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We consider free boundary minimal submanifolds in geodesic balls in thehyperbolic space $\mathbb H^n$ and in the round upper hemisphere $\mathbbS^n_+$. Recently, Lima and Menezes have found a connection between freeboundary minimal surfaces in geodesic balls in $\mathbb S^n_+$ and maximalmetrics for a functional, defined on the set of Riemannian metrics on a givencompact surface with boundary. This connection is similar to the connectionbetween free boundary minimal submanifolds in Euclidean balls and the criticalmetrics of the functional \say{the $k$-th normalized Steklov eigenvalue},introduced by Faser and Schoen. We define two natural functionals on the set ofRiemannian metrics on a compact surface with boundary. One of these functionalsis the high order generalization of the functional, introduced by Lima andMenezes. We prove that the critical metrics for these functionals arise asmetrics induced by free boundary minimal immersions in geodesic balls in$\mathbb H^n$ and in $\mathbb S^n_+$, respectively. We also prove a conversestatement. Besides that, we discuss the (Morse) index of free boundary minimalsubmanifolds in geodesic balls in $\mathbb H^n$ or $\mathbb S^n_+$. We showthat the index of the critical spherical catenoids in these spaces is 4 and theindex of a geodesic $k$-ball is $2(n-k)$. For the proof of these statements weintroduce the notion of spectral index similarly to the case of free boundaryminimal submanifolds in a unit ball in the Euclidean space.

On free boundary minimal submanifolds in geodesic balls in $\mathbb H^n$ and $\mathbb S^n_+$