Parabolic minimal surfaces in $\mathbb{M}^{2}\times\mathbb{R}$

Vanderson Lima

Vol. 288 (2017), No. 1, 171–188

DOI: 10.2140/pjm.2017.288.171

Abstract

Let ${\mathbb{M}}^2$ be a complete noncompact orientable surface of nonnegative curvature. We prove some theorems involving parabolicity of minimal surfaces in ${\mathbb{M}}^2\times ℝ$. First, using a characterization of $\delta$-parabolicity we prove that under additional conditions on $\mathbb{M}$, an embedded minimal surface with bounded Gaussian curvature is proper. The second theorem states that under some conditions on $\mathbb{M}$, if $\Sigma$ is a properly immersed minimal surface with finite topology and one end in $\mathbb{M}\times ℝ$, which is transverse to a slice $\mathbb{M}\times \left\{t\right\}$ except at a finite number of points, and such that $\Sigma \cap \left(\mathbb{M}\times \left\{t\right\}\right)$ contains a finite number of components, then $\Sigma$ is parabolic. In the last result, we assume some conditions on $\mathbb{M}$ and prove that if a minimal surface in $\mathbb{M}\times ℝ$ has height controlled by a logarithmic function, then it is parabolic and has a finite number of ends.

Keywords

minimal surfaces, parabolicity, properness

Mathematical Subject Classification 2010

Primary: 49Q05, 53AXX

Milestones

Received: 11 November 2015

Revised: 26 September 2016

Accepted: 3 October 2016

Published: 8 April 2017