Lipschitz convergence of Riemannian manifolds. (English) Zbl 0646.53038

Greene, R. E.; Wu, H.

Pac. J. Math. 131, No. 1, 119-141 (1988).

Let \({\mathcal M}={\mathcal M}^{K,D,\circ}_{K,\circ,v}(n)\) be the set of all connected, compact n-dimensional Riemannian manifolds, M, whose (sectional) curvature, \(K_ M\), diameter, diam M, and volume, Vol M satisfy: \(k\leq K_ M\leq K\), diam \(M\leq D\), Vol \(M\geq v\). By a theorem of M. Gromov [Structures métriques pour les variétés riemanniennes, Rédigé par J. Lafontaine et P. Pansu (Paris, 1981; Zbl 0509.53034)] \({\mathcal M}\) is precompact relative to the Hausdorff metric and convergence in this metric coincides with convergence relative to the Lipschitz metric. Moreover any limit space \(X\in \bar {\mathcal M}\) is a connected, compact \(C^{\infty}\) n-dimensional manifold with a non- smooth “Riemannian” metric \(g_ x\). Several people have studied the regularity properties of \(g_ x\). The best result is that \(g_ x\) is of class \(C^{1,\alpha}\), any \(0<\alpha <1\). This is proved in the present paper and in a paper by S. Peters [Compos. Math. 62, 3-16 (1987; Zbl 0618.53036)]. The essential tool is the use of “linear” harmonic coordinates due to J. Jost and H. Karcher [Manuscr. Math. 40, 27-77 (1982; Zbl 0502.53036)].

Reviewer: K.Grove

MSC:

53C20

Global Riemannian geometry, including pinching

Keywords:

curvature; diameter; volume; Hausdorff metric; convergence; Lipschitz metric; limit space; regularity properties

Citations:

Zbl 0509.53034; Zbl 0618.53036; Zbl 0502.53036