On the Rigidity for Conformally Compact Einstein Manifolds

In this paper we prove that a conformally compact Einstein manifold with the round sphere as its conformal infinity has to be the hyperbolic space. We do not assume the manifolds to be spin, but our approach relies on the positive mass theorem for asymptotic flat manifolds. The proof is based on understanding of positive eigenfunctions and compactifications obtained by positive eigenfunctions. 0. Introduction In this paper we study the rigidity problem for conformally compact Einstein manifolds with the round sphere as their conformal infinity. Quite recently there has been a great deal of interest in both physics and mathematics community in the so-called Anti-de-Sitter/Conformal Field Theory (in short AdS/CFT) correspondence. Conformally compact Einstein manifolds play an essential role in this correspondence. In mathematics it has been known for a long time that there are close connections between the geometry of Minkowski space-time, hyperbolic space and the round sphere. Notably in a seminal paper [FG] Fefferman and Graham showed this approach to be very fruitful in conformal geometry. In establishing scalar curvature rigidity for asymptotically hyperbolic manifold as a generalization from the work of Witten [W] on positive mass theorem for asymptotically flat manifolds which are spin, in [AD], Andersson and Dahl proved that, if a conformally compact Einstein manifold with the round sphere as its conformal infinity is spin, then it has to be a hyperbolic space (please also see recent related works of X. Zhang [Z], Chruściel and Herzlich [CH], and X. Wang [Wa]). It opens an interesting question whether the spin structure is necessary to assure the rigidity. There is some progress made by Anderson in [An1]. Typeset by AMS-TEX 1 2 CONFORMALLY COMPACT EINSTEIN Before we state our results. let us briefly introduce what is a conformally compact Einstein manifold. Let X be a n + 1 dimensional compact manifold with boundary M = ∂X . (X, g) is said to be a conformally compact Einstein manifold if Ric(g) = −ng and (X, sg) is a compact Riemannian manifold with boundary, where s is a defining function of the boundary M . Clearly the restriction of sg to TM is a metric ĝ on the boundary and ĝ rescales upon changing the defining function s. Thus (M, [ĝ]) is determined by (X, g) and called the conformal infinity of (X, g). Theorem 0.1. Suppose that (X, g) is a conformally compact Einstein manifold with the round sphere as its conformal infinity, and 3 ≤ n ≤ 6. Then (X, g) has to be the hyperbolic space. One simple yet very interesting calculation leading to Theorem 0.1 is the following. Lemma 0.2. Suppose that (X, g) is a conformally compact Einstein manifold. And suppose that u is a positive eigenfunction, i.e. ∆u = (n + 1)u. Then (X, ug) is with scalar curvature (0.1) R = n(n+ 1)(u − |du|). Here ∆ is the trace of the Hessian in metric g. Combining with the Bochner formula for eigenfunction u: (0.2) −∆(u − |du|) = 2|Ddu− ug|, observed by Lee in [L], one may know the scalar curvature for the conformal compactification (X, ug) if one knows the asymptotic behavior of u near the boundary. This turns out to be a very interesting construction for its own sake. The paper is organized as follows. In Section 1 we will introduce notations and do some computations for the hyperbolic spaces. In Section 2, we will introduce conformally compact Einstein manifolds and relevant properties. And we will apply theory of uniformly degenerate elliptic linear PDE to solve for the eigenfunctions and their expansions. Finally in Section 3 we will introduce some conformal compactifications and prove Theorem 0.1. Acknowledgment The author would like to thank Xiao Zhang for bringing my attention to the paper [AD] and many stimulating discussions. The author would also like to thank the referee for many suggestions and advices.