Robin nullity in mode $|k|=1$ and asymptotic radius of the critical hyperbolic catenoid
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918494976606208 |
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| author | Pigazzini, Alexander |
| author_facet | Pigazzini, Alexander |
| contents | For each parameter $a>1/2$, the critical hyperbolic catenoid $Σ_a$ 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 $Σ_a$.
(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 $\mathrm{ind}(Σ_a)\geq 4$ 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 $d_\infty=\log[\sqrt{2}\,Γ(1/4)^2/π^{3/2}]=\log[2\sqrt{2π}/Γ(3/4)^2]$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_11244 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Robin nullity in mode $|k|=1$ and asymptotic radius of the critical hyperbolic catenoid Pigazzini, Alexander Differential Geometry Analysis of PDEs Spectral Theory 53A10, 53C42, 58E12, 58J50 For each parameter $a>1/2$, the critical hyperbolic catenoid $Σ_a$ 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 $Σ_a$. (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 $\mathrm{ind}(Σ_a)\geq 4$ 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 $d_\infty=\log[\sqrt{2}\,Γ(1/4)^2/π^{3/2}]=\log[2\sqrt{2π}/Γ(3/4)^2]$. 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. |
| title | Robin nullity in mode $|k|=1$ and asymptotic radius of the critical hyperbolic catenoid |
| topic | Differential Geometry Analysis of PDEs Spectral Theory 53A10, 53C42, 58E12, 58J50 |
| url | https://arxiv.org/abs/2605.11244 |