Advances in Theoretical and Mathematical Physics

Volume 26 (2022)

Number 9

Renormalized volume of minimally bounded regions in asymptotically hyperbolic Einstein spaces

Pages: 3081 – 3123

DOI: https://dx.doi.org/10.4310/ATMP.2022.v26.n9.a6

Authors

Matthew J. Gursky (Department of Mathematics, University of Notre Dame, Indiana, U.S.A.)

Stephen E. McKeown (Department of Mathematical Sciences, University of Texas at Dallas, Richardson, Tx., U.S.A.)

Aaron J. Tyrrell (Department of Mathematics and Statistics, Texas Tech University, Lubbock, Tx., U.S.A.)

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.

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M.J.G. was supported in part by NSF grant DMS-2105460 and DMS-1547292.

S.E.M. was supported in part by Simons Foundation grant 966614. A.J.T. was supported in part by the National University of Ireland.

Published 30 October 2023