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 $$\mathbb {R}^n$$ . 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 and Soner in J Differ Geom 43(4):694–737, 1996) is non fattening, smooth and attains the initial value in the Hausdorff sense. Our results generalize the ones from Hershkovits (Geom Topol 21(1):441–484, 2017), in which the co-dimension one case was studied. We also prove a general (short time) smooth uniqueness result, generalizing the one for evolution of smooth submanifolds, which may be of independent interest, even in co-dimension one.