The geometry of periodic minimal surfaces. (English) Zbl 0807.53049

Meeks, William H. III; Rosenberg, Harold

Comment. Math. Helv. 68, No. 4, 538-578 (1993).

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.

Reviewer: F.Gackstatter (Berlin)

MSC:

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

Keywords:

Dehn’s lemma; finite total curvature; finite topology