Robin nullity in mode $|k|=1$ and asymptotic radius of the critical hyperbolic catenoid

Technical Summary: Robin Nullity in Mode $|k|=1$ and Asymptotic Radius of the Critical Hyperbolic Catenoid

Problem Statement
This note investigates the analytic properties of the critical hyperbolic catenoid family $\Sigma_a$, a one-parameter family of rotationally symmetric free boundary minimal annuli (FBMS) contained within a geodesic ball $B^3(r(a))$ in hyperbolic space $\mathbb{H}^3$. These surfaces, originally introduced by Mori and later revisited by do Carmo–Dajczer and Medvedev, interpolate between a degenerate configuration as the parameter $a \to (1/2)^+$ and an unbounded configuration as $a \to \infty$.

The primary objectives are to determine the Robin nullity and Morse index of the Jacobi operator restricted to the angular Fourier mode $|k|=1$, and to derive precise asymptotic expansions for the boundary radius $r(a)$ in both the large parameter limit ($a \to \infty$) and the degenerate limit ($a \to (1/2)^+$). This work aims to refine the lower bound on the Morse index established by Medvedev and provide closed-form expressions for geometric constants previously known only implicitly.

Methodology
The analysis relies on the explicit parametrization of $\Sigma_a$ provided by Medvedev, utilizing the conserved quantity $K(a) = \sqrt{a^2 - 1/4}$ associated with the rotational symmetry. The methodology proceeds through three main analytical strategies:

1. Geometric and Spectral Setup: The paper establishes the induced metric, the second fundamental form, and the Jacobi operator $L_{\Sigma_a} = \Delta_g + (|II|^2 - 2)$. It identifies the Killing–Jacobi fields generated by the Lie algebra $\mathfrak{so}(3,1)$ that preserve the bounding geodesic ball. Specifically, it isolates the subalgebra of spatial rotations that fix the geodesic axis of the catenoid.
2. Mode-by-Mode Decomposition: Following the strategy of Devyver for the Euclidean critical catenoid, the Jacobi operator is decomposed into angular Fourier modes. The analysis focuses on the radial ordinary differential equation (ODE) associated with the mode $|k|=1$. The proof of the nullity result utilizes Sturm–Liouville theory, specifically analyzing the zeros of the radial profile and the Wronskian of solutions to distinguish between odd and even parity sectors.
3. Asymptotic Analysis: For the large parameter limit, the implicit free boundary condition is analyzed using Laplace-type asymptotics. This involves evaluating an improper integral related to the angular profile $\phi(s)$ via the Beta function. For the degenerate limit, a perturbation analysis is performed near $a = 1/2$ by scaling the radial coordinate and solving a transcendental equation for the leading-order coefficient.

Key Contributions and Results

• Robin Nullity and Index in Mode $|k|=1$:
  The paper proves that for every $a > 1/2$, the Robin nullity of the Jacobi operator restricted to the mode $|k|=1$ is exactly 2. The kernel is spanned by the Killing–Jacobi fields associated with rotations $L_{12}$ and $L_{13}$ in the $(x_1, x_2)$ and $(x_1, x_3)$ planes.
  A central finding is the identification of the radial profile $f_*(s)$ of these fields in closed form:
  $$f_*(s) = \partial_s \Phi^0_a(s, 0) = \sinh(r(s)) \cdot r'(s),$$
  where $r(s)$ is the hyperbolic distance from the center of the ball. This structural identity links the Jacobi field directly to the derivative of the time coordinate in the hyperboloid model.
  Consequently, the Robin Morse index in mode $|k|=1$ is established to be 2. This result refines Medvedev's lower bound of $\text{ind}(\Sigma_a) \ge 4$ by confirming that the contribution from the $|k|=1$ sector is exactly 2, implying the remaining modes contribute at least 2 to the total index.

• Asymptotic Radius ($a \to \infty$):
  The paper derives the asymptotic expansion of the boundary radius $r(a)$ as $a \to \infty$:
  $$r(a) = \frac{3}{2} \log a + d_\infty + o(1).$$
  The constant $d_\infty$ is computed in closed form using the Beta function evaluation of the integral $\int_0^\infty \cosh(2t)^{-3/2} dt$:
  $$d_\infty = \log \left( \frac{\sqrt{2} \, \Gamma(1/4)^2}{\pi^{3/2}} \right) = \log \left( \frac{2\sqrt{2\pi}}{\Gamma(3/4)^2} \right).$$
  This provides the first explicit closed-form expression for this asymptotic constant.

• Degenerate Limit ($a \to (1/2)^+$):
  As the parameter approaches the lower bound, the radius behaves as:
  $$r(a) = c_* \sqrt{a - 1/2} \, (1 + o(1)).$$
  The constant $c_*$ is given by $c_* = \sigma_* \cosh \sigma_*$, where $\sigma_*$ is the unique positive fixed point of the equation $\sigma = \coth \sigma$. This characterizes the rate at which the surface collapses to a degenerate configuration.

Significance and Claims
The paper claims to provide a rigorous analytic refinement of the spectral properties of the critical hyperbolic catenoid. By explicitly identifying the kernel in the $|k|=1$ mode with specific Killing fields and proving the index count via Sturm–Liouville theory, the work bridges the gap between geometric intuition (isometric deformations) and spectral analysis.

The derivation of the closed-form constant $d_\infty$ is presented as a non-elementary step relying on special function theory (Gamma and Beta functions), offering a precise geometric invariant for the family. The results support the conjecture that the total Morse index is exactly 4, though the paper notes that proving the vanishing of the index in modes $|k| \ge 2$ and determining the exact index in mode $|k|=0$ remains an open problem. The work is framed as a short note establishing these specific analytic results, with further investigations into higher-order asymptotics and higher-dimensional analogs left as open questions.