Singularities of Mean Curvature Flow with Bounded Total Curvature

We prove that every tangent flow at the first singular time of a smooth, closed, embedded mean curvature flow has smooth support, provided that the total curvature satisfies $\sup_{t\in[0,T)}\int_{M_t}|A|^n\,dμ_t<\infty$. This extends Ilmanen's smoothness result for tangent flows to higher dimension