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Minimal surfaces in \(M^2\times \mathbb{R}\). (English) Zbl 1036.53008
This work is about minimal surfaces in the product of a 2-dimensional Riemannian surface \(M\) with the real line \(\mathbb{R}\). Here \(M\) is either (a) the 2-sphere or (b) the hyperbolic plane or (c) a Riemannian surface with a complete non-negative curvature metric. The examples of such minimal surfaces in this paper generally are constructed using Plateau techniques, and the examples given include analogues to 1) Scherk’s periodic minimal surfaces 2) unduloids (found by Pedrosa and Ritore, but proven to exist by a different method here) and 3) helicoids. Some minimal surfaces of higher genus are also found.
There are also a fair number of general results about such minimal surfaces; we list those results here that can be stated without many technical details (although the more technical results are also interesting):
1) If \(M\) is the round sphere, then the only compact minimal surfaces in \(M\times \mathbb{R}\) are of the form \(M \times t\) for some fixed \(t\in \mathbb{R}\).
2) If \(M\) is compact and the minimal surface is topologically a punctured disk, then it is also conformally a punctured disk (and more specific information about the conformal parametrization is given).
3) If \(M\) is the round sphere and the minimal surface is properly embedded with finite topology, then either it has exactly one top end and one bottom end or it is of the form \(M \times t\) for some fixed \(t\in \mathbb{R}\).
4) If \(M\) is the round sphere and there are two given properly embedded minimal surfaces in \(M\times \mathbb{R}\), then the two surfaces either intersect or are of the form \(M \times t_1\) and \(M\times t_2\) for some fixed \(t_1\) and \(t_2\in \mathbb{R}\).
5) If \(M\) is the round sphere, then a properly immersed minimal surface meets every flat vertical annulus.
6) If \(M\) has nonnegative curvature, then a properly immersed minimal surface in \(M\times \mathbb{R}\) is parabolic.
7) If \(M\) is the hyperbolic plane, then any properly immersed minimal surface in a half-space \(M\times [0,\infty)\) must be of the form \(M\times t\) for some fixed \(t\in \mathbb{R}\).
8) When \(M\) is the hyperbolic plane, there are some results on the existence of minimal graphs with given boundary, including a Jenkins-Serrin type result.

MSC:
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
35J99 Elliptic equations and elliptic systems
49Q05 Minimal surfaces and optimization
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