Title: The renormalized volume of conformally compact Einstein manifold
Abstract:
In this talk, I will introduce the renormalized volume of a conformally compact Einstein manifold. The classical volume for any conformally compact manifold is infinite, just like the case for a hyperbolic plane. We are interested in finding an appropriate renormalization. It turns out that under Einstein condition, the zeroth order term in the volume expansion of the complement of a collar neighborhood gives a scalar conformal invariant. In the even-dimensional case, this term is the renormalized volume.
This renormalization is initially motivated by the AdS/CFT correspondence in physics. There are many interesting results of the renormalized volume of a conformally compact manifold. For example, we can link the renormalization to the Chern-Gauss-Bonnet formula and Branson's $Q$-curvature. Furthermore, we may define a renormalized integral and prove a renormalized version of the Atiyah-Singer index theorem.
