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

Abstract

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 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 , 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.