On the characterization of minimal surfaces with finite total curvature in ${\mathbb{H}}^2\times \mathbb{R}$ and ${\widetilde{\mathrm{PSL}}}_2(\mathbb{R},\tau )$

Abstract

It is known that a complete immersed minimal surface with finite total curvature in ${\mathbb{H}}^2\times \mathbb{R}$ is proper, has finite topology and each one of its ends is asymptotic to a geodesic polygon at infinity (Hauswirth and Rosenberg, 2006; Hauswirth, Nelli, Sa Earp and Toubiana, 2015). In this paper we prove that these three properties characterize complete immersed minimal surfaces with finite total curvature in ${\mathbb{H}}^2\times \mathbb{R}$. As corollaries of this theorem we obtain characterizations for minimal Scherk-type graphs and horizontal catenoids in ${\mathbb{H}}^2\times \mathbb{R}$. We also prove that if a properly immersed minimal surface in ${\widetilde{\mathrm{PSL}}}_2(\mathbb{R},\tau )$ has finite topology and each one of its ends is asymptotic to a geodesic polygon at infinity, then it must have finite total curvature.

Accepted for publication on Calculus of Variations and Partial Differential Equations

References in corpus (4)

• Periodic constant mean curvature surfaces in H^2 x R
• Minimal surfaces in finite volume hyperbolic 3-manifolds N and in MxS(1), M a finite area hyperbolic surface
• Minimal ends in H2xR with finite total curvature and a Schoen type theorem
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Cited by in corpus (1)

• Minimal Surfaces in H^2xR