Linear potentials and applications in conformal geometry

TL;DR

This paper develops estimates for linear potentials away from thin sets and applies these to improve understanding of singular sets in conformal geometry, especially in relation to scalar curvature and Q-curvature equations.

Contribution

It introduces new potential estimates and verifies geodesic avoidance of thin sets, leading to improved bounds on singular set dimensions in conformal geometry.

Findings

Enhanced estimates for linear potentials away from thin sets.

Improved bounds on Hausdorff dimensions of singular sets.

A stronger finiteness theorem for Q-curvature singular sets in dimension 4.

Abstract

In this paper we derive estimates for linear potentials that hold away from thin subsets. And, inspired by the celebrated work of Huber, we verify that, for a subset that is thin at a point, there is always a geodesic that reaches to the point and avoids the thin subset in general dimensions. As applications of these estimates on linear potentials, we consider the scalar curvature equations and slightly improve the results of Schoen-Yau and Carron on the Hausdorff dimensions of singular sets which represent the ends of complete conformal metrics on domains in manifolds of dimensions greater than 3. We also study Q-curvature equations in dimensions greater than 4 and obtain stronger results on the Hausdorff dimensions of the singular sets. More interestingly, our approach based on potential theory yields a significantly stronger finiteness theorem on the singular sets for Q-curvature equations in dimension 4, which is a remarkable analogue of Huber's theorem.