부산대학교 도서관

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.
Comment: 31 pages, 3 figures. In v2 we discuss more extensively the range of applicability of our result in the context of the state of the art on the analysis of branched minimal surfaces. This is the final version, to appear on the Journal of Differential Geometry