Isoperimetric problem for the first curl eigenvalue (Q2102096)

In this article, the author considers the problem of characterizing eigenvalues of the curl operator on subdomains in ambient manifolds. The corresponding eigenfunctions naturally appear in many physical contexts, such as Beltrami fields in fluid dynamics. It is thus of great interest to understand how these eigenfunctions and eigenvalues depend on the domain or ambient manifold considered. In particular, one may search for \textit{optimal domains} with given volume contained in an ambient manifold, which extremize the smallest positive or largest negative eigenvalues: \begin{align*} \text{Sub}_c^V(\mathcal{R}) \rightarrow \mathbb{R}, \quad \overline{M} \mapsto \lambda_{+}(\overline{M}). \end{align*} Here \(\mathbb{R}^3\), the hyperbolic space \(\mathcal{H}^3\) and the sphere \(S^3\) serve as natural model cases for the ambient manifold. While it is an open problem whether such extremizers exist, the main results of the article establish properties of optimal domains (assuming existence). In particular, the author investigates whether optimal domains can be rotationally symmetric. Main results of the article include: \begin{itemize} \item In the case of the hyperbolic ambient space the corresponding eigenvalues (if they exist) are simple. \item In the case of \(S^3\) the first positive eigenvalue is either simple or a suitable coordinate projection necessarily satisfies further symmetry. \item In \(\mathcal{H}^3\) there are no rotationally symmetric optimal domains with convex cross sections. \item For \(\mathbb{R}^3\) no rotationally symmetric domain which intersects the \(z\)-axis can be optimal. Furthermore, if a rotationally symmetric optimal domain does not intersect the \(z\)-axis then the lengths of certain parts of boundaries of connected components of the cross sections necessarily have to satisfy certain inequalities. \end{itemize}