Robin nullity and asymptotic geometry of the critical hyperbolic catenoid

Abstract: For each parameter $a>1$, the critical hyperbolic catenoid $Σa$ is a rotationally symmetric, free boundary minimal annulus in a geodesic ball $B^3(r(a))\subset\mathbb{H}^3$. The Morse index of $Σ_a$ is at least $4$ by Medvedev [7], who conjectures equality. In this paper we identify a new geometric and spectral phenomenon for the family ${Σ_a}{a>1}$, which we call "parameter-criticality", and study its consequences for the Robin spectrum. Specifically, we prove two main results: (I) Parameter-criticality (Theorem 1.5). The boundary radius $r(a)$ is non-monotone on $(1,\infty)$: it satisfies $r'(1^+)<0$ and $r(a)=\frac{3}{2}\log a+d_\infty+o(1)$ as $a\to\infty$ with $d_\infty=\log[Γ(1/4)/Γ(3/4)]-\frac{1}{2}\log(2π)$ (Theorem 1.4). Hence there exists a parameter-critical value $a^\sharp\in(1,\infty)$ with $r'(a^\sharp)=0$. (II) Robin nullity jump (Theorem 1.6). At every such $a^\sharp$, the Robin nullity of $Σ{a^\sharp}$ satisfies $\text{nul}(L{Σ{a^\sharp}})\geq 3$, with an additional kernel element in mode $k=0$ generated by the parametric variation field $j_a=\langle\partial_aΦ_a,ν\rangle_L|{a=a^\sharp}$, which we show is non-vanishing at the catenoid neck via the closed-form $j_a(0)=1/(2\sqrt{a^2-1})$. The argument requires the limit $r_0:=\lim_{a\to 1^+}r(a)$ characterized as the unique positive solution of the transcendental equation $\tanh(r_0)\,\tanh(2r_0/\sqrt{3})=\sqrt{3}/2$ (Theorem 1.3), giving a clean parametrization of the degeneration $Σ_a\toΣ_1$. The Robin nullity of $Σ_a$ in mode $|k|=1$ is shown to equal $2$ (Proposition 1.1); this extends to the hyperbolic setting the mode-by-mode Fourier decomposition technique of Devyver [2] for the Euclidean critical catenoid, and is used in the proof of (II) to identify the extra kernel as a mode-$k=0$ phenomenon.

• The paper introduces a parameter-criticality phenomenon where the free boundary truncation radius r(a) is non-monotonic and becomes stationary at a unique critical value a^sharp.
• It employs mode-by-mode separation to demonstrate that the Robin nullity of the Jacobi operator increases from 2 to at least 3 at a^sharp, differing from the Euclidean case.
• Analytic techniques yield explicit asymptotic formulas and closed-form expressions that deepen the understanding of spectral and bifurcation behavior in hyperbolic free boundary minimal surfaces.

Summary of "Robin nullity and asymptotic geometry of the critical hyperbolic catenoid" (2605.00617)

Introduction and Geometric Framework

This work analyzes the spectral and asymptotic geometric properties of a one-parameter family $\{\Sigma_a\}_{a>1}$ of free boundary minimal surfaces (FBMS) in geodesic balls $B^3(r(a))\subset \mathbb{H}^3$. Each $\Sigma_a$ is a rotationally symmetric minimal annulus meeting the boundary orthogonally. The parameter $a$ modulates the neck-width of the catenoid and uniquely fixes the geodesic radius $r(a)$ for the free boundary condition. These objects are natural hyperbolic analogues of critical catenoids in $\mathbb{R}^3$, with important connections to the spectrum and moduli of FBMS.

Parameter-Criticality and Double Critical Structure

A principal finding is a phenomenon termed parameter-criticality, which manifests as a non-monotonic dependence of the critical truncation radius $r(a)$ on the neck parameter $a$. It is shown that:

• As $a \to 1^+$, $r(a)$ approaches a finite value $B^3(r(a))\subset \mathbb{H}^3$0 (given as the positive solution to a transcendental equation).
• As $B^3(r(a))\subset \mathbb{H}^3$1, $B^3(r(a))\subset \mathbb{H}^3$2 increases asymptotically as $B^3(r(a))\subset \mathbb{H}^3$3, where $B^3(r(a))\subset \mathbb{H}^3$4 is explicitly described via gamma functions.

Therefore, $B^3(r(a))\subset \mathbb{H}^3$5 exhibits a critical point $B^3(r(a))\subset \mathbb{H}^3$6 with $B^3(r(a))\subset \mathbb{H}^3$7, in contrast to the Euclidean setting. At this value, the radius is stationary under parametric variations—a feature absent in flat space due to the uniqueness up to scaling. This double criticality—each $B^3(r(a))\subset \mathbb{H}^3$8 being FB-critical and the existence of $B^3(r(a))\subset \mathbb{H}^3$9 where the radius is critical in parameter space—has significant spectral implications.

Robin Spectrum and Nullity Jumps

The Jacobi operator for normal variations under the Robin boundary condition (pertinent to the second variation of area) is central to the analysis. The spectrum of

$\Sigma_a$0

is investigated by a mode-by-mode separation of variables approach.

Main Spectral Results

1. Mode $\Sigma_a$1 (Angular Fourier Modes):
   • For all $\Sigma_a$2, the Robin Jacobi problem in angular mode $\Sigma_a$3 has nullity exactly 2, connected to Killing fields from $\Sigma_a$4. This is shown through Fourier analysis and a Wronskian argument, generalizing Devyver's result for the Euclidean catenoid.
2. Mode $\Sigma_a$5 (Rotationally Invariant):
   • At $\Sigma_a$6, an additional element enters the kernel, generated by the parametric variation field $\Sigma_a$7 evaluated at $\Sigma_a$8. This field is shown analytically to be nontrivial at the catenoid neck, with $\Sigma_a$9. The occurrence of such a kernel element at $a$0 is directly tied to the vanishing of $a$1 due to a variation principle for FBMS moduli.
3. Total Robin Nullity Jump:
   • It is established that the Robin nullity satisfies $a$2 at parameter-critical points, representing a discrete jump from the generic value. The corresponding nullity in the Euclidean case is always 2, thus this is a robust hyperbolic phenomenon.

Asymptotic and Closed-form Descriptions

• The behavior as $a$3 is characterized via detailed matched asymptotics and leads to a closed expression for the limiting degenerate radius $a$4 as the unique solution to $a$5.
• The expansion as $a$6 is controlled by explicit integral formulas involving the Beta and Gamma functions.

Implications and Theoretical Directions

Spectral Theory & Bifurcation Analysis:

The enlargement of Robin nullity at $a$7 indicates deeper bifurcation structure in the moduli space of FBMS. Recent work by Cerezo constructs infinite families of non-rotational annuli via bifurcation from rotational catenoids. Enlarged spectral kernel at $a$8 aligns with the requirements for such bifurcations under general Lyapunov–Schmidt theories; a precise connection is left open.

Structural Conjectures:

The author conjectures uniqueness for $a$9 (i.e., exactly one radius-stationary parameter value), which would establish a single isolated spectral transition in the family. The open problems also include determining the full nullity in angular modes $r(a)$0 and $r(a)$1, exact Morse index computations, and explicit identification of bifurcation loci with spectral jumps.

Analytical Techniques

All proofs are analytic and symbolic, leveraging separation of variables in radial-angular coordinates, explicit computation with hyperbolic trigonometric functions, mode-wise analysis facilitated by symmetry and Killing-field structures, and asymptotic expansion (including matched asymptotics and root-finding for transcendental systems). The study notably avoids numerical or variational approximations, instead employing closed-form computations and direct analysis to relate geometric quantities to spectral data.

Conclusion

This paper discovers and details a parameter-induced critical phenomenon affecting the spectral properties of hyperbolic critical catenoids. It quantitatively links geometric parameterization to the enlargement of the Robin null space for the Jacobi operator, showing the nullity increases from 2 to at least 3 precisely at the parameter-critical value $r(a)$2. This marks a sharp deviation from the Euclidean case and has theoretical implications for the moduli and bifurcation theory of FBMS in hyperbolic space. Future investigations are suggested in the direction of uniqueness of $r(a)$3, full spectral characterization in remaining modes, Morse index determination, and rigorous bifurcation analysis connecting to non-rotational examples.