Convexity estimates for mean curvature flow with free boundary

We prove the estimates of [3] and [2] for finite-time singularities of mean-convex, mean curvature flow with free boundary in a barrier S. Here S   can be any embedded, oriented surface in Rn+1Rn+1 of bounded geometry and positive inscribed radius. We also prove the estimate [4] in the case of convex flows and S=SnS=Sn, which gives an alternative proof to [6].