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Explore a lecture on Lipschitz rigidity for scalar curvature delivered by Bernhard Hanke at the Hausdorff Center for Mathematics. Discover how lower scalar curvature bounds on spin Riemannian manifolds exhibit remarkable extremality and rigidity phenomena determined by spectral properties of Dirac type operators. Learn about Llarull's fundamental result stating that no smooth Riemannian metric on the n-sphere can dominate the round metric while having scalar curvature greater than or equal to the round metric's scalar curvature, except for the round metric itself. Examine how this result extends to smooth comparison maps from spin Riemannian manifolds to round spheres. Delve into joint work with Cecchini-Schick and Cecchini-Schick-Schönlinner that generalizes these results to Riemannian metrics with regularity less than C¹ and Lipschitz comparison maps, addressing a question posed by Gromov. Understand the application of distributional scalar curvature introduced by Lee-LeFloch and spectral properties of Lipschitz Dirac operators, revealing how nonzero harmonic spinor fields force comparison maps to be quasiregular in Reshetnyak's sense, creating an unexpected connection between spin geometry and quasiconformal mapping theory.
