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Some aspects of Cheng-Yau gradient estimates

Qixuan Hu, Chengjie Yu

TL;DR

The paper extends Cheng-Yau gradient estimates to complete manifolds with a Ricci curvature lower bound, obtaining locally sharp bounds for positive harmonic and conformal-Laplacian solutions. It proves explicit gradient inequalities on space forms via conformal reductions and provides sharp two-dimensional surface bounds with rigidity, as well as a higher-dimensional local bound with factor $(2n-3)$ for $n\ge 3$, together with corresponding monotonicity formulas for positive harmonic functions. The results employ the conformal covariance of the conformal Laplacian, comparison geometry with $sn_K$ and $cs_K$, and maximum principle techniques to yield Poisson-kernel rigidity and Harnack-type controls along geodesics. These findings enhance gradient-estimate theory under curvature constraints and offer sharp comparison principles in geometric analysis.

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.

Some aspects of Cheng-Yau gradient estimates

TL;DR

The paper extends Cheng-Yau gradient estimates to complete manifolds with a Ricci curvature lower bound, obtaining locally sharp bounds for positive harmonic and conformal-Laplacian solutions. It proves explicit gradient inequalities on space forms via conformal reductions and provides sharp two-dimensional surface bounds with rigidity, as well as a higher-dimensional local bound with factor for , together with corresponding monotonicity formulas for positive harmonic functions. The results employ the conformal covariance of the conformal Laplacian, comparison geometry with and , and maximum principle techniques to yield Poisson-kernel rigidity and Harnack-type controls along geodesics. These findings enhance gradient-estimate theory under curvature constraints and offer sharp comparison principles in geometric analysis.

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.

Paper Structure

This paper contains 3 sections, 7 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Sharp gradient estimate on space forms and surfaces
  3. Gradient estimates and monotonicity formulas

Key Result

Theorem 1.1

Let $B(R)$ be the ball of radius $R$ centered at the origin of $\mathbb{R}^n$ with $n\geq 2$ and $u$ be a positive harmonic function on $B(R)$. Then, Moreover, the equality of eq-CY-Eu holds for some $x\in B(R)$ if and only if for some $y\in \partial B(R)$ and $\lambda >0$. Here is the Poisson kernel of $B(R)$.

Theorems & Definitions (11)

  • Theorem 1.1: Xu Xu
  • Theorem 1.2: Xu Xu
  • Theorem 1.3: Hu-Xu-Yu HXY
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.1
  • proof : Proof of Theorem \ref{['thm-CY-conf']}
  • proof : Proof of Theorem \ref{['thm-CY-surface']}
  • proof : Proof of Theorem \ref{['thm-CY-manifold']}
  • ...and 1 more