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

Abstract

Let $Σ_a\subset B^3(r(a))\subset\mathbb{H}^3$ ($a>1/2$) be the critical spherical catenoid of the Mori family, a free boundary minimal surface in the geodesic ball. The Medvedev conjecture [15] states ind$(Σ_a)=4$ for all $a>1/2$. We study its strong form: ind$(Σ_a)=4$ and nul$(Σ_a)=2$. The nullity condition nul$(Σ_a)=2$ combines the mode-$|k|=1$ result $\text{nul}_R(Σ_a)|_{|k|=1}=2$ of [17, Cor. 4.4] with vanishing kernel in modes $|k|=0,|k|\ge2$; the latter, not in [17], is established here for $a\in(1/2,1/2+δ_0)$. The main result is the analytic local resolution of the strong Medvedev conjecture: $\existsδ_0>0$ s.t. ind$(Σ_a)=4$, nul$(Σ_a)=2$ for all $a\in(1/2,1/2+δ_0)$. This follows from the expansion $H(a):=\sinh r(a)/K(a)=σ_*\coshσ_*+C_0(a-\frac12)+O((a-\frac12)^2)$ as $a\to(1/2)^+$, with $C_0=\frac{σ_*\coshσ_*(\sinh^2σ_*-1)(3\sinh^2σ_*-2)}{12\sinh^2σ_*}$, where $σ_*>0$ the unique positive root of $σ=\cothσ$, and $C_0>0$ by $σ_*>\log(1+\sqrt2)$. The proof proceeds via three reductions: $(i)$ the strong Medvedev conjecture is equivalent to $μ_0^{\mathrm{even}}(2)>0$ $(E)$ and $μ_2(0)>0$ with non-degeneracy in mode $0$ $(F)$; $(ii)$ $μ_2(0)>0$ reduces, via a Sturm shooting-count argument, to $ϕ_a>0$ of the parametric Jacobi field on the principal branch; $(iii)$ $ϕ_a>0$ reduces, under $B(s_0(a))^2>2K(a)^2$ $(G)$, to $H'(a)>0$ via a const. Wronskian and Sturm separation. Aux results: 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$(Σ_a)\ge4$ via Lorentz ambient coordinates.