Saddle towers in H^2 x R

TL;DR

This paper constructs a family of minimal surfaces called Saddle Towers in H^2 x R, characterized by specific symmetries, ends, and curvature, expanding the understanding of minimal surface configurations in hyperbolic spaces.

Contribution

It introduces a new family of minimal surfaces with adjustable parameters, linking them to Jenkins-Serrin graphs and symmetric surfaces, and explores their geometric limits.

Findings

Constructed (2k-2)-parameter family of Saddle Towers

Connected Saddle Towers to Jenkins-Serrin graphs over polygonal domains

Identified limits leading to symmetric minimal surfaces with planar ends

Abstract

Given k>=2, we construct a (2k-2)-parameter family of properly embedded minimal surfaces in H^2 x R invariant by a vertical translation T, called Saddle Towers, which have total intrinsic curvature 4 pi(1-k), genus zero and 2k vertical Scherk-type ends in the quotient by T. As limits of those Saddle Towers, we obtain Jenkins-Serrin graphs over ideal polygonal domains (with total intrinsic curvature 2 pi(1-k)); we also get properly embedded minimal surfaces which are symmetric with respect to a horizontal slice and have total intrinsic curvature 4 pi(1-k), genus zero and k vertical planar ends.