Calculus of Variations and Geometric Measure Theory

A. Figalli - L. Rifford - C. Villani

Nearly round spheres look convex

created by figalli on 17 Jan 2011
modified on 02 Oct 2011


Accepted Paper

Inserted: 17 jan 2011
Last Updated: 2 oct 2011

Journal: Amer. J. Math.
Year: 2011


We prove that a Riemannian manifold $(M,g)$, close enough to the round sphere in the $C^4$ topology, has uniformly convex injectivity domains - so $M$ appears uniformly convex in any exponential chart. The proof is based on the Ma-Trudinger-Wang nonlocal curvature tensor, which originates from the regularity theory of optimal transport.