Index of the critical catenoid

Abstract

We show that the critical catenoid, as a free boundary minimal surface of the unit ball in R 3 , has index 4. We also prove that a free boundary minimal surface of the unit ball in R 3 , that is not a flat disk, has index at least 4.

Compactness analysis for free boundary minimal hypersurfaces

“…First of all, the work of Fraser and Schoen [15][16][17] has clarified the link of these geometric objects with extremal metrics, of given volume, for the first Steklov eigenvalue of manifolds with boundary. Secondly, one has witnessed a few interesting classification results towards conjectural characterisations of the critical catenoid among free boundary minimal hypersurfaces in the Euclidean unit ball [5,12,28,36,38]. Thirdly, and perhaps most relevantly to this paper, various techniques have been developed to prove existence results and produce novel concrete examples.…”

A New Eigenvalue Problem for Free Boundary Minimal Submanifolds in the Unit Ball

“…which implies that ind(Σ) ≥ n, unless Σ is contained in a subspace of dimension n − 1. In [7], this lower bound was improved in the case k = 2, n = 3, and it was shown there that for Σ 2 ⊂ B 3 orientable which is not a flat disk, ind(Σ) ≥ 4 = 3 + 1. Furthermore, this inequality is sharp, since it was proven there that the so-called critical catenoid has index precisely equal to 4 (see also [17], [18]).…”

“…By [9,Prop. 8.1] (see also [7,Cor. 7.2]), if Σ is a free boundary minimal surface in B 3 with index 4, then λ 0 , the first eigenvalue of the Jacobi operator with Dirichlet boundary conditions, has to be zero.…”

“…We now recall a result concerning the solution of the Dirichlet problem for the Jacobi operator, whose proof follows along the lines of the proof of [7,Lemma 4.1], and thus will be omitted: Lemma 6.4. Assume that λ 0 = 0 and let ξ be a first eigenfunction of the Jacobi operator with Dirichlet boundary conditions.…”

“…A. Fraser and R. Schoen [8] proved that the index of a non-flat free boundary minimal surface in the unit 3-ball is at least 3; later on, it was shown in [7] that actually the index of such a surface is at least 4. It was shown independently in [7], [17] and [18], that the index of the so-called critical catenoid, a minimal annulus with free boundary in the unit 3-ball introduced by A. Fraser and R. Schoen in [8], is equal to 4. A natural question is then whether the critical catenoid is the unique (up to congruence) free boundary minimal surface in the unit 3-ball with index 4.…”