Abstract
We prove that for the mean curvature flow of closed embedded hypersurfaces, the intrinsic diameter stays uniformly bounded as the flow approaches the first singular time, provided all singularities are of neck or conical type. In particular, assuming Ilmanen’s multiplicity one conjecture and no cylinder conjecture, we conclude that in the two-dimensional case the diameter always stays bounded. We also obtain sharp \(L^{n-1}\) bound for the curvature. The key ingredients for our proof are the Łojasiewicz inequalities by Colding–Minicozzi and Chodosh–Schulze, and the solution of the mean-convex neighbourhood conjecture by Choi, Haslhofer, Hershkovits and White. Our results improve the prior results by Gianniotis–Haslhofer, where diameter and curvature control has been obtained under the more restrictive assumption that the flow is globally two-convex.
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The author acknowledges his supervisor Robert Haslhofer for his patient guidance and invaluable support in bringing this paper into fruition over the last year.
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Du, W. Bounded Diameter Under Mean Curvature Flow. J Geom Anal 31, 11114–11138 (2021). https://doi.org/10.1007/s12220-021-00674-5
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DOI: https://doi.org/10.1007/s12220-021-00674-5