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Evolution of convex hypersurfaces by powers of the mean curvature. (English) Zbl 1087.53062
The author studies the evolution of a closed convex hypersurface in $$\mathbb R^{n+1}$$ in direction of its normal vector, where the speed equals a positive power $$k$$ of the mean curvature. More precisely, let $$M^n$$ be a smooth compact manifold without boundary, and let $$F_0 : M^n \to \mathbb R^{n+1}$$ be a smooth immersion which is convex. Then he looks for a smooth family of immersions $$F( \cdot , t) : M^n \times [0, T) \to \mathbb R^{n+1}$$ which satisfies $F( \cdot , 0) = F_0( \cdot ), \qquad \frac{d F}{d t} ( \cdot , t) = - H^k( \cdot , t) \nu( \cdot , t),$ where $$k > 0$$, $$H$$ is the mean curvature and $$\nu$$ is the outer unit normal, such that $$- H \nu = \overset \rightarrow {H}$$ is the mean curvature vector. Such a flow is called an $$H^k$$-flow. For $$k = 1$$, it coincides with the well-known mean curvature flow. Huisken proved that for this flow the surfaces stay convex and contract to a point in finite time [see G. Huisken, J. Differ. Geom. 20, 237–266 (1984; Zbl 0556.53001)].
Under the additional assumption that $$H(F_0 (p)) > 0$$ for every point $$p\in M^n$$, the author shows that the $$H^k$$-flow exists on a maximal finite time interval $$[0, T)$$, and that, for $$t\to T$$, the hypersurfaces $$F_0(M^n, t)$$ contract to a point in $$\mathbb R^{n+1}$$.

##### MSC:
 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
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