On the Morse index of higher-dimensional free boundary minimal catenoids

For all $n$, we define the $n$-dimensional critical catenoid $M_n$ to be the unique rotationally symmetric, free boundary minimal hypersurface of non-trivial topology embedded in the closed unit ball in ${\mathbb{R}}^{n+1}$. We show that the Morse index $\text{MI}(n)$ of $M_n$ satisfies the following asymptotic estimate as $n$ tends to infinity. $$\underset{n\to +\mathrm{\infty }}{\lim }\frac{\text{Log}(\text{MI}(n))}{\sqrt{n}\text{Log}(\sqrt{n})}=1.$$ We also study the numerical problem, providing exact values for the Morse index for $n=2,\cdots ,100$, together with qualitative studies of $\text{MI}(n)$ and related geometric quantities for large values of $n$.