceed with a similar result is true forĨ * and I. Here we will find explicitlyĨ * = 0.

Minimization problems
An interesting question is to know whether the inclusions in Theorems 3.3 and 3.4 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 aspect 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 is inspired by [4,Lemma 5.86 ].
Lemma 4.1. For every positive real numbers L 1 , L 2 , A 1 , A 2 , A 3 , we get Proof. By contradiction, assume that L 1 , L 2 , A 1 , A 2 , A 3 satisfy We rewrite the two preceding inequalities respectively as Summing up these two last inequalities, we get This gives contradiction, because we assumed A 1 , A 2 and A 3 strictly positive.
The use of Lemma 4.1 is for purposes of topological nature. This will be more clear in the following result. Then Ω and M − Ω are connected, ∂Ω separates M and I ♭ = C = D. In particular, if n + 1 ≤ 7, then ∂Ω is a smooth hypersurface that separates M .
Proof. The proof goes by contradiction. Firstly, we show that Ω is connected. In order to do this, we suppose that Ω = Ω 1∪ Ω 2 contains two connected components Ω 1 and Ω 2 such that vol( , and L 2 = P(Ω 2 ). Then Applying Lemma 4.1, we find , which contradicts the minimality of Ω. We conclude that Ω must be connected. We have all the ingredients for the proof of one of our main results.
(ii) I ♭ < C 1 . Then there exists a connected finite perimeter set Ω ⊆ M such that In particular, C = D > 0 and, if n + 1 ≤ 7, ∂Ω separates M and is smooth.
Proof. It is enough to note that I About the functional I(N ), the limit V n may be zero or not, but the previous arguments shall be modified. This is illustrated in the following theorem Theorem 4.6. Let M be a complete riemannian manifold of finite volume, satisfying the following conditions for a positive constant C 2 : Then there exists a finite perimeter set Ω ⊆ M such that I(∂Ω) ≤Ĩ * . In particular, I * > 0. Moreover, if n + 1 ≤ 7, then Ω has smooth boundary.
Proof. By the definition ofĨ * , we may consider a minimizing sequence {Ω i } i∈N such that I(∂Ω i ) tends toĨ * for i running to ∞. Now it is easy to observe that vol(Ω i ) + P(Ω i ) is uniformly bounded for all i ∈ N. Putting vol( The case of the plane gives equality for the functionals in Theorem 4.6. We need of some preliminary results, in order to justify this statement.
We may apply the same argument of Lemma 4.2, in order to have topological information on the connected regions which appear in the proof of Theorem 4.3.

Some examples
The difficulty of applying the argument of Theorem 4.3 is due to the fact that I M (V ) may be or not continuous. This has motivated us to change approach in Theorem 4.6 and Corollary 4.9. In the present section, we provide some examples in order to show different behaviours, when we test the condition (i) of Theorem 4.3 on complete Riemannian manifolds of finite volume. Of course, these behaviours depends on the metric which we are considering.
Example 5.1. The present example illustrates Theorem 4.3. It is 2-dimensional and can be generalized without difficulties to higher dimensions. We take a rotationally symmetric surface M ≃ R 2 , that is, diffeomorphic to the usual plane (see [20]). Fix the origin p ∈ M and a smooth metric g which can be written in normal polar coordinates by g = dt 2 + f (t) 2 dθ 2 , where (t, θ) ∈ ]0, ∞[×S 1 and S 1 = {(x, y) ∈ R 2 | x 2 + y 2 = 1} is endowed with the Riemannian metric induced by that of the Euclidean plane. Here dθ 2 is the Riemannian volume form of S 1 and f : t ∈]0, ∞[ → f (t) ∈]0, ∞[ is such that at the origin we can extend g to a smooth metric on the entire plane. This construction can be done always, see [18, p.13]. Now M has sectional curvature [18] for the details). Note that the length l(c t ) of a geodesic circle c t depends on f (t), namely L(c t ) = 2πf (t). We require the following restrictions on f (t): (j). There is t 1 > 0 such that f ′ (t) > 0 for t < t 1 and f ′ (t) < 0 for t > t 1 ; (jj). K ′ (t) ≤ 0 for enough large t; (v). There exists a τ ∈]0, t 0 [ such that t 0 f (τ ) > f (t 0 ), when t 0 is enough large.

Now if
where h : t ∈]0, t 1 ] → h(t) ∈ ]0, ∞[ is a smooth function with h ′ (t) > 0, then (j) is satisfied, (jj) becomes K(t) = −1 for all t > t 1 and the integrals in (jjj) and (jv) are always well defined. Here the above limit becomes lim V (t)→0 Therefore the condition (i) of Theorem 4.3 is true. It remains to check that also the condition (ii) of Theorem 4.3 is true. Here we use the assumption (v); in fact from the existence of a τ ∈]0, t 0 [ such that t 0 g(τ ) > g(t 0 ), when t 0 is enough large, we conclude that g(t 0 ) < 1 2π 2π This inequality implies a fortiori that the following inequality is true: Example 5.2. In Example 5.1, we may consider a different function g(t) satisfiying the conditions (j)-(v), but not of exponential type for large values of t. Of course, the constant C 1 will be different and the limit lim V →0 I M (V )/V will require a slight different solution, but it will exist and we may argue in the same way, finding further families of examples for Theorem 4.3.