Lipschitz rigidity for scalar curvature

Let M be a closed smooth connected spin manifold of even dimension n , let g be a Riemannian metric of regularity W1,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) → 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.