Some aspects of Cheng-Yau gradient estimates

TL;DR

This paper extends Cheng-Yau gradient estimates to surfaces with lower Ricci curvature bounds and derives new pointwise estimates for higher-dimensional manifolds, leading to applications like monotonicity formulas for harmonic functions.

Contribution

It generalizes Cheng-Yau gradient estimates to broader geometric contexts and provides new pointwise estimates and applications in Riemannian geometry.

Findings

Extended gradient estimates to surfaces with lower Ricci curvature bounds

Derived pointwise gradient estimates for higher-dimensional manifolds

Established monotonicity formulas for positive harmonic functions

Abstract

In this note, we extend the rigidity of Cheng-Yau gradient estimate in \cite{HXY} to surfaces with lower Ricci curvature bound. Motivated by these sharp Cheng-Yau gradient estimates, pointwise Cheng-Yau gradient estimates for higher dimensional Riemannian manifolds are obtained, and as their applications, monotonicity formulas for positive harmonic functions are obtained.