Optimal Hamilton-type gradient estimates for the heat equation on noncompact manifolds

TL;DR

This paper establishes optimal Hamilton-type gradient estimates for the heat equation on noncompact Riemannian manifolds, leading to new sharp inequalities and continuity estimates.

Contribution

It provides the first localized and global noncompact Hamilton gradient estimates with optimal bounds, improving previous results significantly.

Findings

Derived localized and global gradient estimates for the heat equation.

Established a sharp, space-only local pseudo-Harnack inequality.

Provided estimates for the spatial modulus of continuity of solutions.

Abstract

We derive localized and global noncompact versions of Hamilton's gradient estimate for positive solutions to the heat equation on Riemannian manifolds with Ricci curvature bounded below. Our estimates are essentially optimal and significantly improve on all previous estimates of this type. As applications, we derive a new and sharp, space only, local pseudo-Harnack inequality, as well as estimates of the spatial modulus of continuity of solutions.