## Lipschitz convergence of Riemannian manifolds.(English)Zbl 0646.53038

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

### Citations:

Zbl 0509.53034; Zbl 0618.53036; Zbl 0502.53036
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