Positive scalar curvature and isolated conical singularity

TL;DR

This paper proves that certain manifolds with isolated conical singularities cannot support metrics with nonnegative scalar curvature that are positive somewhere, extending classical scalar curvature results to singular settings.

Contribution

It establishes nonexistence results for metrics with positive scalar curvature on manifolds with isolated conical singularities, generalizing Geroch's theorem to singular spaces.

Findings

No metric with nonnegative scalar curvature and positivity at some point exists on manifolds with isolated conical singularities.

Scalar-flat metrics with isolated conical singularities must be flat and extend smoothly.

The cross section of the conical singularity may not be spherical, broadening the class of singularities considered.

Abstract

We prove a Geroch type result for isolated conical singularity. Namely, we show that there is no Riemannian metric $g$ on $ X \# T^n $ with an isolated conical singularity which has nonnegative scalar curvature on the regular part, and is positive at some point. In particular, this implies that there is no metric on tori with an isolated conical singularity and positive scalar curvature. We also prove that a scalar flat Riemannian metric $g$ on $X \# T^n$ with finitely many isolated conical singularities must be flat, and extend smoothly across the singular points. We do not a priori assume that a conically singular point on $X$ is a manifold point; i.e., the cross section of the conical singularity may not be spherical.