Sharp integral bound of scalar curvature on $3$-manifolds

TL;DR

This paper establishes an asymptotic upper bound for the integral of scalar curvature over large geodesic balls in 3-manifolds with nonnegative Ricci curvature, assuming scalar curvature is bounded between positive constants.

Contribution

It provides a sharp integral bound for scalar curvature on 3-manifolds under specific curvature conditions, extending understanding of geometric analysis in such manifolds.

Findings

Integral of scalar curvature bounded by 8πR asymptotically

Bound holds for manifolds with nonnegative Ricci curvature

Scalar curvature bounded between positive constants

Abstract

It is shown that the integral of the scalar curvature on a geodesic ball of radius $R$ in a three-dimensional complete manifold with nonnegative Ricci curvature is bounded above by $8\pi R$ asymptotically for large $R$ provided that the scalar curvature is bounded between two positive constants.