Published Paper

Inserted: 1 sep 2020
Last Updated: 24 jun 2025

Journal: J. Differential Geom.
Volume: 130
Number: 2
Pages: 477-516
Year: 2025
Doi: https://doi.org/10.4310/jdg/1747158946

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

Download:

• Dynamical instability of minimal surfaces at flat singular points