Hadamard variational formula for eigenvalues of the Stokes equations and its application

Based on the explicit representation of the Hadamard variational formula [1] for eigenvalues of the Stokes equations, we investigate the geometry of the domain in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb R^3$$\end{document}. It turns out that if the first variation of some eigenvalue of the Stokes equations for all volume preserving perturbations vanishes, then the domain is necessarily diffeomorphic to the 2-dimensional torus T2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^2$$\end{document}.