Analytic local resolution of Medvedev's Morse index conjecture for the critical hyperbolic catenoid in $\mathbb{H}^3$

TL;DR

This paper analytically resolves the local version of Medvedev's Morse index conjecture for the critical hyperbolic catenoid in hyperbolic space, establishing the index and nullity for a range of parameters.

Contribution

It provides the first analytic proof of the local strong Medvedev conjecture for the hyperbolic catenoid, confirming the index and nullity values near the critical parameter.

Findings

Confirmed ind$(\Sigma_a)=4$ and nul$(\Sigma_a)=2$ for $a$ close to 1/2.

Derived an explicit expansion for $H(a)$ as $a o (1/2)^+$.

Established positivity of the parametric Jacobi field $\phi_a$ under certain conditions.

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] states ind$(\Sigma_a)=4$ for all $a>1/2$. We study its strong form: ind$(\Sigma_a)=4$ and nul$(\Sigma_a)=2$. The nullity condition nul$(\Sigma_a)=2$ combines the mode-$|k|=1$ result $\text{nul}_R(\Sigma_a)|_{|k|=1}=2$ of [10, Cor. 4.4] with vanishing kernel in modes $|k|=0,|k|\ge2$; the latter, not in [10], is established here for $a\in(1/2,1/2+\delta_0)$. The main result is the analytic local resolution of the strong Medvedev conjecture: $\exists\delta_0>0$ s.t. ind$(\Sigma_a)=4$, nul$(\Sigma_a)=2$ for all $a\in(1/2,1/2+\delta_0)$. This follows from the expansion $H(a):=\sinh r(a)/K(a)=\sigma_*\cosh\sigma_*+C_0(a-\frac12)+O((a-\frac12)^2)$ as $a\to(1/2)^+$, with $C_0=\frac{\sigma_*\cosh\sigma_*(\sinh^2\sigma_*-1)(3\sinh^2\sigma_*-2)}{12\sinh^2\sigma_*}$, where $\sigma_*>0$ the unique positive root of $\sigma=\coth\sigma$, and $C_0>0$ by $\sigma_*>\log(1+\sqrt2)$. The proof proceeds via three reductions: $(i)$ the Medvedev conjecture is equivalent to $\mu_0^{\mathrm{even}}(2)>0$ $(E)$ and $\mu_2(0)>0$ with non-degeneracy in mode $0$ $(F)$; $(ii)$ $\mu_2(0)>0$ reduces, via a Sturm shooting-count argument, to $\phi_a>0$ of the parametric Jacobi field on the principal branch; $(iii)$ $\phi_a>0$ reduces, under $\sinh r(a)>2K(a)$ $(G)$, to $H'(a)>0$ via a constant Wronskian and Sturm separation. Auxiliary results: a Picone identity (base $f_*$) closing unconditionally the odd radial sector for $|k|\ge2$; a second Picone identity (base $B$) proving $(E)$ unconditionally on $(1/2,1]$ and, via Hardy estimates, on $(1/2,A_*]$ ($A_*>1$); analytic closure of $(G)$ on $(1/2,1]$ via strict concavity of a transcendental function; an alternative proof of ind$(\Sigma_a)\ge4$ via Lorentz ambient coordinates.