| We obtain diffeomorphism finiteness and orbifold compactness results for a class of smooth, closed Riemannian manifolds with vanishing Fefferman-Graham tensor and constant scalar curvature in dimension six. In particular, our results apply to conformally compact Einstein manifolds and certain product of compact Einstein manifolds which are not conformally Einstein. As a consequence of our finiteness result, we obtain volume growth estimate for the class of manifolds in consideration. As the Fefferman-Graham tensor is the higher dimensional generalization of the Bach tensor in dimension four, our results can be regarded as the higher dimensional version of results for the Bach flat manifolds in dimension four (Invent. Math. 160 (2005), no. 2, 357-415, Adv. Math. 196 (2005), no. 2, 346-372, Math. Ann. 331 (2005), no. 4, 739-778, Gafa (2007), to appear). Similar results have been obtained for the class of Einstein metrics (J. Amer. Math. Soc. 2 (1989), no. 3, 455-490, Invent. Math. 97 (1989) (2), 313-349, Invent. Math. 160 (2005), no. 2, 357-415). Our analysis is different from those in that we study a more complicated sixth-order elliptic system in metrics and only assume some integral bounds on the curvatures and geometric non-collapsing condition. |
