Orbifold Compactness for Spaces of Riemannian Metrics and Applications

is precompact in the C1,α topology. Here R denotes the Riemann curvature tensor, vol the volume and diam the diameter of (M,g). Thus, for any sequence of metrics gi on n-manifolds Mi satisfying (1.1), there is a subsequence, also called gi, and diffeomorphisms φi : M∞ → Mi such that the metrics φi gi converge in the C 1,α topology to a limit metric g∞ onM∞, for any α < 1. In particular, there are only finitely many diffeomorphism types of n-manifolds M which admit Riemannian metrics satisfying (1.1). In addition, the convergence is in the weak L2,p topology, and the limit metric g∞ is L 2,p, for any p < ∞. While conceptually important, this result is of limited applicability in itself, since the bound on the full curvature tensor R is very strong and only realizable in special situations. A direct generalization of this result to the more natural situation where one imposes bounds on the Ricci curvature, i.e.