Vol. 288, No. 1, 2017

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ISSN: 0030-8730
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 M2 be a complete noncompact orientable surface of nonnegative curvature. We prove some theorems involving parabolicity of minimal surfaces in M2 × . First, using a characterization of δ-parabolicity we prove that under additional conditions on M, an embedded minimal surface with bounded Gaussian curvature is proper. The second theorem states that under some conditions on M, if Σ is a properly immersed minimal surface with finite topology and one end in M × , which is transverse to a slice M ×{t} except at a finite number of points, and such that Σ (M ×{t}) contains a finite number of components, then Σ is parabolic. In the last result, we assume some conditions on M and prove that if a minimal surface in M × 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
Authors
Vanderson Lima
Instituto de Matemática e Estatística
Universidade do Estado do Rio de Janeiro (UERJ)
Rua São Francisco Xavier, 524
Pavilhão Reitor João Lyra Filho, 6∘ andar - Bloco B
20550-900 Rio de Janeiro-RJ
Brazil