Preprint

Inserted: 5 oct 2026
Last Updated: 5 oct 2026

Year: 2026

Abstract:

Medvedev [8] proved that the critical spherical catenoid $\Sigma_a\subset B^3(r(a))\subset\mathbb{H}^3$ --- the rotationally symmetric free boundary minimal annulus in the family introduced by Mori [9] and reconsidered by do Carmo--Dajczer [3] --- has Morse index at least $4$, and conjectured equality [8, Rem. 5.6]. In this note we compute the Robin nullity and index of $\Sigma_a$ \emph{exactly} in the angular Fourier mode $
k
=1$, and we determine the boundary radius $r(a)$ in its two limiting regimes. In detail:

(I) Robin nullity and index in mode $
k
=1$. The Robin nullity of the Jacobi operator $L_{\Sigma_a}=\Delta_g+(
II
^2-2)$ in angular Fourier mode $
k
=1$ equals $2$ for every $a>1/2$, with kernel spanned by the Killing--Jacobi fields associated to the rotations $L_{12},L_{13}\in\mathfrak{so}(3,1)$ that fix the geodesic axis of $\Sigma_a$ and send $\partial B^3(r(a))$ to itself. The radial profile of these Jacobi fields admits the closed form \[ f_*(s)=\partial_s\Phi_a^0(s,0)=\frac{d}{ds}\bigl[A(s)\cosh\varphi(s)\bigr] =\sinh r(s)\cdot r'(s), \] where $r(s)=\mathrm{dist}_{\mathbb{H}^3}(p_0,\Phi_a(s,0))$. As a consequence of Sturm--Liouville theory and the structure of the zeros of $f_*$, the Robin Morse index of $\Sigma_a$ in mode $
k
=1$ equals $2$ for every $a>1/2$, refining from the analytic side the lower bound $\mathrm{ind}(\Sigma_a)\geq 4$ of Medvedev [8].

(II) Asymptotic radius. The boundary radius admits the asymptotic expansion \[ r(a)\;=\;\tfrac{3}{2}\log a+d_\infty+o(1)\qquad(a\to\infty), \qquad d_\infty\;=\;\log\!\frac{\sqrt{2}\,\Gamma(1/4)^2}{\pi^{3/2}}\;=\;\log\!\frac{2\sqrt{2\pi}}{\Gamma(3/4)^2}. \] The closed form for $d_\infty$ follows from a closed evaluation of the improper integral $I_\infty=\int_0^{\infty}\cosh(2t)^{-3/2}\,dt$ via the Beta function.

(III) Degenerate limit. As $a\to(1/2)^+$, $r(a)=c_*\sqrt{a-1/2}\,(1+o(1))$ with $c_*=\sigma_*\cosh\sigma_*$, where $\sigma_*$ is the unique positive fixed point of $\sigma=\coth\sigma$.

The proof of (I) follows the mode-by-mode strategy of Devyver [2] for the Euclidean critical catenoid, with the group $\mathfrak{so}(3,1)$ replacing $\mathfrak{so}(3)$, supplemented by the closed-form identification $f_*=\partial_s\Phi^0$ specific to the hyperbolic ambient. The proof of (II) is a Laplace-type asymptotic analysis of the implicit free boundary condition.

Keywords: Free boundary minimal surfaces, Spherical catenoid, Jacobi operator, Robin nullity, Morse index in Fourier modes;, Killing fields, Asymptotic geometry, Gamma function values

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