Bounded Diameter Under Mean Curvature Flow

We prove that for the mean curvature flow of closed embedded hypersurfaces, intrinsic diameter stays uniformly bounded as approaches first singular time, provided all singularities are neck or conical type. In particular, assuming Ilmanen’s multiplicity one conjecture and no cylinder conjecture, we conclude in two-dimensional case always bounded. also obtain sharp $$L^{n-1}$$ bound curvature. The key ingredients our proof Łojasiewicz inequalities by Colding–Minicozzi Chodosh–Schulze, solution mean-convex neighbourhood Choi, Haslhofer, Hershkovits White. Our results improve prior Gianniotis–Haslhofer, where control has been obtained under more restrictive assumption is globally two-convex.