Control of the isoperimetric deficit by the Willmore deficit (Q2907397)

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 is the existence of a conformal parametrization \(\psi:S^2\to\Sigma\) such that the \(W^{2,2}(S^2)\)-norm of \(\psi-\text{Id}\) is bounded above by a universal constant times the integral over \(\Sigma\) of \(|\vec{H}|^2-4K\), where \(K\) is the Gauss curvature of \(\Sigma\) [\textit{C. de Lellis} and \textit{S. Müller}, J. Differ. Geom. 69, No. 1, 75--110 (2005; Zbl 1087.53004)].