Parabolic Minimal Surfaces in $\mathbb{M}^{2}\times\mathbb{R}$

TL;DR

This paper investigates the parabolicity and properness of minimal surfaces in the product space of a non-compact surface with non-negative curvature and the real line, establishing conditions for these properties and their implications.

Contribution

It provides new theorems linking parabolicity, properness, and geometric conditions of minimal surfaces in $ ext{M}^2 imes ext{R}$, extending understanding of their global behavior.

Findings

Properness of minimal surfaces with bounded Gaussian curvature under certain conditions.

Parabolicity of minimal surfaces with finite topology and one end, transverse to slices.

Minimal surfaces with logarithmic height growth are parabolic with finitely many ends.

Abstract

Let $\mathbb{M}^{2}$ be a complete non compact orientable surface of non negative curvature. We prove in this paper some theorems involving parabolicity of minimal surfaces in $\mathbb{M}^{2}\times\mathbb{R}$. 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 $\mathbb{M}\times\mathbb{R}$, which is transverse to a slice $\mathbb{M}\times\{t\}$ except at a finite number of points, and such that $\Sigma\cap(\mathbb{M}\times\{t\})$ 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 $\mathbb{M}\times\mathbb{R}$ has height controlled by a logarithmic function, then it is parabolic and has a finite number of ends.