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Minimal surfaces in $$\mathbb{H}^2\times\mathbb{R}$$. (English) Zbl 1038.53011
This paper is a study of minimal surfaces in the product $$\mathbb{H}^2 \times \mathbb{R}$$ of the hyperbolic plane $$\mathbb{H}^2$$ with the real line $$\mathbb{R}$$, consisting of four parts:
(1) minimal catenoids (surfaces of revolution), helicoids (surfaces foliated by geodesics that are all images of a horizontal geodesic which is translated and rotated along a vertical line) and Scherk type surfaces (created by extending Jenkins-Serrin type graphs over domains in $$\mathbb{H}^2$$ to complete surfaces) are constructed.
(2) An analog of the Jenkins-Serrin result for minimal graphs in Euclidean 3-space is proven here for $$\mathbb{H}^2 \times \mathbb{R}$$. Consider a convex domain $$D$$ in $$H\to \mathbb{H}^2$$ with boundary consisting of open geodesic arcs $$A_i$$, $$B_j$$, $$C_j$$ and their endpoints. Suppose that no two $$A_i$$ (resp. no two $$B_j$$) have a common endpoint. The result in this paper gives a necessary and sufficient condition for a minimal graph over $$D$$ to have boundary data that is $$+\infty$$ on each $$A_i$$, $$-\infty$$ on each $$B_j$$ and finite continuous data on the $$C_k$$. This condition is analogous to the condition in the original Jenkins-Serrin result for Euclidean 3-space.
(3) It is shown that for any continuous Jordan curve in the boundary of $$\mathbb{H}^2 \times \mathbb{R}$$ that is a vertical graph over the boundary of $$\mathbb{H}^2$$, there exists a minimal vertical graph over the entire $$\mathbb{H}^2$$ that has that Jordan curve as its boundary data. This shows that the Bernstein theorem does not hold in the context of $$\mathbb{H}^2 \times \mathbb{R}$$, and this extends a result of Duong Minh Duc and Nguyen Van Hieu [Bull. Lond. Math. Soc. 27, 353–358 (1995; Zbl 0840.53007)].
(4) Finally, there is a regularity result showing that minimal graphs cannot have isolated singularities.

##### MSC:
 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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