Singular Yamabe problem for scalar flat metrics on the sphere

TL;DR

This paper characterizes when a conformal scalar-flat metric on the sphere is complete outside a domain, using Bessel capacity and potential theory, extending the understanding of the singular Yamabe problem.

Contribution

It establishes a precise capacity criterion for the existence of complete scalar-flat conformal metrics on the sphere with prescribed singularities.

Findings

Existence of scalar-flat conformal metrics is equivalent to zero Bessel capacity of the complement.

Utilizes properties of capacity, Wolff potentials, and a Hopf-Rinow type theorem.

Provides a new link between geometric analysis and potential theory in the context of the Yamabe problem.

Abstract

Let $\Omega$ be a domain on the unit $n$-sphere $ \mathbb S^n$ and $\mathring{g}$ the standard metric of $\mathbb S^n$, $n\ge 3$. We show that there exists a conformal metric $g$ with vanishing scalar curvature $R(g)=0$ such that $(\Omega, g)$ is complete if and only if the Bessel capacity $\mathcal C_{\alpha, q}(\mathbb S^n\setminus \Omega)=0$, where $\alpha=1+\frac2n$ and $q=\frac n2$. Our analysis utilizes some well known properties of capacity and Wolff potentials, as well as a version of the Hopf-Rinow theorem for the divergent curves.