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The geometry, topology, and existence of doubly periodic minimal surfaces. (English) Zbl 0642.53062

In the present paper, the authors announce interesting and important results for properly embedded doubly-periodic minimal surfaces. Let \(f: M\to T\times R\) be a proper minimal immersion of a surface of finite topological type, where T is a flat 2-dimensional torus. The main result is as follows: 1. M is conformally diffeomorphic to a closed Riemann surface \(\bar M\) which has been punctured in a finite number of points. 2. The Gauss map \(g: M\to C\cup \{\infty \}\) is a meromorphic function on M. A necessary and sufficient condition for the total curvature c(M) to be finite is that g extends meromorphically to \(\bar M.\) 3. \(c(M)\leq 2\pi \chi (M).\) 4. If M is embedded properly then M has finite total curvature \(c(M)=2\pi \chi (M).\) The top ends and the bottom ends of M converge to distinct parallel flat annuli.
Using the embeddedness assumption and the finite total curvature property, the authors also give topological and geometric obstructions for embedding a given surface of finite topological type as a minimal surface in \(T\times R\).
Reviewer: T.Ishihara

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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