I. Dorofeev - A. Pigazzini

Index and nullity of critical spherical catenoids in hyperbolic three-space

preprint

Inserted: 27 oct 2026
Last Updated: 10 nov 2026

Year: 2026

18 pages. v3: added Theorem 2 and Section 9: the energy index of every critical spherical catenoid in H³ is four, disproving the inequality in Medvedev 10, Remark 7.

Abstract:

We prove that every critical spherical catenoid in a geodesic ball of hyperbolic three-space has Morse index four and nullity two. This resolves Medvedev’s index conjecture throughout the parameter range of the catenoid family. The kernel consists precisely of the normal components of rotations that preserve the center of the ball and tilt the axis of the catenoid. As consequences, the radius of the supporting ball increases strictly along the family, and each critical spherical catenoid admits a smooth family of free boundary constant mean curvature deformations in its supporting ball. The proof combines a global boundary inequality obtained from a Wronskian, a positive Dirichlet eigenfunction supplied by the time component of the unit normal, and a Riccati comparison for the angular mode k = 2. A ground-state identity then yields positivity of every angular mode k ≥ 2. We also show that the energy index of every critical spherical catenoid in H^3, that is, the index of the second variation of its energy, equals four; hence the inequality between the energy index and the spectral index conjectured by Medvedev for geodesic balls of Hⁿ fails.

Keywords: Free boundary minimal surfaces, Morse index, Robin eigenvalue problem, nullity, hyperbolic space, critical catenoid, equivariant nondegeneracy, constant mean curvature., energy index

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• 2610.35651v3.pdf