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

Abstract: We consider free boundary minimal submanifolds in geodesic balls in the hyperbolic space $\mathbb H^n$ and in the round upper hemisphere $\mathbb S^n_+$. Recently, Lima and Menezes have found a connection between free boundary minimal surfaces in geodesic balls in $\mathbb S^n_+$ and maximal metrics for a functional, defined on the set of Riemannian metrics on a given compact surface with boundary. This connection is similar to the connection between free boundary minimal submanifolds in Euclidean balls and the critical metrics of the functional "the $k$-th normalized Steklov eigenvalue", introduced by Faser and Schoen. We define two natural functionals on the set of Riemannian metrics on a compact surface with boundary. One of these functionals is the high order generalization of the functional, introduced by Lima and Menezes. We prove that the critical metrics for these functionals arise as metrics induced by free boundary minimal immersions in geodesic balls in $\mathbb H^n$ and in $\mathbb S^n_+$, respectively. We also prove a converse statement. Besides that, we discuss the (Morse) index of free boundary minimal submanifolds in geodesic balls in $\mathbb H^n$ or $\mathbb S^n_+$. We show that the index of the critical spherical catenoid in a geodesic ball in $\mathbb S^3_+$ is 4 and the index of the critical spherical catenoid in a geodesic ball in $\mathbb H^3$ is at least 4. We prove that the index of a geodesic $k$-ball in a geodesic $n$-ball in $\mathbb H^n$ or $\mathbb S^n_+$ is $n-k$. For the proof of these statements we introduce the notion of spectral index similarly to the case of free boundary minimal submanifolds in a unit ball in the Euclidean space.