Sharp Li-Yau type gradient estimates on hyperbolic spaces

TL;DR

This paper derives sharp Li-Yau type gradient estimates for the heat equation on hyperbolic spaces, extending previous results to spaces with negative curvature using explicit heat kernel formulas.

Contribution

It introduces a general framework for Li-Yau estimates and provides explicit sharp estimates on three-dimensional hyperbolic space and other hyperbolic spaces.

Findings

Sharp Li-Yau gradient estimate on 3D hyperbolic space derived.

General form of Li-Yau estimate reduces to heat kernel analysis.

Optimal estimates obtained for hyperbolic spaces.

Abstract

In this paper, motivated by the works of Bakry et. al in finding sharp Li-Yau type gradient estimate for positive solutions of the heat equation on complete Riemannian manifolds with nonzero Ricci curvature lower bound, we first introduce a general form of Li-Yau type gradient estimate and show that the validity of such an estimate for any positive solutions of the heat equation reduces to the validity of the estimate for the heat kernel of the Riemannian manifold. Then, a sharp Li-Yau type gradient estimate on the three dimensional hyperbolic space is obtained by using the explicit expression of the heat kernel and some optimal Li-Yau type gradient estimates on general hyperbolic spaces are obtained.