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The global theory of doubly periodic minimal surfaces. (English) Zbl 0676.53068

The authors study properly embedded minimal surfaces M in \({\mathbb{T}}\times {\mathbb{R}}\) where \({\mathbb{T}}\) is a flat two-dimensional torus. The main result is that the total Gaussian curvature C(M) of M is finite and equal to \(2\pi\) \(\chi\) (M) if M is homeomorphic to a closed surface punctured in finitely many points. Various consequences of this result are proved, e.g. the number of ends of M must be even. Examples are constructed in every \({\mathbb{T}}\times {\mathbb{R}}\) of an infinite family of nonhomotopic Klein bottles minus two points.
Reviewer: G.Thorbergsson

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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