Lipschitz rigidity for scalar curvature

I will discuss the following scalar curvature rigidity result for the round sphere in the low regularity setting. Let M be a closed smooth connected spin manifold of dimension n, and let g be a Riemannian metric on M of Sobolev regularity W1,p, for p>n, whose distributional scalar curvature in the sense of Lee-LeFloch is bounded below by n(n−1). Let f:(M,g)→Sn be a Lipschitz continuous map, that is area-nonincreasing almost everywhere and has non-zero degree. Then f is a metric isometry. In this talk, I will focus on the odd-dimensional case, using the analysis of Lipschitz Dirac operators on manifolds with conical singularities.

This is based on joint work with Bernhard Hanke, Thomas Schick, and Lukas Schoenlinner