A scalar-mean curvature comparison theorem for manifolds with iterated conical singularities

TL;DR

This paper establishes a scalar-mean curvature comparison theorem for spin manifolds with iterated conical singularities using Dirac operator index theory, leading to rigidity and positive mass theorems.

Contribution

It introduces a novel approach employing Dirac operator index theory to analyze manifolds with complex singularities, extending classical comparison results.

Findings

Proves a scalar-mean curvature comparison theorem for singular manifolds.

Establishes a rigidity theorem for Euclidean domains with singularities.

Proves a spin positive mass theorem for asymptotically flat manifolds with singularities.

Abstract

We use the Dirac operator method to prove a scalar-mean curvature comparison theorem for spin manifolds which carry iterated conical singularities. Our approach is to study the index theory of a twisted Dirac operator on such singular manifolds. A dichotomy argument is used to prove the comparison theorem without knowing precisely the index of the twisted Dirac operator. This framework also enables us to prove a rigidity theorem of Euclidean domains and a spin positive mass theorem for asymptotically flat manifolds with iterated conical singularities.