[Paper Review] The Morse index of the critical catenoid

Graham Smith, Detang Zhou|arXiv (Cornell University)|Sep 6, 2016

Geometric Analysis and Curvature Flows6 citations

TL;DR

This paper proves that the critical catenoid—a rotationally symmetric free boundary minimal surface in the unit ball of R³—has Morse index exactly 4, using separation of variables and Riccati equation analysis on eigenfunctions of the Jacobi operator under Robin boundary conditions. The result resolves a key spectral property of this fundamental example in free boundary minimal surface theory.

ABSTRACT

We show that the rotationally symmetric free boundary minimal catenoid in the unit ball in $\Bbb{R}^3$ has Morse index equal to $4$.

Motivation & Objective

• To determine the Morse index of the critical catenoid, a canonical example of a free boundary minimal surface in the unit ball of R³.
• To resolve the open problem of computing the Morse index for this fundamental surface, which has been previously unknown despite its central role in the theory.
• To extend spectral understanding of free boundary minimal surfaces, analogous to results for minimal surfaces in spheres but significantly more challenging due to the absence of Wilmore energy constraints.
• To provide a complete spectral analysis of the Jacobi operator on the critical catenoid, identifying the number of negative eigenvalues and their geometric origin.

Proposed method

• Applying separation of variables to the Jacobi operator J = -Δ - Tr(A²) on the critical catenoid, decomposing eigenfunctions into Fourier modes.
• Reducing the spectral problem to solving Riccati-type ordinary differential equations for each Fourier mode, with initial conditions derived from geometric variations.
• Analyzing the behavior of solutions to the Riccati equation for each mode to determine the existence and number of negative eigenvalues via the intermediate value theorem.
• Using the Robin boundary condition f = ∂ₙf to ensure perturbations preserve the free boundary condition up to first order.
• Classifying eigenfunctions by their Fourier modes (0, ±1) and analyzing even and odd solutions separately for each mode.
• Establishing that only modes 0, +1, and -1 contribute negative eigenvalues, with two in mode 0 and one each in ±1, leading to total index 4.

Experimental results

Research questions

• RQ1 What is the Morse index of the critical catenoid as a free boundary minimal surface in the unit ball of R³?
• RQ2 How many negative eigenvalues does the Jacobi operator have on the critical catenoid, and what is their geometric origin?
• RQ3 Can the spectral properties of the critical catenoid be fully analyzed using separation of variables and Riccati equation techniques under Robin boundary conditions?
• RQ4 Does the critical catenoid satisfy a uniqueness property analogous to Urbano’s theorem for Clifford tori in S³, based on Morse index?
• RQ5 Why is the free boundary case significantly more difficult than the closed case (e.g., minimal tori in S³), despite apparent similarities?

Key findings

• The Morse index of the critical catenoid is exactly 4, as proven via spectral analysis of the Jacobi operator.
• There are two negative eigenvalues in the 0-th Fourier mode, corresponding to dilatations and translations along the axis of revolution.
• There is one negative eigenvalue in the ±1 Fourier modes, corresponding to translations and rotations in directions orthogonal to the axis.
• The eigenfunctions in the 0-th mode are associated with the two-dimensional family of geometric variations: dilations and axial translations.
• The eigenfunctions in the ±1 modes are associated with the two-dimensional family of variations: translations and rotations in the plane perpendicular to the axis.
• The analysis confirms that no further negative eigenvalues exist beyond these four, due to the behavior of Riccati solutions and the intermediate value theorem applied to the boundary value condition.