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On free boundary minimal submanifolds in geodesic balls in ℍn and 𝕊n+
Cornell University
,
2023.
We consider free boundary minimal submanifolds in geodesic balls in the hyperbolic space ℍn and in the round upper hemisphere 𝕊n+. 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 ℍn and in 𝕊n+, respectively. We also prove a converse statement. Besides that, we discuss the (Morse) index of free boundary minimal submanifolds in geodesic balls in ℍn or 𝕊n+. 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.