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

TL;DR

This paper analyzes the Robin nullity, index, asymptotic radius, and degenerate limits of the critical hyperbolic catenoid, revealing explicit formulas and asymptotic behaviors in hyperbolic space.

Contribution

It provides the first detailed analysis of the Robin nullity, index, and asymptotic properties of hyperbolic catenoids, extending Euclidean results to hyperbolic geometry.

Findings

Robin nullity in mode |k|=1 equals 2, with kernel spanned by Killing--Jacobi fields.

Boundary radius grows as (3/2) log a + constant as a→∞.

As a approaches 1/2 from above, r(a) scales with sqrt(a-1/2) with explicit constants.

Abstract

For each parameter $a>1/2$, the critical hyperbolic catenoid $\Sigma_a$ is a rotationally symmetric, free boundary minimal annulus in a geodesic ball $B^3(r(a))\subset\mathbb{H}^3$, in the family of Mori, do Carmo--Dajczer, and Medvedev. We establish three analytic results about $\Sigma_a$. (I) Robin nullity and index in mode $|k|=1$. The Robin nullity of the Jacobi operator $L_{\Sigma_a}=\Delta_g+(|II|^2-2)$ 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 $\Sigma_a$ and send $\partial B^3(r(a))$ to itself. The radial profile admits the closed form $f_*(s)=\partial_s\Phi_a^0(s,0)=\frac{d}{ds}[A(s)\cosh\varphi(s)]=\sinh r(s)\cdot r'(s)$, where $r(s)$ is the geodesic distance from $p_0=(1,0,0,0)$. By Sturm--Liouville theory, the Robin Morse index of $\Sigma_a$ in mode $|k|=1$ also equals $2$, refining the lower bound $\mathrm{ind}(\Sigma_a)\geq 4$ 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 $d_\infty=\log[\sqrt{2}\,\Gamma(1/4)^2/\pi^{3/2}]=\log[2\sqrt{2\pi}/\Gamma(3/4)^2]$. 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 $c_*=\sigma_*\cosh\sigma_*$, where $\sigma_*$ is the unique positive fixed point of $\sigma=\coth\sigma$. The proof of (I) follows the mode-by-mode strategy of Devyver for the Euclidean critical catenoid, with $\mathfrak{so}(3,1)$ 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.