Lukas Schoenlinner | Searcharxiv

arXiv2026

Lipschitz rigidity for scalar curvature on singular manifolds in odd dimensions

The main result of this article is a Llarull-type rigidity statement for scalar curvature on Riemannian spin manifolds with cone-like singularities in odd dimensions. The even dimensional analog was proven in an earlier work together with Simone Cecchini, Bernhard Hanke and Thomas Schick using index theory and the analysis of abstract cone operators, which applies to Dirac operators associated with generalized cone metrics. We will extend the analysis of abstract cone operators, apply it to twisted Dirac operators on singular manifolds and combine it with a spectral flow argument to prove the main result.

math.DG↗

arXiv2025

Abstract cone operators and Lipschitz rigidity for scalar curvature on singular manifolds

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-type 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 Baer using an upper estimate for the smallest Dirac eigenvalue.

math.DG↗