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Starshaped hypersurfaces and the mean curvature flow. (English) Zbl 0903.53039
The author generalizes a result of G. Huisken and C. Sinestrari [‘Mean curvature flow singularities for mean convex surfaces’, Calc. Var. Partial Differ. Equ. 8, No.1, 1-14 (1999)], giving conditions such that under the mean curvature flow a type II singularity develops. Whereas Huisken-Sinestrari’s paper gets the result assuming that the compact initial hypersurface is mean convex, the present paper poses a priori estimates on a whole family of closed hypersurfaces evolving by mean curvature. The proof uses essentially the same methods as Huisken-Sinestrari’s paper. The second part of the paper shows that the a priori estimates are satisfied for mean convex hypersurfaces and some variety hereof and for starshaped hypersurfaces.

MSC:
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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