Gradient estimates for scalar curvature
Tobias Holck Colding, William P. Minicozzi
TL;DR
The paper studies sharp average gradient bounds for the Green's-function-based regularized distance on open 3-manifolds with nonnegative scalar curvature. By defining $A_1(r) = r^{-2} \int_{b=r} |\nabla b|^2$, along with auxiliary quantities $B_1$, $B_2$, and $S_1$, the authors develop differential inequalities and level-set geometry analyses that connect the gradient of $b$ to curvature and Hessian data. They prove that $A_1(r) \le 4\pi$ under $S \ge 0$, with equality implying rigidity $M \cong \mathbb{R}^3$, and they establish quadratic growth of $A_1$ if it ever exceeds $4\pi$, via a vector-field method and integral-inequality arguments. Additionally, they derive an averaged bound for the scalar curvature along level sets: $(1/r) \int_0^r S_1(s) ds \le 48\pi$, linking gradient control to global curvature distribution. These results fuse monotonicity formulas with geometric-analytic tools to yield sharp, rigidity-informing gradient estimates in three dimensions.
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 $\geq 0$. 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 $\bf{R}^n$, the level sets of $b$ are spheres and $|\nabla b|=1$. If $\text{Ric} \geq 0$, then [C3] proved the sharp gradient estimate $|\nabla b| \leq 1$. We show that the average of $|\nabla b|$ is $\leq 1$ 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 $M=\bf{R}^3$.