Lipschitz rigidity for scalar curvature

Abstract

Let $M$ be a closed smooth connected spin manifold of even dimension $n$, let $g$ be a Riemannian metric of regularity $W^{1,p}$, $p>n$, on $M$ whose distributional scalar curvature in the sense of Lee–LeFloch is bounded below by $n(n-1)$, and let $f:(M,g)\to {\mathbb{S}}^n$ be a $1$-Lipschitz continuous (not necessarily smooth) map of nonzero degree to the unit $n$-sphere. Then $f$ is a metric isometry. This generalizes a result of Llarull (1998) and answers in the affirmative a question of Gromov (2019) in his Four lectures. Our proof is based on spectral properties of Dirac operators for low regularity Riemannian metrics and twisted with Lipschitz bundles. We argue that the existence of a nonzero harmonic spinor field forces $f$ to be quasiregular in the sense of Reshetnyak, and in this way connect the powerful theory for quasiregular maps to the Atiyah–Singer index theorem.