Control of the isoperimetric deficit by the Willmore deficit
type: Item
instance of: scholarly article
title: Control of the isoperimetric deficit by the Willmore deficit
zbMATH Open document ID: 1252.49069
author: Matthias Röger
author: Reiner Schätzle
publication date: 2012-09-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/1106.6213
Mathematics Subject Classification ID: 49Q10
Mathematics Subject Classification ID: 53A05
Mathematics Subject Classification ID: 53A30
zbMATH DE Number: 6079880
zbMATH Keywords: isoperimetric inequality
zbMATH Keywords: Willmore functional
zbMATH Keywords: isoperimetric deficit
zbMATH Keywords: Willmore deficit
MaRDI profile type: Publication
OpenAlex ID: W2949172317
arXiv ID: 1106.6213
cites work: Optimal rigidity estimates for nearly umbilical surfaces
cites work: A mass transportation approach to quantitative isoperimetric inequalities
DOI: 10.1524/ANLY.2012.1140
published in: Analysis (München)
review text: Given an embedded sphere \(\Sigma\subset \mathbb R^3\) of area \(\text{ar}(\Sigma)\) enclosing a region \(\Omega_\Sigma\subset \mathbb R^3\) of volume \(\text{vol}(\Omega_\Sigma)\), the authors consider the isoperimetric ratio NEWLINE\[NEWLINE I(\Sigma)={\text{ar}(\Sigma)\over {\text{vol}(\Omega_\Sigma)^{2/3}}}, NEWLINE\]NEWLINE and the Willmore functional NEWLINE\[NEWLINE W(\Sigma)={1\over 4} \int_\Sigma |\vec{H}|^2\,dH^2, NEWLINE\]NEWLINE where the mean curvature vector \(\vec{H}\) is defined as the sum of the principal curvatures times the unit inner normal to \(\Sigma\). Both the isoperimetric ratio and the Willmore functional are invariant by dilations and translations and have the round spheres as the only minimizers. The \textit{deficit} of both functionals is the difference from their optimal values.NEWLINENEWLINEThe main result of the paper, Theorem 1.1, is an estimate of the isoperimetric ratio in terms of the Willmore functional: the authors prove that for all \(c_0>0\), there exists a universal constant \(C>0\) such that NEWLINE\[NEWLINE I(\Sigma)-I(S^2)\leq C\,(W(\Sigma)-W(S^2)), NEWLINE\]NEWLINE whenever \(\Sigma\) is an embedded sphere with \(I(\Sigma)-I(S^2)\leq c_0\). The authors show by example in Remark 1.2 that the linear growth in the right hand side of the inequality is optimal. They also remark that the isoperimetric deficit cannot be estimated from below by the Willmore deficit simply using a \(C^1\) deformation of a round sphere with large \(C^2\) norm. In Remark 1.3, the authors compare their result, using the Alexandrov-Fenchel inequalities, to the ones of Zhou, who obtained a lower bound for the Willmore functional for convex hypersurfaces [\textit{J. Zhou}, in: Suh, Young Jin (ed.) et al., Proceedings of the 10th international workshop on differential geometry, Taegu, Korea, November 10--11, 2005. Taegu: Kyungpook National University. 57--71 (2006; Zbl 1158.49046)].NEWLINENEWLINEThe main technical ingredient in the proof of Theorem 1.1 i…
reviewed by: Manuel Ritoré
description: scientific article; zbMATH DE number 6079880