Calculus of Variations and Geometric Measure Theory

S. Stuvard - Y. Tonegawa

Dynamical instability of minimal surfaces at flat singular points

created by stuvard on 01 Sep 2020
modified on 03 Sep 2020


Submitted Paper

Inserted: 1 sep 2020
Last Updated: 3 sep 2020

Year: 2020

ArXiv: 2008.13728 PDF
Links: arXiv webpage


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