Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds

TL;DR

This paper establishes sharp localized gradient estimates for positive solutions to the heat equation on noncompact manifolds, extending classical results and proving a Liouville theorem under growth conditions, with implications for heat kernel analysis.

Contribution

It introduces a localized gradient estimate for the heat equation on noncompact manifolds and generalizes Yau's Liouville theorem under growth conditions, addressing open problems in long-time heat kernel estimates.

Findings

Derived sharp localized gradient estimates for heat solutions.

Generalized Yau's Liouville theorem with growth conditions.

Proved a long-time gradient estimate for heat kernel.

Abstract

We derive a sharp, localized version of elliptic type gradient estimates for positive solutions (bounded or not) to the heat equation. These estimates are akin to the Cheng-Yau estimate for the Laplace equation and Hamilton's estimate for bounded solutions to the heat equation on compact manifolds. As applications, we generalize Yau's celebrated Liouville theorem for positive harmonic functions to positive eternal solutions of the heat equation, under certain growth condition. Surprisingly, this Liouville theorem for the heat equation does not hold even in ${\bf R}^n$ without such a condition. We also prove a sharpened long time gradient estimate for the log of heat kernel on noncompact manifolds. This has been an open problem in view of the well known estimates in the compact, short time case.