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

Abstract

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 \(\mathrm{MI}(n)\) of \(M_n\) satisfies the following asymptotic estimate as n tends to infinity. $$\begin{aligned} \mathop {{\text {Lim}}}_{n\rightarrow +\infty }\frac{\text {Log}(\mathrm{MI}(n))}{\sqrt{n}\text {Log}(\sqrt{n})} = 1. \end{aligned}$$We illustrate our results with an in-depth study of the numerical problem, providing exact values for the Morse index for \(n=2,\ldots ,100\), together with qualitative studies of \(\mathrm{MI}(n)\) and related geometric quantities for large values of n.