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

Abstract

Medvedev proved that the critical spherical catenoid $Sigma_a$, the rotationally symmetric free boundary minimal annulus in a geodesic ball $B^3(r(a))$ in $H^3$ from the family of Mori and do Carmo-Dajczer, has Morse index at least 4, and conjectured equality. For each $a>1/2$ we establish three analytic results about $Sigma_a$. (I) Robin nullity and index in mode $|k|=1$. The Robin nullity of the Jacobi operator $L_{Σ_a}=Δ_g+(|II|^2-2)$ in angular Fourier mode $|k|=1$ equals $2$, 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 $Σ_a$ and send $\partial B^3(r(a))$ to itself. The radial profile admits the closed form $f_*(s)=\partial_sΦ_a^0(s,0)=\frac{d}{ds}[A(s)\coshφ(s)]=\sinh r(s)\cdot r'(s)$, where $r(s)$ is the geodesic distance from $p_0=(1,0,0,0)$. By Sturm--Liouville theory, the Robin Morse index of $Σ_a$ in mode $|k|=1$ also equals $2$, refining the lower bound of Medvedev. (II) Asymptotic radius. The boundary radius satisfies $r(a)=\tfrac{3}{2}\log a+d_\infty+o(1)$ as $a\to\infty$, with $d_\infty=\log[\sqrt{2}\,Γ(1/4)^2/π^{3/2}]=\log[2\sqrt{2π}/Γ(3/4)^2]$. The closed form for $d_\infty$ follows from a Beta-function evaluation of $I_\infty=\int_0^{\infty}\cosh(2t)^{-3/2}\,dt$. (III) Degenerate limit. As $a\to(1/2)^+$, $r(a)=c_*\sqrt{a-1/2}\,(1+o(1))$ with $c_*=σ_*\coshσ_*$, where $σ_*$ is the unique positive fixed point of $σ=\cothσ$. The proof of (I) follows the mode-by-mode strategy of Devyver for the Euclidean critical catenoid, with $\mathfrak{so}(3,1)$ replacing $\mathfrak{so}(3)$, supplemented by the closed-form identification $f_*=\partial_sΦ^0$ specific to the hyperbolic ambient. The proof of (II) is a Laplace-type asymptotic analysis of the implicit free boundary condition.