Dynamical instability of minimal surfaces at flat singular points
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866911005425008640 |
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| author | Stuvard, Salvatore Tonegawa, Yoshihiro |
| author_facet | Stuvard, Salvatore Tonegawa, Yoshihiro |
| contents | Suppose that a countably $n$-rectifiable set $Γ_0$ is the support of a multiplicity-one stationary varifold in $\mathbb{R}^{n+1}$ with a point admitting a flat tangent plane $T$ of density $Q \geq 2$. We prove that, under a suitable assumption on the decay rate of the blow-ups of $Γ_0$ towards $T$, there exists a non-constant Brakke flow starting with $Γ_0$. This shows non-uniqueness of Brakke flow under these conditions, and suggests that the stability of a stationary varifold with respect to mean curvature flow may be used to exclude the presence of flat singularities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2008_13728 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Dynamical instability of minimal surfaces at flat singular points Stuvard, Salvatore Tonegawa, Yoshihiro Analysis of PDEs Differential Geometry 53E10, 49Q05 Suppose that a countably $n$-rectifiable set $Γ_0$ is the support of a multiplicity-one stationary varifold in $\mathbb{R}^{n+1}$ with a point admitting a flat tangent plane $T$ of density $Q \geq 2$. We prove that, under a suitable assumption on the decay rate of the blow-ups of $Γ_0$ towards $T$, there exists a non-constant Brakke flow starting with $Γ_0$. This shows non-uniqueness of Brakke flow under these conditions, and suggests that the stability of a stationary varifold with respect to mean curvature flow may be used to exclude the presence of flat singularities. |
| title | Dynamical instability of minimal surfaces at flat singular points |
| topic | Analysis of PDEs Differential Geometry 53E10, 49Q05 |
| url | https://arxiv.org/abs/2008.13728 |