Regularity of solutions of the isoperimetric problem that are close to a smooth manifold

In this work we consider a question in the calculus of variations motivated by riemannian geometry, the isoperimetric problem. We show that solutions to the isoperimetric problem, close in the flat norm to a smooth submanifold, are themselves smooth and $C^{2,\alpha}$-close to the given sub manifold. We show also a version with variable metric on the manifold. The techniques used are, among other, the standards outils of linear elliptic analysis and comparison theorems of riemannian geometry, Allard's regularity theorem for minimizing varifolds, the isometric immersion theorem of Nash and a parametric version due to Gromov.Comment: 75 pages, 2 figures, corrected typos, and some minor errors, added more detailed proofs. Accepted in Bulletin of the Brazilian Mathematical Society, 201

Generalized existence of isoperimetric regions in non-compact Riemannian manifolds and applications to the isoperimetric profile

For a complete noncompact connected Riemannian manifold with bounded geometry, we prove the existence of isoperimetric regions in a larger space obtained by adding finitely many limit manifolds at infinity. As one of many possible applications, we extend properties of the isoperimetric profile from compact manifolds to such noncompact manifolds.Comment: 40 pages, to appear in Asian Journal of Mathematic

A discontinuous isoperimetric profile for a complete Riemannian manifold

The first known example of a complete Riemannian manifold whose isoperimetric profile is discontinuous is given

Existence of isoperimetric regions in non-compact Riemannian manifolds under Ricci or scalar curvature conditions

We prove existence of isoperimetric regions for every volume in non-compact Riemannian $n$-manifolds $(M,g)$, $n\geq 2$, having Ricci curvature $Ric_g\geq (n-1) k_0 g$ and being locally asymptotic to the simply connected space form of constant sectional curvature $k_0$; moreover in case $k_0=0$ we show that the isoperimetric regions are indecomposable. We also discuss some physically and geometrically relevant examples. Finally, under assumptions on the scalar curvature we prove existence of isoperimetric regions of small volume.Comment: 17 page

Two bounds on the noncommuting graph

Erd\H{o}s introduced the noncommuting graph, in order to study the number of commuting elements in a finite group. Despite the use of combinatorial ideas, his methods involved several techniques of classical analysis. The interest for this graph is becoming relevant in the last years for various reasons. Here we deal with a numerical aspect, showing for the first time an isoperimetric inequality and an analytic condition in terms of Sobolev inequalities. This last result holds in the more general context of weighted locally finite graphs.Comment: Submitte

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

We contribute to an original problem studied by Hamilton and others, in order to understand the behaviour of maximal solutions of the Ricci flow both in compact and non-compact complete orientable Riemannian manifolds of finite volume. The case of dimension two has peculiarities, which force us to use different ideas from the corresponding higher dimensional case. We 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. This is possible via an argument of compactness in geometric measure theory. Indeed we develop a definitive theory, which allows us to circumvent the shortening curve flow approach of previous authors at the cost of some applications of geometric measure theory and Ascoli-Arzela's Theorem.Comment: Example 5.4 is new; Theorem 4.5 is reformulated; 29 pages; 7 figure

Local H\"older continuity of the isoperimetric profile in complete noncompact Riemannian manifolds with bounded geometry

For a complete noncompact connected Riemannian manifold with bounded geometry $M^n$, we prove that the isoperimetric profile function $I_{M^n}$ is a locally $\left(1-\frac{1}{n}\right)$-H\"older continuous function and so in particular it is continuous. Here for bounded geometry we mean that $M$ have $Ricci$ curvature bounded below and volume of balls of radius $1$, uniformly bounded below with respect to its centers. We prove also the equivalence of the weak and strong formulation of the isoperimetric profile function in complete Riemannian manifolds which is based on a lemma having its own interest about the approximation of finite perimeter sets with finite volume by open bounded with smooth boundary ones of the same volume. Finally the upper semicontinuity of the isoperimetric profile for every metric (not necessarily complete) is shown.Comment: 17 pages. This is an improvement of the result about the continuity of the isoperimetric profile function contained in arXiv:1404.3245. The arguments used here are a slight modification of the ones already used in arXiv:1404.3245. The paper is already accepted in Geometriae Dedicat

Continuity and differentiability properties of the isoperimetric profile in complete noncompact Riemannian manifolds with bounded geometry

For a complete noncompact connected Riemannian manifold with bounded geometry $M^n$, we prove that the isoperimetric profile function $I_{M^n}$ is continuous. Here for bounded geometry we mean that $M$ have $Ricci$ curvature bounded below and volume of balls of radius $1$, uniformly bounded below with respect to its centers. Then under an extra hypothesis on the geometry of $M$, we apply this result to prove some differentiability property of $I_M$ and a differential inequality satisfied by $I_M$, extending in this way well known results for compact manifolds, to this class of noncompact complete Riemannian manifolds with bounded geometry.Comment: 31 pages, 8 figures. A new entire section, namely section $2$ is added to give the details of the equivalence between the weak and strong formulation of the isoperimetric problem, some typos are correcte

The isoperimetric problem of a complete Riemannian manifolds with a finite number of C0C^0-asymptotically Schwarzschild ends

We study the problem of existence of isoperimetric regions for large volumes, in $C^0$-locally asymptotically Euclidean Riemannian manifolds with a finite number of $C^0$-asymptotically Schwarzschild ends. Then we give a geometric characterization of these isoperimetric regions, extending previous results contained in [EM13b], [EM13a], and [BE13]. Moreover strengthening a little bit the speed of convergence to the Schwarzschild metric we obtain existence of isoperimetric regions for all volumes for a class of manifolds that we named $C^0$-strongly asymptotic Schwarzschild, extending results of [BE13]. Such results are of interest in the field of mathematical general relativity.Comment: Revised version, 23 pages, 6 figure

Generalized compactness for finite perimeter sets and applications to the isoperimetric problem

For a complete noncompact connected Riemannian manifold with bounded geometry, we prove a compactness result for sequences of finite perimeter sets with uniformly bounded volume and perimeter in a larger space obtained by adding limit manifolds at infinity. We extends previous results contained in [Nar14a], in such a way that the generalized existence theorem, Theorem 1 of [Nar14a] is actually a generalized compactness theorem. The suitable modifications to the arguments and statements of the results in [Nar14a] are non-trivial. As a consequence we give a multipointed version of Theorem 1.1 of [LW11], and a simple proof of the continuity of the isoperimetric profile function.Comment: Existence of isoperimetric region, isoperimetric profile, metric geometry. arXiv admin note: substantial text overlap with arXiv:1210.1328, arXiv:1503.0236