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

TL;DR

This paper investigates the behavior of maximal solutions to the Ricci flow on complete Riemannian manifolds of finite volume, introducing new methods for analyzing isoperimetric ratios, especially in higher dimensions.

Contribution

It develops a novel geometric measure theory approach to study Hamilton's isoperimetric ratio in non-compact manifolds, bypassing traditional curve flow techniques.

Findings

Existence of separating regions in manifolds of dimension two.

Reduction of the isoperimetric problem to an auxiliary minimization problem in higher dimensions.

Development of a comprehensive geometric measure theory framework for this problem.

Abstract

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.