Minimal surfaces with positive genus and finite total curvature in $\mathbb{H}^2 \times \mathbb{R}$

TL;DR

This paper constructs new examples of complete minimal surfaces with positive genus and finite total curvature in hyperbolic space, expanding understanding of their geometry and topology.

Contribution

It introduces the first known examples of such surfaces in  imes \u211d, constructed via gluing techniques, and proves nondegeneracy of horizontal catenoids.

Findings

First examples of minimal surfaces with positive genus and finite total curvature in  imes 

Horizontal catenoids are proven to be nondegenerate

Existence of minimal surfaces with infinitely many ends asymptotic to geodesic planes

Abstract

We construct the first examples of complete, properly embedded minimal surfaces in $\mathbb{H}^2 \times \mathbb{R}$ with finite total curvature and positive genus. These are constructed by gluing copies of horizontal catenoids or other nondegenerate summands. We also establish that every horizontal catenoid is nondegenerate. Finally, using the same techniques, we are able to produce properly embedded minimal surfaces with infinitely many ends. Each annular end has finite total curvature and is asymptotic to a vertical totally geodesic plane.