Uniqueness, symmetry, and embeddedness of minimal surfaces.

*(English)*Zbl 0575.53037M. Shiffman [Ann. Math., II. Ser. 63, 77-90 (1956; Zbl 0070.168)] proved that an immersed minimal surface M in \(R^ 3\) of genus zero with \(\partial M=\Gamma_ 1\cup \Gamma_ 2\), where \(\Gamma_ 1\) and \(\Gamma_ 2\) are convex curves lying in parallel planes, meets each intermediate plane transversally in a convex curve. In the first part of this paper the author extends this result into various directions. For example the topological restriction on M is removed and the ambient space may have higher dimension. In addition to this the (nontrivial) problem is discussed under which assumptions symmetries of the boundary are inherited by the spanning minimal hypersurface. Details are too complicated to be given here.

The second part of this paper introduces minimal hypersurfaces which are ’regular at infinity’. It is shown that reasonable behavior of the normal vectors at infinity implies this property. Finally the author proves that any complete minimal hypersurface which is regular at infinity and has two ends is a catenoid or a pair of planes.

The second part of this paper introduces minimal hypersurfaces which are ’regular at infinity’. It is shown that reasonable behavior of the normal vectors at infinity implies this property. Finally the author proves that any complete minimal hypersurface which is regular at infinity and has two ends is a catenoid or a pair of planes.

Reviewer: Bernd Wegner

##### MSC:

53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |

53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |

49Q05 | Minimal surfaces and optimization |