Conformally prescribed scalar curvature on orbifolds

TL;DR

This paper investigates the prescribed scalar curvature problem on orbifolds with isolated singularities, establishing compactness and existence results in dimensions four and higher, and analyzing the degree for specific orbifolds.

Contribution

It extends scalar curvature prescription results to orbifolds, addressing challenges posed by singularities and positive mass theorem limitations.

Findings

Proved a compactness theorem in dimension 4.

Established an existence theorem for dimensions ≥4.

Determined the U(2)-invariant Leray-Schauder degree for certain negative-mass orbifolds.

Abstract

We study the prescribed scalar curvature problem in a conformal class on orbifolds with isolated singularities. We prove a compactness theorem in dimension $4$, and an existence theorem which holds in dimensions $n \geq 4$. This problem is more subtle than the manifold case since the positive mass theorem does not hold for ALE metrics in general. We also determine the $\rm{U}(2)$-invariant Leray-Schauder degree for a family of negative-mass orbifolds found by LeBrun.