Renormalizing curvature integrals on Poincaré-Einstein manifolds
Riemannian Einstein manifolds (Ricci tensor constant proportional to the metric tensor) form a class next interesting beyond manifolds of constant curvature. A further extension arises from conformal changes of the metric. This paper, part of the author's thesis, studies such changes for compact manifolds with boundary \( \overline{M} = M \cup \partial M \), where the conformal change of the metric is of the form \( \overline{g} = x^{2}g \), the non-negative and smooth function \( x: \overline{M} \to \mathbb R \) defining the boundary, i.e. the boundary is the (regular) zero set of \( x \). Such functions \( x \) are called boundary defining functions (bdf), and the manifolds equipped in this way bear the name Poincaré-Einstein manifolds (PE-manifolds). The author gives an overview of the recent developments in this field, including the relations to string theory, and contributes to it by several own results. The hypothesis that \( \overline{g} \) extends smoothly to \( \overline{M} \) is too restrictive for this research. Instead, \( \overline{g} \) is assumed to be of a certain expansion type including finitely many terms of the form \( a\cdot x^{k} \cdot \log^{p} x \), \(a\) a smooth function independent of \(x\), the metric \( \overline{g} \) thus being `polyhomogeneous conormal'. Accordingly, restrictions on `special' bdfs must be considered. First some renormalization schemes on PE-manifolds are compared. Under such assumptions, the author studies the renormalization of scalar curvature integral invariants. While, in even dimensions, the renormalized integrals are independent of the special bdf, in odd dimensions there are residual integrals on the boundary, also independent of the choice of special bdf. In particular, the renormalized integral of the Pfaffian of the curvature operator yields the Euler characteristic as in the standard closed situation for even dimensions. In addition, variations of integrals of scalar invariants are calculated. For classical variation formulae of integrals of scalar invariants depending on the first two derivatives of the metric see \textit{H. Rund} [Abh. Math. Semin. Univ. Hamburg 29, 243--262 (1966; Zbl 0146.35701)] and, especially for the Gauss-Bonnet-integral, \textit{M. Armsen} [Result. Math. 4, 128--140 (1981; Zbl 0479.53033)]. The present author studies, in his context, more general characteristic forms and their behavior under a variation of the PE-structure.
asymptotically hyperbolic Einstein metric
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