  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 $f \colon (M,g) \to \mathbb{S}^n$ be a $1$-Lipschitz
continuous (not necessarily smooth) map of non-zero 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 non-zero 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.

Differential Geometry;Metric Geometry