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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

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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