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

TL;DR

This paper characterizes complete immersed minimal surfaces with finite total curvature in hyperbolic product spaces, establishing key properties and providing new characterizations for specific minimal surfaces.

Contribution

It proves that properness, finite topology, and asymptotic behavior characterize finite total curvature surfaces in these spaces, extending previous results.

Findings

Characterization of finite total curvature surfaces in $ ext{H}^2 imes ext{R}$

New criteria for minimal Scherk-type graphs and catenoids

Finite topology plus asymptotic conditions imply finite total curvature in $ ilde{ m PSL}_2( ext{R}, au)$

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 $\widetilde{\rm PSL}_2(\mathbb{R},\tau)$ has finite topology and each one of its ends is asymptotic to a geodesic polygon at infinity, then it must have finite total curvature.