Uniform gradient estimates on manifolds with a boundary and applications

TL;DR

This paper develops uniform gradient estimates for heat semigroups on manifolds with boundary and applies these results to derive isoperimetric inequalities and spectral projection estimates.

Contribution

It provides new uniform gradient estimates for heat semigroups with boundary conditions and applies them to isoperimetric and spectral analysis on manifolds.

Findings

Established uniform gradient bounds for Dirichlet and Neumann heat semigroups.

Derived isoperimetric inequalities using Ledoux's argument.

Obtained gradient estimates for spectral projection operators.

Abstract

We revisit the problem of obtaining uniform gradient estimates for Dirichlet and Neumann heat semigroups on Riemannian manifolds with boundary. As applications, we obtain isoperimetric inequalities, using Ledoux's argument, and uniform quantitative gradient estimates, firstly for $C^2_b$ functions with boundary conditions and then for the unit spectral projection operators of Dirichlet and Neumann Laplacians.