A renormalized index theorem for some complete asymptotically regular metrics: The Gauss–Bonnet theorem

We study the Gauss–Bonnet theorem as a renormalized index theorem for edge metrics. These metrics include the Poincaré–Einstein metrics of the AdS/CFT correspondence and the asymptotically cylindrical metrics of the Atiyah–Patodi–Singer index theorem. We use renormalization to make sense of the curvature integral and the dimensions of the L2-cohomology spaces as well as to carry out the heat equation proof of the index theorem. For conformally compact metrics even mod xm, we show that the finite time supertrace of the heat kernel on conformally compact manifolds renormalizes independently of the choice of special boundary defining function.