Abstract: First, we state a conformal Gauss-Bonnet theorem for four-manifolds with corner. Then we review the definition of the renormalized volume of asymptotically hyperbolic Einstein metrics, which came from the AdS/CFT correspondence in physics; and review the Gauss-Bonnet theorem for such manifolds in the ordinary case. We then state a new Gauss-Bonnet formula for half of an asymptotically hyperbolic Einstein space that has been partitioned into two by a minimal surface, and use this to derive a variation formula for its renormalized volume under variations of the minimal surface. The latter two results are joint work with Matthew J. Gursky and Aaron J. Tyrrell.
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