Geometry of the magnetic Steklov problem on Riemannian annuli

We study the geometry of the first two eigenvalues of a magnetic Steklov problem on an annulus Sigma (a compact Riemannian surface with genus zero and two boundary components), the magnetic potential being the harmonic one-form having flux nu is an element of & Ropf; around any of the two boundary components. The resulting spectrum can be seen as a perturbation of the classical, non-magnetic Steklov spectrum, obtained when nu = 0 and studied e.g., by Fraser and Schoen in Schoen in [9, 10]. We obtain sharp upper bounds for the first and the second normalized eigenvalues and we discuss the geometry of the maximizers. Concerning the first eigenvalue, we isolate a noteworthy class of maximizers which we call alpha-surfaces: they are free-boundary surfaces which are stationary for a weighted area functional (depending on the flux) and have proportional principal curvatures at each point; in particular, they belong to the class of linear Weingarten surfaces. Inspired by [9], we then study the second normalized eigenvalue for a fixed flux nu and prove the existence of a maximizer for rotationally invariant metrics. Moreover, the corresponding eigenfunctions define a free-boundary immersion in the unit ball of & Ropf;(3). Finally, we prove that the second normalized eigenvalue associated to a flux nu has an absolute maximum when nu = 0, the corresponding maximizer being the critical catenoid.