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The geometry of periodic minimal surfaces. (English) Zbl 0807.53049
The authors demonstrate relationships between the topology of periodic minimal surfaces and its global geometry. The main theorem is: A properly embedded minimal surface in a complete nonsimply connected flat three- manifold has finite total curvature if and only if it has finite topology. This result has topological and analytical consequences, for example the following uniqueness theorem can be deduced: The plane and the helicoid are the only properly embedded simply connected minimal surfaces in \(\mathbb{R}^ 3\) with infinite symmetry group.

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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