  Given a compact surface with boundary, we introduce a family of functionals
on the space of its Riemannian metrics, defined via eigenvalues of a
Steklov-type problem. We prove that each such functional is uniformly bounded
from above, and we characterize maximizing metrics as induced by free boundary
minimal immersions in some geodesic ball of a round sphere. Also, we determine
that 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 geodesic
balls of round spheres which are immersed by first eigenfunctions.
