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

TL;DR

This paper investigates the Morse index of higher-dimensional free boundary minimal catenoids, providing asymptotic estimates, exact numerical values for dimensions 2 to 100, and qualitative analysis of geometric properties.

Contribution

It introduces the critical $n$-dimensional catenoid, derives its Morse index asymptotics, and offers extensive numerical data and qualitative insights for large dimensions.

Findings

Asymptotic estimate of Morse index as n grows large

Exact Morse index values for n=2 to 100

Qualitative analysis of geometric quantities for large n

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 $n$.