Renormalized volume of minimally bounded regions in asymptotically hyperbolic Einstein spaces

TL;DR

This paper introduces a renormalized volume concept for regions in asymptotically hyperbolic Einstein spaces bounded by minimal surfaces, establishing a Gauss-Bonnet theorem and analyzing volume variations.

Contribution

It defines a new renormalized volume for such regions, proves a Gauss-Bonnet theorem, and computes how this volume changes under minimal surface variations.

Findings

Established a Gauss-Bonnet theorem for the renormalized volume.

Derived formulas for volume derivatives under hypersurface variations.

Provided explicit computations in asymptotically hyperbolic Einstein spaces.

Abstract

We define a renormalized volume for a region in an asymptotically hyperbolic Einstein manifold that is bounded by a Graham-Witten minimal surface and the conformal infinity. We prove a Gauss-Bonnet theorem for the renormalized volume, and compute its derivative under variations of the minimal hypersurface.