2025/01/29 by Tobias Colding, Colding, Tobias Holck, William P. Minicozzi +1 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Advanced Operator Algebra Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.2501.17947
openalex publication_date 2025/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ≥ 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(1)/(2-n) to the pole. On \bfRn, the level sets of b are spheres and |∇ b|=1. If Ric ≥ 0, then [C3] proved the sharp gradient estimate |∇ b| ≤ 1. We show that the average of |∇ b| is ≤ 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=\bfR3.