In this paper, we generalize White's regularity and structure theory for mean-convex mean curvature flow [45, 46, 48] to the setting with free boundary. A major new challenge in boundary is derive an a priori bound ratio between norm of second fundamental form curvature. We establish such via maximum principle triple-approximation scheme, which combines ideas from Edelen [9], Haslhofer-Hershkov...

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In this note we establish that finite-time singularities of the mean curvature flow of compact Riemannian submanifolds M t →֒ (N, h) are characterised by the blow up of the mean curvature.