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

TL;DR

This paper constructs a broad class of non-proper complete minimal surfaces in H^2 x R, demonstrating new behaviors and counterexamples to classical conjectures, expanding understanding of minimal surface embeddings in hyperbolic product spaces.

Contribution

It introduces a large family of non-proper embedded minimal surfaces in H^2 x R, invariant under various isometries, with finite total curvature, challenging existing conjectures.

Findings

Constructed non-proper complete minimal disks invariant under isometries.

Surfaces have finite total curvature in the quotient space.

Counterexamples to Calabi-Yau conjectures in H^2 x R.

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.