Liouville-type properties for embedded minimal surfaces
type: Item
instance of: scholarly article
title: Liouville-type properties for embedded minimal surfaces
zbMATH Open document ID: 1117.53009
author: Joaquín Pérez
author: Antonio Ros
author: William H. III Meeks
published in: Communications in Analysis and Geometry
publication date: 2007-02-05T00:00:00Z
review text: The following properties of a noncompact Riemann surface \(M\) without boundary are considered {\parindent=8mm \begin{itemize}\item[(1)]\(M\) admits a nonconstant positive superharmonic function (equivalently, \(M\) is transient for Brownian motion); \item[(2)] \(M\) admits a nonconstant positive harmonic function; \item[(3)] \(M\) admits a nonconstant bounded harmonic function. \end{itemize}} A noncompact Riemann surface \(M\) without boundary is said to be recurrent for Brownian motion if it does not satisfy property (1). Clearly \((3)\Rightarrow(2)\Rightarrow(1)\). In this paper, it is determined which of the above three properties is satisfied by a complete embedded minimal surface \(M\subset\mathbb{R}^3\), in terms of its topology and geometry: For example, the following theorem is proved: Theorem 1.4. Let \(M\subset\mathbb{R}^3\) be a nonflat \(k\)-periodic complete embedded minimal surface such that \(M/G\) has finite topology. Then, every positive harmonic function on \(M\) is constant. \(M\) is said to be \(k\)-periodic if its group of isometries \(G\) has rank \(k=0,1,2,3\). These results are applied to study the question of existence of complete embedded minimal surfaces which are \(a\)-stable for some \(a> 0\).
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 5121964
zbMATH Keywords: minimal surfaces
zbMATH Keywords: flat three-manifolds
zbMATH Keywords: harmonic function
zbMATH Keywords: Brownian motion
zbMATH Keywords: \(a\)-stable surfaces
reviewed by: Alexander A. Borisenko
MaRDI profile type: Publication
full work available at URL: cag.2006.v14.n4.a5
OpenAlex ID: W2032229349
DOI: 10.4310/CAG.2006.V14.N4.A5
description: scientific article; zbMATH DE number 5121964