A geometric Ginzburg-Landau problem
type: Item
instance of: scholarly article
title: A geometric Ginzburg-Landau problem
zbMATH Open document ID: 1266.49091
author: Roger Moser
published in: Mathematische Zeitschrift
publication date: 2013-03-28T00:00:00Z
review text: Let \(M\) be a closed surface smoothly embedded in \(\mathbb R^3\). If \(\nu\) is its normal vector, \(A\) is its second fundamental form, and \({\mathcal H}^n\) is the \(n\)-dimensional Hausdorff measure, then for \(\varepsilon > 0\), the integral \[ E_\varepsilon(M)=\frac{\sqrt\varepsilon}2\int\limits_M\left(|A|^2+\frac{\nu_1^2}{\varepsilon^2}\right)d{\mathcal H}^2 \] is the Wilmore-like energy functional. In this paper, the author studies the behavior of this functional, and of surfaces with reasonably small energy, as the parameter \(\varepsilon\) tends to \(0\). It is shown that for any \(L>0\), \[ 2^{3/4}\pi\sqrt L\leq\liminf\limits_{\varepsilon\searrow 0}\inf\limits_{\Lambda(M)\geq L}E_\varepsilon(M)\leq\limsup\limits_{\varepsilon\searrow 0}\inf\limits_{\Lambda(M)\geq L}E_\varepsilon(M)\leq c\sqrt L, \] where \(\Lambda(M)=\sup\limits_{x,y\in M}(|x-y|)\) and \(c=\frac{2\,\pi^{5/4}(6\,b-a)}{\sqrt {3\,a}}\), with \(a=\frac{\Gamma(7/6)}{\Gamma(2/3)}\) and \(b=\frac{\Gamma(13/18)}{\Gamma(2/9)}\). Moreover, for embedded spheres, a compactness result is obtained under appropriate energy bounds.
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 49Q20
zbMATH DE Number: 6149052
zbMATH Keywords: Ginzburg-Landau problem
zbMATH Keywords: Willmore energy
zbMATH Keywords: minimal energy
MaRDI profile type: Publication
full work available at URL: s00209-012-1029-5
OpenAlex ID: W2056617233
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DOI: 10.1007/S00209-012-1029-5
description: scientific article; zbMATH DE number 6149052