Index characterization for free boundary minimal surfaces

Abstract

In this paper, we compute the Morse index of a free boundary minimal submanifold from data of two simpler problems. The first is the fixed boundary problem and the second is concered with the Dirichlet-to-Neumann map associated with the Jacobi operator. As an application, we show that the Morse index of a free boundary minimal annulus is equal to 4 if and only if it is the critical catenoid.

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

“…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]).…”

“…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.…”

“…Steklov, Robin, Dirichlet), and each of them is relevant for the study of the index. Indeed, in [7] and [18] the Steklov and Dirichlet spectrum of the critical catenoid are studied in order to compute the index, while in [17] the Robin spectrum is used for the same purpose. Nonetheless, while for the critical catenoid the components of the unit normal are Steklov eigenfunctions for the Jacobi operator, this is not the case in general; in [19], the critical catenoid is even characterized as the unique non-flat free boundary minimal surface of the unit 3-ball, for which the components of the unit normal are Steklov eigenfunctions for the Jacobi operator.…”

Minimal hypersurfaces in the ball with free boundary

“…One reason for this is that incredibly powerful complex analytical techniques that apply for surfaces do not seem to carry over to hypersurfaces. Nevertheless, progress continues to be made: see Sargent [21], Ambrozio, Carlotto-Sharp [3], Smith-Stern-Tran-Zhou [23] and Tran [25] for some new index bounds for minimal hypersurfaces with free boundary, Mondino-Spadaro [18] for a new characterisation of free boundary minimal submanifolds, and Li-Zhou [14,15] for far-reaching min-max and regularity theory, including an extension of the classical program of Almgren [1,2] (see Pitts [20] and Schoen-Simon [22] for further classical theory) to the case of minimal hypersurfaces with free boundary, for example.…”

Index of the critical catenoid

“…Remark 1.4. While this article was in preparation, the author has been informed of related works by G. Smith and D. Zhou [18] on the one hand, and by H. Tran [19] on the other, in which the index of the critical catenoid is also computed. Our proof and H. Tran's use the Steklov spectrum of the Jacobi operator, while G. Smith and D. Zhou's use the Robin spectrum.…”