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A property of complete minimal surfaces. (English) Zbl 0538.53057

The authors consider isometric immersions Φ : M 2 2+k . Any point p in the complement W of all tangent planes canonically determines a unit normal field e on M 2 . Result: ”Suppose that M 2 is complete and that Φ is minimal. If there exists a point pW whose associated normal field e is parallel in the normal bundle, then Φ is an embedding into a 2-dimensional affine subspace.” In particular the set of tangent planes of a non-flat complete minimal surface in 3 covers all of 3 ·

Finally the authors pass to the standard sphere S 2+k and replace the tangent planes to M 2 by tangential great 2-spheres in S 2+k . In general one can only conclude that Φ immerses M 2 into a great hypersphere in S 2+k . However, a complete minimal surface M 2 S 3 must be a great S 2 , provided the spherical image of Φ lies in a closed hemisphere. All the proofs are based on some analysis of the support function associated with p and e.

Reviewer: U.Abresch

MSC:
53C42Immersions (differential geometry)
53A10Minimal surfaces, surfaces with prescribed mean curvature
53A07Higher-dimensional and -codimensional surfaces in Euclidean n-space