Volume renormalization for singular Yamabe metrics

TL;DR

This paper introduces a volume renormalization for singular Yamabe metrics, generalizing Poincare-Einstein volume renormalization, and defines a new conformally invariant energy related to boundary geometry.

Contribution

It extends volume renormalization techniques to singular Yamabe metrics and introduces a new conformally invariant energy generalizing the Willmore energy.

Findings

Defines a volume renormalization for singular Yamabe metrics

Introduces a conformally invariant energy related to boundary geometry

Answers a question by Gover and Waldron about existence of such an energy

Abstract

This paper carries out a renormalization of the volume of the Loewner-Nirenberg singular Yamabe metric in a given conformal class on a compact manifold-with-boundary. This generalizes the usual volume renormalization for Poincare-Einstein metrics. The coefficient of the log term in the volume expansion defines a conformally invariant energy generalizing the Willmore energy of a surface whose variational derivative with respect to variations of the boundary hypersurface is a multiple of the obstruction to smoothness of the singular Yamabe metric itself. The existence of such an energy answers a question raised by Gover and Waldron.