Scalar and mean curvature comparison via the Dirac operator

We use the Dirac operator technique to establish sharp distance estimates for compact spin manifolds under lower bounds on the scalar curvature in the interior and on the mean curvature of the boundary.In the situations we consider, we thereby give refined answers to questions on metric inequalities recently proposed by Gromov.This includes optimal estimates for Riemannian bands and for the long neck problem.In the case of bands over manifolds of non-vanishing A-genus, we establish a rigidity result stating that any band attaining the predicted upper bound is isometric to a particular warped product over some spin manifold admitting a parallel spinor.Furthermore, we establish scalar-and mean curvature extremality results for certain log-concave warped products.The latter includes annuli in all simply-connected space forms.On a technical level, our proofs are based on new spectral estimates for the Dirac operator augmented by a Lipschitz potential together with local boundary conditions.