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Deforming convex hypersurfaces by the square root of the scalar curvature. (English) Zbl 0608.53005
Let $$F_ 0: S^ n\to {\mathbb{R}}^{n+1}$$ be a smooth parametrization of a strictly convex hypersurface $$M_ 0\subset {\mathbb{R}}^{n+1}$$, $$n\geq 2$$, and consider the initial value problem $(*)\quad \partial F/\partial t (x,t)=-R^{1/2}(x,t)\cdot \nu (x,t)$ $F(x,0)=F_ 0(x),\quad x\in S^ n.$ Here R denotes the scalar curvature and $$\nu$$ the outward normal of $$M_ t$$, the corresponding hypersurface at time t. Under the additional assumption that $$R(p)/H^ 2(p)>C(n)>0$$ for all $$p\in M_ 0$$, where H is the mean curvature and the constant C(n) is chosen so that the inequality will imply $$M_ 0$$ is strictly convex, the author proves the following result: Problem (*) has a unique solution on a maximum time interval $$0\leq t<T$$ and the $$M_ t's$$ converge to a point as $$t\to T$$. Moreover, the shape of the $$M_ t's$$ approaches that of the standard round sphere. Similar results, with $$R^{1/2}$$ replaced by H or the n-th root of the Gaussian curvature K, respectively, are due to G. Huisken [J. Differ. Geom. 20, 237-266 (1984; Zbl 0556.53001)] and the author [ibid. 22, 117-138 (1985; Zbl 0589.53005)].
Reviewer: R.Schneider

##### MSC:
 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related $$n$$-spaces 52A20 Convex sets in $$n$$ dimensions (including convex hypersurfaces) 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations
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