preprint
Inserted: 13 may 2026
Last Updated: 14 may 2026
Year: 2026
Abstract:
Let $\Sigma_a\subset B^3(r(a))\subset\mathbb{H}^3$ ($a>1/2$) be the critical hyperbolic catenoid of the Mori family, a free boundary minimal surface in the geodesic ball. The Medvedev conjecture 8 asserts that $\text{ind}(\Sigma_a)=4$ for every $a>1/2$. We consider here the $\textit{strong form}$ of this conjecture, namely $\text{ind}(\Sigma_a)=4$ and $\text{nul}(\Sigma_a)=2$. The nullity condition $\text{nul}(\Sigma_a)=2$ combines the mode-$
k
=1$ result $\text{nul}_R(\Sigma_a)\vert_{
k
=1}=2$ of 10, (Cor. 4.4) with the additional requirement of vanishing kernel in modes $
k
=0$ and $
k
\geq 2$; this latter requirement, not addressed in 10, is part of the analytic content of the present paper and is established in the local regime $a\in(1/2,1/2+\delta_0)$.
The principal quantitative result of the paper is the analytic local resolution of the strong Medvedev conjecture: there exists $\delta_0>0$ such that $\text{ind}(\Sigma_a)=4$ and $\text{nul}(\Sigma_a)=2$ for every $a\in(1/2,1/2+\delta_0)$. This follows from an explicit closed form for the leading asymptotic coefficient of $H(a):=\sinh r(a)/K(a)$ as $a\to(1/2)^+$, \[ H(a)=\sigma_*\cosh\sigma_*+C_0\,(a-\tfrac{1}{2})+O\bigl((a-\tfrac{1}{2})^2\bigr),\qquad C_0=\frac{\sigma_*\cosh\sigma_*\,(\sinh^2\sigma_*-1)(3\sinh^2\sigma_*-2)}{12\sinh^2\sigma_*}, \] where $\sigma_*>0$ is the unique positive root of $\sigma=\coth\sigma$, together with the analytic proof that $C_0>0$ via $\sigma_*>\log(1+\sqrt{2})$.
The route to the local resolution proceeds through three analytic reductions of independent interest: $\textit{(i)}$ the Medvedev conjecture is shown to be equivalent to the conjunction of two spectral conditions, $\mu_0^{\mathrm{even}}(2)>0$ (condition (E)) and $\mu_2(0)>0$ together with spectral non-degeneracy in mode $0$ (condition (F)); $\textit{(ii)}$ the eigenvalue inequality $\mu_2(0)>0$ is reduced, via a Sturm shooting-count argument, to the geometric positivity $\phi_a>0$ of the parametric Jacobi field on the principal branch; $\textit{(iii)}$ the positivity $\phi_a>0$ is in turn reduced, under the strict geometric inequality $\sinh r(a)>2K(a)$ (condition (G)), to the one-dimensional scalar differential inequality $H'(a)>0$, via a constant Wronskian identity and a Sturm separation argument.
Auxiliary results include: a Picone identity with base $f_*$ yielding the unconditional closure of the odd radial sector in modes $
k
\geq 2$; a second Picone identity with base $B$ proving (E) unconditionally on $(1/2,1]$ and, via explicit Hardy estimates, on $(1/2,A_*]$ for some $A_*>1$; the analytic closure of (G) on $(1/2,1]$ via strict concavity of a transcendental function; and a short alternative proof of the lower bound $\text{ind}(\Sigma_a)\geq 4$ via the four Lorentz ambient coordinates as test functions.
Keywords: Free boundary minimal surfaces, Jacobi operator, Morse index, hyperbolic catenoid, Robin eigenvalue problem, Picone identity, Sturm-Liouville theory
Download: