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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

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
Full Text: DOI EuDML
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