Self-shrinkers whose asymptotic cones fatten
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909397990506496 |
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| author | Ketover, Daniel |
| author_facet | Ketover, Daniel |
| contents | For each positive integer $g$ we use variational methods to construct a genus $g$ self-shrinker $Σ_g$ in $\mathbb{R}^3$ with entropy less than $2$ and prismatic symmetry group $\mathbb{D}_{g+1}\times\mathbb{Z}_2$. For $g$ sufficiently large, the self-shrinker $Σ_g$ has two graphical asymptotically conical ends and the sequence $Σ_g$ converges on compact subsets to a plane with multiplicity two as $g\to\infty$. Angenent-Chopp-Ilmanen conjectured the existence of such self-shrinkers in 1995 based on numerical experiments. Using these surfaces as initial conditions for large $g$, we obtain examples of mean curvature flows in $\mathbb{R}^3$ with smooth initial non-compact data that evolve non-uniquely after their first singular time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_01240 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Self-shrinkers whose asymptotic cones fatten Ketover, Daniel Differential Geometry For each positive integer $g$ we use variational methods to construct a genus $g$ self-shrinker $Σ_g$ in $\mathbb{R}^3$ with entropy less than $2$ and prismatic symmetry group $\mathbb{D}_{g+1}\times\mathbb{Z}_2$. For $g$ sufficiently large, the self-shrinker $Σ_g$ has two graphical asymptotically conical ends and the sequence $Σ_g$ converges on compact subsets to a plane with multiplicity two as $g\to\infty$. Angenent-Chopp-Ilmanen conjectured the existence of such self-shrinkers in 1995 based on numerical experiments. Using these surfaces as initial conditions for large $g$, we obtain examples of mean curvature flows in $\mathbb{R}^3$ with smooth initial non-compact data that evolve non-uniquely after their first singular time. |
| title | Self-shrinkers whose asymptotic cones fatten |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2407.01240 |