Renormalized volume of minimally bounded regions 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.

arXiv comment39 pages, typos have been fixed and the main calculation in section 4 has been streamlined