On the Renormalized Volumes for Conformally Compact Einstein Manifolds

We study the renormalized volume of a conformally compact Einstein manifold. In even dimenions, we derive the analogue of the Chern-Gauss-Bonnet formula incorporating the renormalized volume. When the dimension is odd, we relate the renormalized volume to the conformal primitive of the Q-curvature. 0. Introduction Recently, there is a series of work ([GZ],[FG-2] and [FH]) exploring the connection between scattering theory on asymptotically hyperbolic manifolds, the Q-curvature in conformal geometry and the ”renormalized volume” of conformally compact Einstein manifolds. In particular, in [FG-2], a notion of Q-curvature was introduced for an odd-dimensional manifold as the boundary of a conformally compact Einstein manifold of even-dimension. In this note, in section 2 below, we will clarify the relation between the work of [FG-2] and the notion of Q-curvature in our earlier work [CQ] for the special case when the manifold is of dimension three. We then explore this relation to give a different proof of a result of Anderson [A] writing the Chern-Gauss-Bonnet formula for conformally compact Einstein 4-manifold with the renormalized volume. Our proof makes use of the special exhaustion function introduced in [FG-2] that yields remarkable simplification in computing the Q curvature. In section 3, using some recent result of Alexakis on Q-curvature, we generalize the Chern-Gauss-Bonnet formula involving the renormalized volume to all even dimensional conformally compact Einstein manifolds. The formula includes as special Princeton University, Department of Mathematics, Princeton, NJ 08544-1000, supported by NSF Grant DMS–0245266. University of California, Department of Mathematics, Santa Cruz, CA 95064, supported in part by NSF Grant DMS–0103160. Princeton University, Department of Mathematics, Princeton, NJ 08544-1000, supported by NSF Grant DMS–0245266. Typeset by AMS-TEX 1 2 RENORMALIZED VOLUME case the formula of Epstein (appendix A in [E]) for conformally compact hyperbolic manifolds. The formula also shows that the renormalized volume is a conformal invariant of the conformally compact structure when the dimension is odd. Finally in section 4, we obtain a formula similar to that in [FG-2] expressing the renormalized volume of a odd dimensional conformally compact Einstein manifold as the conformal primitive of the Q curvature, and in terms of the data of the scattering matrix. 1. Conformally compact Einstein manifolds and renormalized volumes In this section, we will first recall some basic definitions and facts of conformally compact Einstein manifolds. We then state the main result in [FG-2]. Given a smooth manifold X of dimension n+1 with smooth boundary ∂X = M. Let x be a defining function for M in X as follows: x > 0 in X; x = 0 on M ; dx 6= 0 on M. A Riemannian metric g on X is conformally compact if (X, xg) is a compact Riemannian manifold with boundary. Conformally compact manifold (X, g) carries a well-defined conformal structure [ĝ] on the boundary M; where each ĝ is the restriction of xg for a defining function x. We call (M, [ĝ]) the conformal infinity of the conformally compact manifold (X, g). If, in addition, g satisfies Ricg = −ng, where Ricg denotes the Ricci tensor of the metric g, then we call (X, g) a conformally compact Einstein manifold. A conformally compact metric is said to be asymptotically hyperbolic if its sectional curvature approach −1 at ∂X = M . It was shown ([FG-1], [GL]) that if g is an asymptotically hyperbolic metric on X, then a choice of metric ĝ in [ĝ] on M uniquely determines a defining function x near the boundary M and an identification of a neighborhood of M in X with M × (0, ) such that g has the normal form (1.1) g = x(dx + gx) where gx is a 1-parameter family of metrics on M . As a conformally compact Einstein metric g is clearly asymptotically hyperbolic, we have, as computed in [G-1] by Graham, (1.2) gx = ĝ + g x + (even powers of x) + gx + gx + · · · , CHANG, QING AND YANG 3 when n is odd, and (1.3) gx = ĝ + g x + (even powers of x) + gx + hx log x + · · · , when n is even. Where ĝ = xg|x=0, g are determined by ĝ for 2i < n. The trace part of g is zero when n is odd; the trace part of g is determined by ĝ and h is traceless and determined by ĝ too when n is even. As a realization of the holography principle proposed in physics, one considers the asymptotic of the volume of a conformally compact Einstein manifold (X, g). Namely, if denote by x the defining function associated with a choice of a metric ĝ ∈ [ĝ], we have (1.4) Volg({x > }) = c0 −n + c2 −n+2 + · · ·+ cn−1 −1 + V + o(1) for n odd, and (1.5) Volg({x > }) = c0 −n + c2 −n+2 + · · ·+ cn−2 −2 + L log 1 + V + o(1) for n even. We call the constant term V in all dimensions the renormalized volume for (X, g). We recall that V in odd dimension and L in even dimension are independent of the choice ĝ in the class [ĝ] (cf. [HS] [G-1]). Based on the work of [GZ], Fefferman and Graham [FG-2] introduced the following formula to calculate the renormalized volume V for a conformally compact Einstein manifold. Here we will quote a special case of their result. For odd n, upon a choice of a special defining function x, one sets v = − d ds |s=n℘(s)1, where ℘(s) denotes the Possion operator (see [GZ] or section 4 below for the definition of the operator) on X. The v solves (1.6) −∆v = n in X, and has the asymptotic behavior (1.7) v = log x + A + Bx in a neighborhood of M, where A, B are functions even in x, and A|x=0 = 0. Then Fefferman and Graham in [FG-2] defined (1.8) (Qn)(g,ĝ) = knB|x=0 where kn = 2 n Γ( n 2 ) Γ(− n2 ) . 4 RENORMALIZED VOLUME Theorem 1.1. ([FG-2]) When n is odd, (1.9) V (X, g) = 1 kn ∫