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

Abstract

We construct the first examples of complete, properly embedded minimal surfaces in ${ℍ}^2\times ℝ$ 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.