Suppose that a countably $n$-rectifiable set $Gamma_0$ 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 $Q≥2$. We prove that, under a suitable assumption on the decay rate of the blow-ups of $Gamma_0$ towards $T$, there exists a non-constant Brakke flow starting with $Gamma_0$. 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.
Dynamical instability of minimal surfaces at flat singular points / S. Stuvard, Y. Tonegawa. - (2020 Aug 31).
Dynamical instability of minimal surfaces at flat singular points
S. Stuvard
;
2020
Abstract
Suppose that a countably $n$-rectifiable set $Gamma_0$ 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 $Q≥2$. We prove that, under a suitable assumption on the decay rate of the blow-ups of $Gamma_0$ towards $T$, there exists a non-constant Brakke flow starting with $Gamma_0$. 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.File | Dimensione | Formato | |
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