Maximum principles for minimal surfaces in \(\mathbb{R}^ 3\) having noncompact boundary and a uniqueness theorem for the helicoid
type: Item
instance of: scholarly article
title: Maximum principles for minimal surfaces in \(\mathbb{R}^ 3\) having noncompact boundary and a uniqueness theorem for the helicoid
zbMATH Open document ID: 0838.53014
DOI: 10.1007/BF02570484
published in: Manuscripta Mathematica
publication date: 1996-06-10T00:00:00Z
full work available at URL: https://eudml.org/doc/156097
review text: For the sake of simplicity two main results will be quoted: ``Let \(M_1\) be a complete and \(M_2\) a properly immersed minimal surface in \(\mathbb{R}^3\) such that the principal curvatures of \(M_1\) and \(M_2\) are bounded in modulus by some constant \(k\). We assume furthermore \(M_1\cap M_2= \emptyset\), \(\min\{d(M_1, \partial M_2), d(M_2, \partial M_1)\}\geq \delta> 0\), and that \(M_1\) is ``bandlike'', i.e., the radius of geodesic discs contained in the interior of \(M_1\) is bounded by some fixed constant \(L\). Then it follows that \(d(M_1, M_2)> 0\)'' (where \(d(A, B)= \inf\{|p- q|\mid p\in A, q\in B\}\), \(A, B\subset \mathbb{R}^3\)). This theorem is derived from the Harnack inequality for elliptic differential equations. The second result, which is an application of the first one, reads as follows: ``Let \(S\) be a line segment through the origin in the \((x, y)\)-plane with endpoints \(p_1\) and \(p_2\) such that \(\max\{|p_1|, |p_2|\}\leq \varepsilon_0\). Let \(M\) be a complete properly immersed minimal surface, with bounded curvature and with boundary \(h_1\cup h_2\), where \(h_1\) and \(h_2\) are the helices generated by \(p_1\) and \(p_2\), and such that \[ \lim_{(x, y, z)\to \infty} {x^2+ y^2\over z^2}= 0,\qquad (x, y, z)\in M. \] Then \(M\) coincides with the helicoidal domain generated by \(S\).'' Here \(\varepsilon_0= \text{sinh}(u_0)= 1.50888\dots\) is defined by \(\text{cosh}(u_0)= u_0\text{sinh}(u_0)\).
reviewed by: Erich Hoy
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 824169
zbMATH Keywords: helicoidal surface
zbMATH Keywords: minimal surface
author: Jaime Ripoll
author: F. Tomi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 824169