Mean Curvature Flow of Arbitrary Co-Dimensional Reifenberg Sets

Abstract

We study the existence and uniqueness of smooth mean curvature flow, in arbitrary dimension and co-dimension, emanating from so called $k$-dimensional $(\varepsilon,R)$ Reifenberg flat sets in $\mathbb{R}^n$. Our results generalize the ones from a previous paper by the author, in which the co-dimension one case (i.e. $k=n-1$) was studied. For $\varepsilon$ fixed, this class is general enough to include (i) all $C^2$ sub-manifolds (ii) all Lipschitz sub-manifolds with Lipschitz constant less than $\varepsilon$ (iii) some sets with Hausdorff dimension larger than $k$. The Reifenberg condition, roughly speaking, says that the set has a weak metric notion of a $k$-dimensional tangent plane at every point and scale, but those tangents are allowed to tilt as the scales vary. We show that if the Reifenberg parameter $\varepsilon$ is small enough, the (arbitrary co-dimensional) level set flow (in the sense of Ambrosio-Soner ) is non fattening, smooth and attains the initial value in the Hausdorff sense. In particular, our result generalizes a result of Wang and, in fact, all known existence and uniqueness results for smooth mean curvature flow in arbitrary co-dimension. The largest deviation from the proof of the co-dimension case comes in the proof of uniqueness (i.e. non-fattening), where one is forced to work with the viscosity notion of the high co-dimensional level set flow, rather than Ilmanen's more geometric definition. This study leads to a general (short time) smooth uniqueness result, generalizing the one for evolution of smooth sub-manifolds, which may be of independent interest, even in co-dimension one.