Nonproper Complete Minimal Surfaces Embedded in H2 x R

Abstract

Examples of complete minimal surfaces properly embedded in H 2 × R have been extensively studied and the literature contains a plethora of nontrivial ones. In this paper we construct a large class of examples of complete minimal surfaces embedded in H 2 × R, not necessarily proper, which are invariant by a vertical translation or by a hyperbolic or parabolic screw motion. In particular, we construct a large family of non-proper complete minimal disks embedded in H 2 × R invariant by a vertical translation and a hyperbolic screw motion and whose importance is twofold. They have finite total curvature in the quotient of H 2 × R by the isometry, thus highlighting a different behaviour from minimal surfaces embedded in R 3 satisfying the same properties. They show that the Calabi-Yau conjectures do not hold for embedded minimal surfaces in H 2 × R.

Saddle towers in Heisenberg space

“…In H 2 × R, new examples have been constructed such as for instance doubly periodic minimal surfaces by Mazet, Rodríguez and Rosenberg [12] or genus one constant mean curvature 1/2 surfaces by Plehnert [19]. Other examples are involved in important results such as the resolution of Alexandrov problem in a quotient space of H 2 × R by Menezes [15] or the fact the Calabi-Yau conjectures do not hold for embedded minimal surface in H 2 × R by Rodríguez and Tinaglia [21]. In Heisenberg space, new examples are mostly of Scherk type like the Jenkins-Serrin theorem for compact domains obtained by Pinheiro [18].…”

Mean curvature, volume and properness of isometric immersions

“…Other example of surfaces which do not satisfy the hypothesis on the volume of the extrinsic balls in our theorem are the helicoidal-Scherk examples constructed in [30]. They are simply connected minimal surfaces ϕ : M 2 n,h → H 2 × R embedded in H 2 × R which are not properly embedded.…”

Non-properly embedded $H$-planes in $\mathbb{H}^3$

“…Given H ∈ [0, 1 2 ), does there exist a complete, non-properly embedded H-surface of finite topology in H 2 × R ? When H = 0, Rodríguez and Tinaglia [12] have constructed non-proper, complete minimal planes embedded in H 2 × R. However, their construction does not generalize to produce complete, non-proper planes embedded in H 2 × R with non-zero constant mean curvature.…”