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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 \(\mathbb R^{n+1}\).

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
Full Text: DOI
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