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Using the index theory for twisted Dirac operators acting on sections of Lipschitz bundles over non-compact manifolds, we prove  Llarull-type comparison results in scalar curvature geometry. They apply to spin Riemannian manifolds with cone-like singularities and Lipschitz comparison maps to spheres. We use the  language of abstract cone operators which are introduced and studied in a general functional analytic setting and which may be of independent interest. Applying our discussion to spherical suspensions of odd-dimensional closed manifolds, we  generalize a  Lipschitz rigidity  result of the first three named authors from even to odd dimensions. Under stronger conditions, this has already been shown by Lee-Tam using geometric flows and by Bär using an upper estimate for the smallest Dirac eigenvalue.

Related project(s):
15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II