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

Abstract

Let ${\Sigma }_a\subset B^3(r(a))\subset {\mathbb{H}}^3$ ($a>1/2$) be the critical spherical catenoid of the Mori family, a free boundary minimal surface in the geodesic ball. The Medvedev conjecture [15] states ind$({\Sigma }_a)=4$ 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 [17, Cor. 4.4] with vanishing kernel in modes $|k|=0,|k|\ge 2$; the latter, not in [17], 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 $H(a):=\sinh r(a)/K(a)={\sigma }_{\ast }\cosh {\sigma }_{\ast }+C_0(a-\frac{1}{2})+O((a-\frac{1}{2}{)}^2)$ as $a\to (1/2{)}^{+}$, with $C_0=\frac{{\sigma }_{\ast }\cosh {\sigma }_{\ast }({\sinh }^2{\sigma }_{\ast }-1)(3{\sinh }^2{\sigma }_{\ast }-2)}{12{\sinh }^2{\sigma }_{\ast }}$, where ${\sigma }_{\ast }>0$ the unique positive root of $\sigma =\coth \sigma$, and $C_0>0$ by ${\sigma }_{\ast }>\log (1+\sqrt{2})$. The proof proceeds via three reductions: $(i)$ the strong 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 ${\varphi }_a>0$ of the parametric Jacobi field on the principal branch; $(iii)$ ${\varphi }_a>0$ reduces, under $B(s_0(a){)}^2>2K(a{)}^2$ $(G)$, to $H^{\prime }(a)>0$ via a const. Wronskian and Sturm separation. Aux results: Picone identity (base $f_{\ast }$) closing unconditionally the odd radial sector for $|k|\ge 2$; a second Picone identity (base $B$) proving $(E)$ unconditionally on $(1/2,1]$ and, via Hardy estimates, on $(1/2,A_{\ast }]$ ($A_{\ast }>1$); analytic closure of $(G)$ on $(1/2,1]$ via strict concavity of a transcendental function; an alternative proof of ind$({\Sigma }_a)\ge 4$ via Lorentz ambient coordinates.

v7: two auxiliary proofs repaired (asymptotic condition (G) near $a=1/2$, now self-contained; large-$a$ asymptotics of $y(a)$, via an exact identity) and statements adjusted accordingly; Main results unchanged