Mumford's compactness theorem

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In mathematics, Mumford's compactness theorem states that the space of compact Riemann surfaces of fixed genus g > 1 with no closed geodesics of length less than some fixed ε > 0 in the Poincaré metric is compact. It was proved by David Mumford (1971) as a consequence of a theorem about the compactness of sets of discrete subgroups of semisimple Lie groups generalizing Mahler's compactness theorem.


  • Mumford, David (1971), "A remark on Mahler's compactness theorem", Proceedings of the American Mathematical Society 28: 289–294, JSTOR 2037802, MR 0276410