Nonproper complete minimal surfaces embedded in ℍ<sup>2</sup> × ℝ

Magdalena Rodríguez*, Giuseppe Tinaglia

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)

Abstract

Examples of complete minimal surfaces properly embedded in ℍ<sup>2</sup> × 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<sup>2</sup> × ℝ, not necessarily proper, which are invariant by a vertical translation or by a screw motion. In particular, we construct a large family of nonproper complete minimal disks embedded in ℍ<sup>2</sup> × ℝ invariant by a vertical translation and a screw motion and whose importance is two-fold. They have finite total curvature in the quotient of ℍ<sup>2</sup> × ℝ by the isometry, thus highlighting a very different behavior from minimal surfaces embedded in R3 satisfying the same properties. They show that the Calabi-Yau conjectures for embedded minimal surfaces do not hold in ℍ<sup>2</sup> × ℝ.

Original languageEnglish
Pages (from-to)4322-4331
Number of pages10
JournalInternational Mathematics Research Notices
Volume2015
Issue number12
DOIs
Publication statusPublished - 2015

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