Renormalized Curvature Integrals on Poincare-Einstein manifolds

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Poincare-Einstein (PE) manifolds such as Poincare ball model are complete Einstein manifolds with a well-defined conformal boundary. There is a rich interplay between the conformal geometry of the boundary of a PE manifold and the Riemannian geometry of its interior. A first step in studying the moduli space of PE manifolds is to develop a good understanding of its global invariants. In even dimensions, renormalized curvature integrals give many such invariants. In this talk, I will discuss a general procedure for computing renormalized curvature integrals on PE manifolds. In particular, this explains the connection between the Gauss-Bonnet-type formulas of Albin and Chang-Qing-Yang for the renormalized volume, and explicitly identify a scalar conformal invariant in the latter formula. Our procedure also produces similar formulas for compact Einstein manifolds. This talk is based on joint works with Jeffrey Case, Ayush Khaitan, Aaron Tyrrell, and Wei Yuan.

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