Volume 18, issue 1 (2014)

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

Francisco Martín, Rafe Mazzeo and M Magdalena Rodríguez

Geometry & Topology 18 (2014) 141–177
Abstract

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

Keywords
properly embedded minimal surfaces, finite total curvature, gluing constructions, moduli spaces, minimal surfaces, positive genus
Mathematical Subject Classification 2010
Primary: 49Q05, 53A10, 53C42