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Deforming convex hypersurfaces by the $$n$$th root of the Gaussian curvature. (English) Zbl 0589.53005
The following theorem is proved: If the map $$F_ 0: S^ n\to M_ 0\subset {\mathbb{R}}^{n+1}$$ represents a strictly convex smooth hypersurface in $${\mathbb{R}}^{n+1}$$, $$n\geq 2$$, then the initial value problem $\partial F(x,t)/\partial t=-K(x,t)^{\beta}\cdot \nu (x,t),\quad F(x,0)=F_ 0(x),\quad x\in S^ n,$ has a unique solution on a maximum finite time interval [0,T) such that the $$M_ t's$$ converge to a point as $$t\to T$$. Here K denotes the Gaussian curvature and $$\nu$$ the outward normal of M; $$\beta$$ is a positive constant. Moreover, if $$\tilde M_ t$$ is $$M_ t$$ rescaled by a homothetic expansion so that $$Vol(\tilde M_ t)=Vol(M_ 0)$$, then as $$t\to T$$ the $$\tilde M_ t's$$ converge to a smooth hypersurface $$\tilde M_ T$$ in the $$C^{\infty}$$- topology. In the case $$\beta =1/n$$, $$\tilde M_ T$$ is a round sphere. - This extends earlier work of M. Gage and R. S. Hamilton [J. Differ. Geom. 23, 69-96 (1986)] for $$n=2$$, $$\beta =1$$ and of Tso [Commun. Pure Appl. Math. (to appear)] for $$\beta =1$$. A similar result for the mean curvature instead of $$K^{\beta}$$ is due to G. Huisken [J. Differ. Geom. 20, 237-266 (1984; Zbl 0556.53001)].
Reviewer: R.Schneider

##### MSC:
 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related $$n$$-spaces 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations 52A40 Inequalities and extremum problems involving convexity in convex geometry
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