The spaces of index one minimal surfaces and stable constant mean curvature surfaces embedded in flat three manifolds

Ritoré, Manuel; Ros, Antonio

doi:10.1090/S0002-9947-96-01496-1

The spaces of index one minimal surfaces and stable constant mean curvature surfaces embedded in flat three manifolds
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by Manuel Ritoré and Antonio Ros

Trans. Amer. Math. Soc. 348 (1996), 391-410

DOI: https://doi.org/10.1090/S0002-9947-96-01496-1

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Abstract:

It is proved that the spaces of index one minimal surfaces and stable constant mean curvature surfaces with genus greater than one in (non fixed) flat three manifolds are compact in a strong sense: given a sequence of any of the above surfaces we can extract a convergent subsequence of both the surfaces and the ambient manifolds in the $C^k$ topology. These limits preserve the topological type of the surfaces and the affine diffeomorphism class of the ambient manifolds. Some applications to the isoperimetric problem are given.

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