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Singularity profile in the mean curvature flow. (English) Zbl 1184.53071

The paper is concerned with the evolution of a closed smooth hypersurface \(F_t\) in \(\mathbb R^{n+1}\), \(n\geq 2\), governed by the mean curvature flow \({\frac{dx}{dt} = H \nu}\). The authors are interested in the geometry of the first time singularities.
The main result states that if \(F_t\), \(t\in [0,T)\), is mean convex, \(H>0\), then it is \(k\)-noncollapsing up to the first singular time \(T\), for some constant \(k>0\) depending only on \(n\) and the initial surface \(F_0\).
Here, the \(k\)-noncollapsing means a pinching of \(F_t\) by a sphere from inside [cf. B. White, J. Am. Math. Soc. 13, No. 3, 665–695 (2000; Zbl 0961.53039), J. Am. Math. Soc. 16, No. 1, 123–138 (2003; Zbl 1027.53078); G. Huisken and C. Sinestrari, Acta Math. 183, No. 1, 45–70 (1999; Zbl 0992.53051)].

MSC:

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
35K55 Nonlinear parabolic equations
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