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A lecture by Tobias Colding from MIT explores gradient estimates for scalar curvature, explaining how these estimates control the rate of change of functions on manifolds to enable deeper geometric analysis. Learn about the Cheng-Yau gradient bounds on manifolds with non-negative Ricci curvature and why they aren't sharp. Discover how Green's functions can define regularized distances to poles, and how on Euclidean space the level sets form spheres with gradient magnitude equal to one. Follow Colding's explanation of his earlier work showing that non-negative Ricci curvature leads to the sharp gradient estimate where the gradient magnitude is less than or equal to one. The lecture culminates in presenting joint work with Minicozzi demonstrating that on three-manifolds with non-negative scalar curvature, the average of the gradient magnitude over any level set is less than or equal to one—with equality on even one level set implying the manifold must be Euclidean space.
