A geometric Ginzburg-Landau problem
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.
scientific article; zbMATH DE number 6149052