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arXiv · 1709.00977

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

Graham Smith·Ari Stern·Hung Tran·Detang Zhou
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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 $\Bbb{R}^{n+1}$. We show that the Morse index $\text{MI}(n)$ of $M_n$ satisfies the following asymptotic estimate as $n$ tends to infinity. $$ \lim_{n\rightarrow+\infty}\frac{\text{Log}(\text{MI}(n))}{\sqrt{n}\text{Log}(\sqrt{n})} = 1. $$ 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 $n$.

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Graham Smith, Ari Stern, Hung Tran, Detang Zhou. 2017-09-04. On the Morse index of higher-dimensional free boundary minimal catenoids. https://doi.org/10.1007/s00526-021-02049-8

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