Lipschitz convergence of Riemannian manifolds

Robert E. Greene, Hung-Hsi Wu

Type: Article

Publication Date: 1988-01-01

Citations: 199

DOI: https://doi.org/10.2140/pjm.1988.131.119

Abstract

Let #s <g(n,A,8 O9 V o ) be the set of all connected compact C°°dimensional Riemannian manifolds with |sectional curvature | < Λ 2 , diameter < 8 0 , and volume > V o .The main result of this paper is that this class ^ has certain compactness, or more precisely, precompactness properties.The class # consists of only finitely many diffeomorphism classes so the precompactness properties can be thought of as dealing with the set of metrics satisfying the class # requirements on a fixed differentiate manifold.The main theorem of this paper is then that a sequence of such metrics always has a subsequence which, after application of suitable diffeomorphisms of M, converges to a limit metric.The regularity of the limit can be taken to be C lα , for all a with 0 < a < 1 and the convergence to be in the C ι ' a norm.

Locations

Pacific Journal of Mathematics - View - PDF
Project Euclid (Cornell University) - View - PDF

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