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