Invariant singular minimal surfaces
type: Item
instance of: scholarly article
title: Invariant singular minimal surfaces
zbMATH Open document ID: 1396.53010
author: Rafael López
published in: Annals of Global Analysis and Geometry
publication date: 2018-06-26T00:00:00Z
review text: Let \(\alpha\) denote a real number, and for a unit vector \({\mathbf a} \in {\mathbb R}^3\), let \({\mathbb R}^3_+({\mathbf a}) := \{p\in {\mathbb R}^3\,:\, \langle p, {\mathbf a}\rangle >0\}\) be the half-space determined by \({\mathbf a}\). For a smooth immersion \(\phi: M \to {\mathbb R}^3_+({\mathbf a})\) of an oriented connected compact surface \(M\), the potential \(\alpha\)-energy of \(\phi\) in the direction \({\mathbf a}\) is defined by \[ E(\phi) = \int_M \langle p, {\mathbf a}\rangle^\alpha\, dM. \] It is well known that the immersion \(\phi\) is a critical point of the energy \(E\) for all compactly supported variations if and only if \[ 2H(p) = \alpha \frac{\langle N(p), {\mathbf a}\rangle}{\langle \phi(p), {\mathbf a}\rangle} \quad (p\in M),\tag{1} \] where \(N\) and \(H\) denote the Gauss map and the mean curvature of \(\phi\). \(M\) is said to be a \textit{singular minimal surface} or \(\alpha\)-\textit{minimal surface} if the mean curvature \(H\) satisfies (1). \(M\) is a minimal surface when \(\alpha = 0\), and from now on, we will assume \(\alpha \neq 0\) unless stated otherwise. In this paper, the author classifies all singular minimal surfaces in Euclidean space that are invariant by a uniparametric group of translations and rotations. Among them, the author proves that the only cylindrical singular minimal surfaces are planes parallel to the vector \({\mathbf a}\) or \(\alpha\)-catenary cylinder where the rulings are orthogonal to \({\mathbf a}\). An \(\alpha\)-catenary curve \(\gamma : I \to {\mathbb R}^2\) can be parametrized by a positive function \(f\) in the form \(f(s)f''(s) = \alpha (1+f'(s)^2)\), and it gives an example of \(\alpha\)-minimal surface \(M\) by considering the generalized cylinder with basis \(\gamma\). In case of \({\mathbf a} = (0,0,1)\), such \(M\) is called an \(\alpha\)-catenary cylinder. The second main result of the paper is the following. If an \(\alpha\)-minimal surface \(M\) with respect to the direction \({\mat…
reviewed by: Gabjin Yun
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 49Q05
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C44
zbMATH DE Number: 6894770
zbMATH Keywords: minimal surface
zbMATH Keywords: singular minimal surface
zbMATH Keywords: cylindrical surface
zbMATH Keywords: rotational surface
zbMATH Keywords: potential energy
MaRDI profile type: Publication
full work available at URL: s10455-017-9586-9
OpenAlex ID: W2773592598
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DOI: 10.1007/S10455-017-9586-9
description: scientific article; zbMATH DE number 6894770