Scalar curvature and Dirac operator for some singular spaces

TL;DR

This paper investigates the conditions under which scalar curvature measures and Dirac operators can be defined on smooth manifolds with potentially singular Riemannian metrics, expanding geometric analysis into singular spaces.

Contribution

It provides sufficient conditions for defining scalar curvature measures and Dirac operators on manifolds with singular metrics, advancing the understanding of geometric analysis in singular settings.

Findings

Established criteria for scalar curvature measures on singular spaces

Defined Dirac operators in the presence of metric singularities

Extended geometric analysis tools to broader classes of manifolds

Abstract

We look at smooth manifolds equipped with a possibly singular Riemannian metric. We give sufficient conditions for the existence of scalar curvature measures and Dirac operators.