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Given a compact surface with boundary, we introduce a family of functionalson the space of its Riemannian metrics, defined via eigenvalues of aSteklov-type problem. We prove that each such functional is uniformly boundedfrom above, and we characterize maximizing metrics as induced by free boundaryminimal immersions in some geodesic ball of a round sphere. Also, we determinethat the maximizer in the case of a disk is a spherical cap of dimension two,and we prove rotational symmetry of free boundary minimal annuli in geodesicballs of round spheres which are immersed by first eigenfunctions.

Eigenvalue problems and free boundary minimal surfaces in spherical caps