Llarull's theorem on punctured sphere with $L^\infty$ metric

TL;DR

This paper extends Llarull's rigidity theorem to $L^{ abla}$ metrics on punctured spheres, showing that scalar curvature bounds imply the metric must be standard, addressing a question posed by Gromov.

Contribution

It proves Llarull's theorem for $L^{ abla}$ metrics on punctured spheres, broadening the scope of the original rigidity result.

Findings

Llarull's theorem holds for $L^{ abla}$ metrics on punctured spheres.

The result confirms Gromov's question on scalar curvature and metric rigidity.

The theorem applies to metrics with finitely many punctures, generalizing classical results.

Abstract

The classical Llarull theorem states that a smooth metric on $n$-sphere cannot have scalar curvature no less than $n(n-1)$ and dominate the standard spherical metric at the same time unless it is the standard spherical metric. In this work, we prove that Llarull's rigidity theorem holds for $L^{\infty}$ metrics on spheres with finitely many points punctured. This is related to a question of Gromov.