Geometry & Topology

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

Francisco Martín, Rafe Mazzeo, and M Magdalena Rodríguez

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Abstract

We construct the first examples of complete, properly embedded minimal surfaces in 2× 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.

Article information

Source
Geom. Topol., Volume 18, Number 1 (2014), 141-177.

Dates
Received: 30 August 2012
Revised: 6 May 2013
Accepted: 19 July 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732721

Digital Object Identifier
doi:10.2140/gt.2014.18.141

Mathematical Reviews number (MathSciNet)
MR3158774

Zentralblatt MATH identifier
1280.49062

Subjects
Primary: 49Q05: Minimal surfaces [See also 53A10, 58E12] 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42] 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

Keywords
properly embedded minimal surfaces finite total curvature gluing constructions moduli spaces minimal surfaces positive genus

Citation

Martín, Francisco; Mazzeo, Rafe; Rodríguez, M Magdalena. Minimal surfaces with positive genus and finite total curvature in $\mathbb{H}^2 \times \mathbb{R}$. Geom. Topol. 18 (2014), no. 1, 141--177. doi:10.2140/gt.2014.18.141. https://projecteuclid.org/euclid.gt/1513732721


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