Isoperimetric problem for the first curl eigenvalue

TL;DR

This paper investigates an isoperimetric problem related to curl eigenvalues on various manifolds, establishing eigenvalue simplicity, boundary properties, and connections to Killing fields, extending recent Euclidean and hyperbolic results.

Contribution

It generalizes recent results on eigenvalue simplicity and boundary properties to broader settings, including hyperbolic spaces, and links the existence of Killing-Beltrami fields to ambient geometry.

Findings

Eigenvalues on optimal domains are simple in Euclidean and hyperbolic spaces.

Points closest to the symmetry axis in optimal domains disconnect the boundary.

Second variation inequalities relate Killing-Beltrami fields to ambient space geometry.

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 getting rid of the connected boundary assumption. Lastly, we show how a second variation inequality related to the isoperimetric problem may be used in order to relate the existence of Killing-Beltrami fields to the geometry of the ambient space.