Lipschitz rigidity for scalar curvature

Lower scalar curvature bounds on spin Riemannian manifolds exhibit remarkable rigidity properties determined by spectral properties of Dirac operators. For instance, a fundamental result of Llarull states that there is no smooth Riemannian metric on the n-sphere which dominates the round metric and whose scalar curvature is greater than or equal to the scalar curvature of the round metric, except the round metric itself. A similar result holds for smooth comparison maps from spin Riemannian manifolds to round spheres. In a joint work with Simone Cecchini and Thomas Schick, we generalize this result to Riemannian metrics with regularity less than C^1 and Lipschitz comparison maps, answering a question of Gromov in his "Four Lectures". To this end, we rely on a notion of scalar curvature in the distributional sense introduced by Lee-LeFloch and on spectral properties of Lipschitz Dirac operators. It turns out that the existence of a nonzero harmonic spinor field - guaranteed by the Atiyah-Singer index theorem - forces the given comparison map to be quasiregular in the sense of Reshetnyak. Thus we build an unexpected bridge from spin geometry to the theory of quasiconformal mappings.