Lipschitz convergence of Riemannian manifolds

Robert Greene and Hung-Hsi Wu

Vol. 131 (1988), No. 1, 119–141

DOI: 10.2140/pjm.1988.131.119

Abstract

Let 𝒞≡𝒞(n,Λ,δ₀,V ₀) be the set of all connected compact C^∞ n-dimensional Riemannian manifolds with |sectional curvature| < Λ², diameter < δ₀, and volume > V ₀. The main result of this paper is that this class 𝒞 has certain compactness, or more precisely, precompactness properties. The class 𝒞 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 𝒞 requirements on a fixed differentiable 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^1,α, for all α with 0 < α < 1 and the convergence to be in the C^1,α norm.

Mathematical Subject Classification 2000

Primary: 53C20

Milestones

Received: 19 August 1986

Published: 1 January 1988