Renormalized Area for Minimal Hypersurfaces of 5D Poincar\'e-Einstein Spaces

TL;DR

This paper derives a Gauss-Bonnet formula for the renormalized area of minimal hypersurfaces in 5D Poincaré-Einstein spaces, linking geometric invariants to conformal geometry at infinity.

Contribution

It introduces a new Gauss-Bonnet formula for renormalized area and characterizes minimal hypersurfaces with L^2 second fundamental form via conformal geometry.

Findings

Gauss-Bonnet formula for renormalized area derived

Characterization of minimal hypersurfaces with L^2 second fundamental form

Expresses renormalized area through scalar Riemannian invariants

Abstract

In this paper we derive a Gauss-Bonnet formula for the renormalized area of Graham-Witten minimal hypersurfaces of 5-dimensional Poincar\'e-Einstein spaces. The formula we derive expresses the renormalized area in terms of integrals of pointwise scalar Riemannian invariants. We also prove a result which gives a characterization of minimal hypersurfaces with $L^2$ second fundamental form in terms of conformal geometry at infinity.