Non-uniqueness of Brakke flows starting from minimal surfaces with singularities

Abstract

We prove the existence of a genuinely time-dependent Brakke flow starting from $Γ_0 \subset \mathbb{R}^{n+1}$ whose associated multiplicity-one varifold is stationary, provided that, at some singular point, the scale-invariant $L^2$ distance of $Γ_0$ from an $n$-dimensional plane has sufficiently small limsup as the scale tends to zero. This yields the dynamical instability of $Γ_0$, a notion recently introduced by Stuvard and Tonegawa, and hence the non-uniqueness of Brakke flows starting from $Γ_0$. A notable feature of our result is that it holds without assuming the uniqueness of tangent cones at the singular point.