On free boundary minimal submanifolds in geodesic balls in Hn and Sn+

Mathematische Zeitschrift. 2025. Vol. 310. No. 1. Article 10.

Medvedev V.

We consider free boundary minimal submanifolds in geodesic balls in the hyperbolic space Hn and in the round upper hemisphere Sn+. Recently, Lima and Menezes have found a connection between free boundary minimal surfaces ingeodesic balls in Sn+ and maximal metrics for a functional, defined on the setof Riemannian metrics on a given compact surface with boundary. This connec-tion is similar to the connection between free boundary minimal submanifolds inEuclidean balls and the critical metrics of the functional ”the k-th normalized Steklov eigenvalue”, introduced by Fraser and Schoen. We define two naturalfunctionals on the set of Riemannian metrics on a compact surface with bound-ary. One of these functionals is the high order generalization of the functional,introduced by Lima and Menezes. We prove that the critical metrics for thesefunctionals arise as metrics induced by free boundary minimal immersions ingeodesic balls in Hn and in Sn+, respectively. We also prove a converse statement.Besides that, we discuss the (Morse) index of free boundary minimal subman-ifolds in geodesic balls in Hn or Sn+. We show that the index of the criticalspherical catenoid in a geodesic ball in S3+ is 4 and the index of the critical spher-ical catenoid in a geodesic ball in H3 is at least 4. We prove that the index of ageodesic k-ball in a geodesic n-ball in Hn or Sn+ is n− k. For the proof of thesestatements we introduce the notion of spectral index similarly to the case of freeboundary minimal submanifolds in a unit ball in the Euclidean space.

Keywords: Morse index, индекс Морса, free boundary minimal submanifolds, минимальные подмногообразия со свободной границей, spectral index, Robin problem, спектральный индекс, задача Робэна

Publication based on the results of: Integrability in mathematical physics, representation theory, algebraic and differential geometry (2025)