2012/11/24 by Rodriguez, Magdalena, Tinaglia, Giuseppe
#49Q05 #53A10 #53C42 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1211.5692
Examples of complete minimal surfaces properly embedded in H2 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 H2 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 H2 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 H2 x R by the isometry, thus highlighting a different behaviour from minimal surfaces embedded in R3 satisfying the same properties. And they show that the Calabi-Yau conjectures do not hold for embedded minimal surfaces in H2 x R.