Sharp modulus of continuity for parabolic equations on manifolds and lower bounds for the first eigenvalue

TL;DR

This paper establishes sharp continuity estimates for heat equation solutions on manifolds with Ricci curvature bounds and uses these results to derive optimal lower bounds for the first Laplacian eigenvalue based on the manifold's diameter.

Contribution

It provides new sharp modulus of continuity estimates for parabolic equations on manifolds and simplifies the proof of eigenvalue lower bounds.

Findings

Sharp modulus of continuity estimates for heat solutions

Optimal lower bounds for the first eigenvalue based on diameter

Simplified proof techniques for eigenvalue bounds

Abstract

We derive sharp estimates on modulus of continuity for solutions of the heat equation on a compact Riemannian manifold with a Ricci curvature bound, in terms of initial oscillation and elapsed time. As an application, we give an easy proof of the optimal lower bound on the first eigenvalue of the Laplacian on such a manifold as a function of diameter.