Cheeger constants of surfaces and isoperimetric inequalities

TL;DR

This paper investigates bounds on Cheeger constants of surfaces, explores isoperimetric inequalities in various dimensions, and generalizes Gromov's results, providing new insights into geometric analysis and isoperimetric profiles.

Contribution

It establishes bounds on Cheeger constants based on area, extends Gromov's isoperimetric results to non-compact surfaces, and analyzes the filling volume function in dimension 3 under specific conditions.

Findings

Cheeger constant of compact surfaces is bounded by a function of area

Isoperimetric profile growth rate is at least linear if faster than sqrt(t)

Filling function in dimension 3 is almost linear under certain conditions

Abstract

We show that the Cheeger constant of compact surfaces is bounded by a function of the area. We apply this to isoperimetric profiles of bounded genus non-compact surfaces, to show that if their isoperimetric profile grows faster than $\sqrt t$, then it grows at least as fast as a linear function. This generalizes a result of Gromov for simply connected surfaces. We study the isoperimetric problem in dimension 3. We show that if the filling volume function in dimension 2 is Euclidean, while in dimension 3 is sub-Euclidean and there is a $g$ such that minimizers in dimension 3 have genus at most $g$, then the filling function in dimension 3 is `almost' linear.