A renormalized index theorem for some complete asymptotically regular metrics: the Gauss-Bonnet theorem

Abstract

The Gauss-Bonnet Theorem is studied for edge metrics as a renormalized index theorem. These metrics include the Poincaré-Einstein metrics of the AdS/CFT correspondence. Renormalization is used to make sense of the curvature integral and the dimensions of the $L^2$-cohomology spaces as well as to carry out the heat equation proof of the index theorem. For conformally compact metrics even mod $x^m$, the finite time supertrace of the heat kernel on conformally compact manifolds is shown to renormalize independently of the choice of special boundary defining function.

3 figures

References in corpus (1)

• Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds

Cited by in corpus (1)

• Generalized Krein formula and determinants for even dimensional Poincare-Einstein manifolds