Renormalizing Curvature Integrals on Poincare-Einstein Manifolds

TL;DR

This paper investigates the process of renormalizing integrals of scalar Riemannian invariants on Poincaré-Einstein manifolds, extending classical theorems like Gauss-Bonnet through this framework.

Contribution

It introduces a systematic approach to renormalize scalar Riemannian invariants and extends the Gauss-Bonnet theorem using these renormalized integrals.

Findings

Renormalized integrals of scalar invariants behave similarly to volume.

Extension of Gauss-Bonnet theorem via renormalized Pfaffian.

Analysis of characteristic forms under Poincaré-Einstein structure variations.

Abstract

After analyzing renormalization schemes on a Poincar\'e-Einstein manifold, we study the renormalized integrals of scalar Riemannian invariants. The behavior of the renormalized volume is well-known, and we show any scalar Riemannian invariant renormalizes similarly. We consider characteristic forms and their behavior under a variation of the Poincar\'e-Einstein structure, and obtain, from the renormalized integral of the Pfaffian, an extension of the Gauss-Bonnet theorem.