## 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

### Keywords:

mean curvature flow; singularity; collapsing

### Citations:

Zbl 0961.53039; Zbl 1027.53078; Zbl 0992.53051
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