Ricci Curvature Bounds and Einstein Metrics on Compact Manifolds

dL(Mo,M1) = inf[llogdil/l + Ilogdil/-II], f where I: Mo -+ MI is a homeomorphism and dil I is the dilatation of I given by dill = SUPXt#2 dist(f(x l ), l(x2))/ dist(x1 ,x2). If Mo and MI are not homeomorphic, define dL(Mo,M1) = +00. Gromov [20] proves the remarkable result that the space of compact Riemannian manifolds L(A,t5 ,D) of sectional curvature IKI :::; A, injectivity radius i M 2: t5 > 0, and diameter dM :::; D, is CI,1 compact with respect to the Lipschitz topology. By d ,I compact we mean that any sequence in L(A,t5 ,D) has a subsequence which converges, in the Lipschitz topology, to a COO manifold M with CO Riemannian metric and C 1 ,I distance function p: M x M -+ R. Related but different proofs of this result obtaining a limit d ,n, a < 1, Riemannian metric on M appear in [19, 25]. A number of applications of the Gromov compactness theorem have now been obtained, for example in [4, 25]. For an interesting discussion of this result in the context of more general studies, we refer to [30]. An important antecedent of Gromov's compactness theorem is Cheeger's finiteness theorem [8] that the set L (A, v, D) of compact Riemannian manifolds of curvature IKI :::; A, volume V M 2: v , and diameter d M :::; D, has only finitely many diffeomorphism types (cf. also [31]). A basic step in this theorem is a lower bound estimate for the injectivity radius i M 2: c(IKI , d M ' V;; I). In particular, Gromov's compactness theorem may be strengthened to the statement that L (A, v, D) is Cl,1 compact in the Lipschitz topology. In this paper, we study the question of Lipschitz convergence of compact Riemannian manifolds with bounds imposed on the Ricci curvature Ric in