An Arzela-Ascoli theorem for immersed submanifolds

Graham Smith

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Submission date: 21. Nov. 2005
Pages: 45
published in: Annales de la Faculté des Sciences de Toulouse, 16 (2008) 4, p. 817-866 
Bibtex
MSC-Numbers: 57.20, 53.40
Keywords and phrases: cheeger/gromov, arzela ascoli, compactness
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Abstract:
The classical Arzela-Ascoli theorem is a compactness result for families of functions depending on bounds on the derivatives of the functions, and is of invaluable use in many fields of mathematics. In this paper, inspired by a result of Corlette, we prove an analogous compactness result for families of immersed submanifolds which depends only on bounds on the derivatives of the second fundamental forms of these submanifolds. We then show how the result of Corlette may be obtained as an immediate corollary