Integrals and Rigidity on Manifolds with Nonnegative Ricci Curvature

TL;DR

This paper establishes a sharp mean value inequality for superharmonic functions on manifolds with nonnegative Ricci curvature, providing new rigidity results and an explicit asymptotic formula, with applications to Hamilton's pinching conjecture.

Contribution

It generalizes classical inequalities by removing radius restrictions and derives an explicit asymptotic formula for weighted scalar curvature on such manifolds.

Findings

Proves a sharp mean value inequality for superharmonic functions.

Derives an explicit asymptotic formula for weighted scalar curvature.

Provides a new proof of Hamilton's pinching conjecture in this setting.

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.