Isoperimetric problem for the first curl eigenvalue

Abstract

We consider an isoperimetric problem involving the smallest positive and largest negative curl eigenvalues on abstract ambient manifolds, with a focus on the standard model spaces. We in particular show that the corresponding eigenvalues on optimal domains, assuming optimal domains exist, must be simple in the Euclidean and hyperbolic setting. This generalises a recent result by Enciso and Peralta-Salas who showed the simplicity for axisymmetric optimal domains with connected boundary in Euclidean space. We then generalise another recent result by Enciso and Peralta-Salas, namely that the points of any rotationally symmetric optimal domain with connected boundary in Euclidean space which are closest to the symmetry axis must disconnect the boundary, to the hyperbolic setting, as well as strengthen it in the Euclidean case by removing the connected boundary assumption.