Volume growth and puncture repair in conformal geometry

TL;DR

This paper proves that the only way to conformally compactify a punctured compact Riemannian manifold is by filling in the puncture, using volume growth estimates around the missing point.

Contribution

It introduces a method based on volume growth estimates to characterize the unique conformal compactification of punctured manifolds.

Findings

The conformal compactification of a punctured manifold is unique and equals the original manifold.

Volume growth estimates around a point determine the conformal structure at infinity.

The result applies to arbitrary points in compact Riemannian manifolds.

Abstract

Suppose $M$ is a compact Riemannian manifold and $p\in M$ an arbitrary point. We employ estimates on the volume growth around $p$ to prove that the only conformal compactification of $M\setminus\{p\}$ is $M$ itself.