Appears in collection : Not Only Scalar Curvature Seminar

We 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 non-negative Ricci curvature. 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 $b=u^{2−n}$ to the pole. On $ {\bf{R}}^n$, the level sets of $b$ are spheres and $|\nabla b|=1$. If the Ricci curvature is non-negative, then I showed earlier that this leads to the sharp gradient estimate $|\nabla b|\leq 1$. In joint work with Minicozzi we show that the average of $|\nabla b|$ is $\leq 1$ on a three manifold with non-negative scalar curvature. The average is over any level set of $b$ and if the average is one on even one level set, then $M={\bf{R}}^3$.