On parabolicity and area growth of minimal surfaces
type: Item
instance of: scholarly article
title: On parabolicity and area growth of minimal surfaces
zbMATH Open document ID: 1281.53015
author: Robert W. Neel
published in: The Journal of Geometric Analysis
publication date: 2013-09-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/1004.4544
review text: The paper establishes parabolicity and quadratic area growth for minimal surfaces-with-boundary contained in regions of \(\mathbb R^3\) which are within a sublogarithmic factor of the exterior of a cone. The author is interested in controlling the geometry and conformal structure of such surfaces under the assumption that they are contained in certain rotationally symmetric regions. The motivation for this comes in part from the fact that such surfaces arise as representatives of ends of complete minimal surfaces. No use is made of universal superharmonic functions. Instead, stochastic methods are used, which have the additional feature of giving a type of parabolicity in a more general context than Brownian motion on a minimal surface.
reviewed by: Kaarin Riives
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 58J65
Mathematics Subject Classification ID: 60H30
zbMATH DE Number: 6203568
zbMATH Keywords: minimal surfaces
zbMATH Keywords: parabolicity
zbMATH Keywords: area growth
zbMATH Keywords: Brownian motion
MaRDI profile type: Publication
OpenAlex ID: W2025321990
arXiv ID: 1004.4544
cites work: Minimal surfaces in \(\mathbb{R}^3\) properly projecting into \(\mathbb{R}^2\)
cites work: The topological, geometry and conformal structure of properly embedded minimal surfaces
cites work: Q3059475
cites work: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds
cites work: The classical theory of minimal surfaces
DOI: 10.1007/S12220-011-9280-2
description: scientific article; zbMATH DE number 6203568