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 $\mathbb S^n_+$. Similarly to 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. 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 catenoids in these spaces is 4 and the index of a geodesic $k$-ball is $2(n-k)$. For the proof of this 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.