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Integrals and Rigidity on Manifolds with Nonnegative Ricci Curvature

Zixuan Chen, Guoyi Xu, Shuai Zhang

TL;DR

The paper addresses sharp mean value control on manifolds with nonnegative Ricci curvature, removing the radius limitation of Schoen–Yau and linking mean value equality to Euclidean rigidity. It introduces a weighted framework via the Green function, proves a radius-free sharp mean value inequality and its rigidity, and develops a detailed asymptotic study of level-set geometry for the Green-function-derived height function $b$. A central achievement is an explicit scale-invariant formula for the weighted integral of scalar curvature in dimension three, $\lim_{r\to\infty} \frac{\int_{b\le r} R\cdot |\nabla b|}{r}=8\pi[1-\mathrm{V}_M]$, which yields an alternative proof of Hamilton’s pinching conjecture in this setting. The results hinge on new tools: a generalized divergence theorem for geodesic balls, a monotone function $F(t)$, and a careful asymptotic analysis of $b$-level sets, tying curvature, volume growth, and topology together in nonparabolic, nonnegatively curved manifolds.

Abstract

We prove the general sharp mean value inequality for non-negative superharmonic functions and its corresponding rigidity, which removes the radius restriction of Schoen-Yau's classical result about this inequality. And we obtain an explicit formula of the asymptotic scaling invariant integral of weighted scalar curvature, on three dimensional complete Riemannian manifolds with non-negative Ricci curvature and maximal volume growth. As an application, we use this formula to give another proof of Hamilton's pinching conjecture in this case.

Integrals and Rigidity on Manifolds with Nonnegative Ricci Curvature

TL;DR

The paper addresses sharp mean value control on manifolds with nonnegative Ricci curvature, removing the radius limitation of Schoen–Yau and linking mean value equality to Euclidean rigidity. It introduces a weighted framework via the Green function, proves a radius-free sharp mean value inequality and its rigidity, and develops a detailed asymptotic study of level-set geometry for the Green-function-derived height function . A central achievement is an explicit scale-invariant formula for the weighted integral of scalar curvature in dimension three, , which yields an alternative proof of Hamilton’s pinching conjecture in this setting. The results hinge on new tools: a generalized divergence theorem for geodesic balls, a monotone function , and a careful asymptotic analysis of -level sets, tying curvature, volume growth, and topology together in nonparabolic, nonnegatively curved manifolds.

Abstract

We prove the general sharp mean value inequality for non-negative superharmonic functions and its corresponding rigidity, which removes the radius restriction of Schoen-Yau's classical result about this inequality. And we obtain an explicit formula of the asymptotic scaling invariant integral of weighted scalar curvature, on three dimensional complete Riemannian manifolds with non-negative Ricci curvature and maximal volume growth. As an application, we use this formula to give another proof of Hamilton's pinching conjecture in this case.
Paper Structure (6 sections, 29 theorems, 188 equations)

This paper contains 6 sections, 29 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. Mean value formula
  3. Divergence theorem for geodesic balls
  4. Sharp mean value inequality
  5. Weighted volume and its asymptotic behavior
  6. Integral formula for scalar curvature

Key Result

Theorem 1.1

Let $(M^n, g)$ be a complete Riemannian manifold with $\operatorname{Ric} \ge 0$. Suppose $f\in C^\infty(M^n)- \{0\}$ satisfies $f\geq 0$ and $\Delta \leq 0$. Then Furthermore, if equality holds for some $x$ and $r>0$, then the geodesic ball $B_x(r)$ is isometric to the Euclidean ball $B^n(r) \subseteq \mathbb{R}^n$.

Theorems & Definitions (42)

  • Theorem 1.1: Sharp Mean Value Inequality and Rigidity
  • Remark 1.2
  • Example 1.4
  • Theorem 1.5
  • Remark 1.6
  • Conjecture 1.7
  • Remark 1.8
  • Conjecture 1.9
  • Corollary 1.10
  • Remark 1.11
  • ...and 32 more