Gradient estimates for scalar curvature

Abstract

A gradient estimate is a crucial tool used to control the rate of change of a function on a manifold, paving the way for deeper analysis of geometric properties. A celebrated result of Cheng and Yau gives gradient bounds on manifolds with Ricci curvature $\ge 0$. The Cheng-Yau bound is not sharp, but there is a gradient sharp estimate. To explain this, a Green's function $u$ on a manifold can be used to define a regularized distance $b=u^{\frac{1}{2-n}}$ to the pole. On ${\mathbf{R}}^{\mathbf{n}}$, the level sets of $b$ are spheres and $|\nabla b|=1$. If $\text{Ric}\ge 0$, then [C3] proved the sharp gradient estimate $|\nabla b|\le 1$. We show that the average of $|\nabla b|$ is $\le 1$ on a three manifold with nonnegative scalar curvature. The average is over any level set of $b$ and if the average is one on even one level set, then $M={\mathbf{R}}^3$.