Thinness for Scalar-Negative Singular Yamabe Metrics

TL;DR

This paper investigates the conformal deformation of standard sphere metrics to complete metrics with constant negative scalar curvature, applying nonlinear potential theory to extend known results in geometric analysis.

Contribution

It introduces a novel approach using nonlinear potential theory to analyze scalar-negative singular Yamabe metrics, expanding the class of domains where such deformations are understood.

Findings

Solved the deformation problem for constant negative scalar curvature

Extended the description of domains allowing such deformations

Applied nonlinear potential theory to geometric analysis

Abstract

This paper deals with the conformal deformation of the standard metric in a domain on the sphere to a complete metric with the constant scalar curvature. The problem of description of domains allowing such deformation originates in the works of Loewner and Nirenberg, and Schoen and Yau concerned with the locally conformally flat manifolds. The goal of this work is to apply ideas from the nonlinear potential theory to the problem. They allow, in particular, to solve the problem in the case of the constant negative scalar curvature.