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

The authors consider isometric immersions ${\Phi }$ : ${M}^{2}\to {ℝ}^{2+k}$. 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 ${\Phi }$ is minimal. If there exists a point $p\in W$ whose associated normal field e is parallel in the normal bundle, then ${\Phi }$ is an embedding into a 2-dimensional affine subspace.” In particular the set of tangent planes of a non-flat complete minimal surface in ${ℝ}^{3}$ covers all of ${ℝ}^{3}·$

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 ${\Phi }$ immerses ${M}^{2}$ into a great hypersphere in ${S}^{2+k}$. However, a complete minimal surface ${M}^{2}\to {S}^{3}$ must be a great ${S}^{2}$, provided the spherical image of ${\Phi }$ 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:
 53C42 Immersions (differential geometry) 53A10 Minimal surfaces, surfaces with prescribed mean curvature 53A07 Higher-dimensional and -codimensional surfaces in Euclidean $n$-space
##### Keywords:
isometric immersions; minimal surface; support function