Parabolic rectifiability of the Brakke flow

We prove that the support of the canonical space-time measure for a Brakke flow is a parabolic $(k+2)$-rectifiable set. As a consequence, we obtain that at almost all points along the flow, with respect to this canonical space-time measure, there exists a unique, static, planar tangent flow, and that various notions of density for the flow agree at these points. Moreover, following on from our previous work The space-time-Grassmann measure of the Brakke flow', we continue to develop the approach to the Brakke flow as a space-time-Grassmann measure. We prove that the standard notion of convergence for Brakke flows, coming from the compactness theorem of Ilmanen (7.1 of Elliptic regularization and partial regularity for motion by mean curvature’), is equivalent to the convergence of these space-time-Grassmann Radon measures. This gives an alternate notion of varifold convergence to the one exhibited in 7.1(ii) of `Elliptic regularization and partial regularity for motion by mean curvature’.