Sharp Hamilton's Laplacian estimate for the heat kernel on complete manifolds

TL;DR

This paper extends Hamilton's Laplacian estimates for the heat equation to complete noncompact manifolds with nonnegative Ricci curvature, providing sharp bounds on the heat kernel's Laplacian form.

Contribution

It introduces Hamilton's Laplacian estimates for the heat equation on such manifolds and derives sharp heat kernel bounds using Li-Yau estimates.

Findings

Laplacian estimates for the heat equation on noncompact manifolds

Sharp bounds on the heat kernel's Laplacian form

Results are optimal in the order of the time parameter

Abstract

In this paper we give Hamilton's Laplacian estimates for the heat equation on complete noncompact manifolds with nonnegative Ricci curvature. As an application, combining Li-Yau's lower and upper bounds of the heat kernel, we give an estimate on Laplacian form of the heat kernel on complete manifolds with nonnegative Ricci curvature that is sharp in the order of time parameter for the heat kernel on the Euclidean space.