Isoperimetric Inequality on Manifolds with Quadratically Decaying Curvature

TL;DR

This paper explores how quadratic decay in Ricci curvature influences Sobolev and isoperimetric inequalities on manifolds, revealing new links between geometry and analysis.

Contribution

It establishes conditions linking heat kernel decay to isoperimetric inequalities and demonstrates the existence of isoperimetric sets in generalized Grushin spaces.

Findings

Heat kernel decay implies isoperimetric inequalities

Existence of isoperimetric sets in Grushin spaces

New geometric-functional analysis connections

Abstract

In this paper, we investigate the reverse improvement property of Sobolev inequalities on manifolds with quadratically decaying Ricci curvature. Specifically, we establish conditions under which the uniform decay of the heat kernel implies the validity of an isoperimetric inequality. As an application, we demonstrate the existence of isoperimetric sets in generalized Grushin spaces. Our approach is built on a weak-type Sobolev inequality, gradient estimates on remote balls, and a Hardy-type gluing technique. This method provides new insights into the deep connections between geometric and functional analysis.