Scalar curvature rigidity of spheres with subsets removed and $L^\infty$ metrics

TL;DR

This paper establishes scalar curvature rigidity results for $L^ abla$ metrics on spheres with certain subsets removed, introducing the wrapping property, and extends these results to tori, with implications for positive mass theorems.

Contribution

It introduces the wrapping property and proves scalar curvature rigidity for $L^ abla$ metrics on spheres and tori with subsets removed, extending previous rigidity results.

Findings

Scalar curvature rigidity holds for $L^ abla$ metrics on spheres with subsets satisfying the wrapping property.

The techniques apply to tori with similar subset conditions, extending rigidity results.

A positive mass theorem is derived for asymptotically flat spin manifolds with $L^ abla$ metrics.

Abstract

We prove the scalar curvature rigidity for $L^\infty$ metrics on $\mathbb S^n\backslash\Sigma$, where $\mathbb S^n$ is the $n$-dimensional sphere with $n\geq 3$ and $\Sigma$ is a closed subset of $\mathbb S^n$ of codimension at least $\frac{n}{2}+1$ that satisfies the wrapping property. The notion of wrapping property was introduced by the second author for studying related scalar curvature rigidity problems on spheres. For example, any closed subset of $\mathbb S^n$ contained in a hemisphere and any finite subset of $\mathbb S^n$ satisfy the wrapping property. The same techniques also apply to prove an analogous scalar rigidity result for $L^\infty$ metrics on tori that are smooth away from certain subsets of codimension at least $\frac{n}{2}+1$. As a corollary, we obtain a positive mass theorem for complete asymptotically flat spin manifolds with arbitrary ends for $L^\infty$ metrics.