The Gauss-Bonnet theorem and index theory on conformally compact manifolds.

摘要

In the first part of this thesis, after analyzing renormalization schemes on a Poincare-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 Poincare-Einstein structure, and obtain, from the renormalized integral of the Pfaffian, an extension of the Gauss-Bonnet theorem.; A renormalized index is defined for generalized Dirac operators for some complete asymptotically regular metrics (MICE), and the corresponding index theorem is proved in the second part of the thesis. To carry out a suitable modification of the heat kernel proof of the index theorem, an adapted pseudo-differential calculus is constructed and shown to contain the heat kernel of these metrics. The renormalized integral of the Pfaffian is shown to equal the renormalized index of the de Rham operator, and a "soft" index formula is developed to relate this to the Euler characteristic. The renormalized trace of the heat kernel on forms of a Poincare-Einstein metric is shown to renormalize independently of the choice of special boundary defining function. Remaining problems include the contribution of extended solutions and the analysis of the eta invariant in general.