An εε-regularity theorem for the mean curvature flow

In this paper, we will derive a small energy regularity theorem for mean curvature flow of arbitrary dimension and codimension. It says that if the parabolic integral of |A|2|A|2 around a point in space–time is small, then the mean curvature flow cannot develop singularity at this point. We prove, as an application, that the two-dimensional Hausdorff measure of the singular set of the mean curvature flow from a surface to a Riemannian manifold must be zero.