Non-proper complete minimal surfaces embedded in H^2 x R

Magdalena Rodriguez; Giuseppe Tinaglia

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arXiv · 1211.5692

Non-proper complete minimal surfaces embedded in H^2 x R

Magdalena Rodriguez·Giuseppe Tinaglia

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Abstract

Examples of complete minimal surfaces properly embedded in H^2 x 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 x 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 x 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 x R by the isometry, thus highlighting a different behaviour from minimal surfaces embedded in R^3 satisfying the same properties. And they show that the Calabi-Yau conjectures do not hold for embedded minimal surfaces in H^2 x R.

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Magdalena Rodriguez, Giuseppe Tinaglia. 2012-11-24. Non-proper complete minimal surfaces embedded in H^2 x R. https://arxiv.org/abs/1211.5692

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