Accepted Paper

Inserted: 1 sep 2020
Last Updated: 7 may 2024

Journal: J. Differential Geom.
Year: 2024

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 $Q \geq 2$. 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.

Keywords: varifolds, mean curvature flow, singularities of minimal surfaces

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• Dynamical instability of minimal surfaces at flat singular points