Minimal surfaces in \(\mathbb{H}^2\times\mathbb{R}\).

*(English)*Zbl 1038.53011This 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.

(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.

Reviewer: Wayne Rossman (Kobe)

##### MSC:

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