Singular metrics with negative scalar curvature

TL;DR

This paper extends the understanding of scalar curvature rigidity to singular metrics with negative Yamabe invariant, showing that certain singular metrics still exhibit Einstein-like properties in various dimensions.

Contribution

It demonstrates that metrics with edge singularities and negative Yamabe invariant retain scalar curvature rigidity, generalizing known results from smooth to singular settings.

Findings

Edge singularities with cone angles ≤ 2π preserve scalar curvature rigidity.

In three dimensions, isolated point singularities also maintain Einstein-like properties.

The results hold in all dimensions for metrics with negative Yamabe invariant.

Abstract

Motivated by the work of Li and Mantoulidis, we study singular metrics which are uniformly Euclidean $(L^\infty)$ on a compact manifold $M^n$ ($n\ge 3$) with negative Yamabe invariant $\sigma(M)$. It is well-known that if $g$ is a smooth metric on $M$ with unit volume and with scalar curvature $R(g)\ge \sigma(M)$, then $g$ is Einstein. We show, in all dimensions, the same is true for metrics with edge singularities with cone angles $\leq 2\pi$ along codimension-2 submanifolds. We also show in three dimension, if the Yamabe invariant of connected sum of two copies of $M$ attains its minimum, then the same is true for $L^\infty$ metrics with isolated point singularities.