Isoperimetric Inequality on Manifolds with Quadratically Decaying Curvature
Dangyang He
TL;DR
The paper investigates how uniform heat-kernel decay enforces isoperimetric inequalities on manifolds with quadratically decaying Ricci curvature, addressing the reverse-improvement problem for Sobolev inequalities. It develops a framework based on weak-type Sobolev inequalities, remote-ball gradient estimates, and a Hardy-type gluing technique to patch local geometric-analytic control into global isoperimetric bounds. The results cover both doubling and non-doubling manifolds, including connected sums like $\mathbb{R}^n \# \mathbb{R}^n$ and manifolds with ends, and extend to Grushin-type spaces where the Grushin operator and its curvature-dimension structure yield generalized isoperimetric inequalities with explicit homogeneous dimensions. This work deepens the connection between geometric curvature decay, heat-kernel regularity, and functional-analytic inequalities, offering tools to tackle isoperimetric questions in noncompact and singular geometric settings.
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.