On the Morse index of higher-dimensional free boundary minimal catenoids
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2017
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| _version_ | 1866913866173120512 |
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| author | Smith, Graham Stern, Ari Tran, Hung Zhou, Detang |
| author_facet | Smith, Graham Stern, Ari Tran, Hung Zhou, Detang |
| contents | 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 $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1709_00977 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | On the Morse index of higher-dimensional free boundary minimal catenoids Smith, Graham Stern, Ari Tran, Hung Zhou, Detang Differential Geometry 53A10 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 $n$. |
| title | On the Morse index of higher-dimensional free boundary minimal catenoids |
| topic | Differential Geometry 53A10 |
| url | https://arxiv.org/abs/1709.00977 |