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Plongements radiaux $$S^ n\,\hookrightarrow \,{\mathbb{R}}^{n+1}$$ à courbure de Gauss positive prescrite. (Radial embeddings $$S^ N\,\hookrightarrow \,{\mathbb{R}}^{n+1}$$ with prescribed positive Gauss curvature). (French) Zbl 0594.53039
The Gauss curvature problem consists in finding conditions on a given function $$K: R^{n+1}\to R$$ which assure existence of a closed convex hypersurface F in $$R^{n+1}$$ such that it is a radial graph over a unit sphere $$S^ n$$ and its Gauss-Kronecker curvature at the point $$x\in S^ n$$ is K(X), where $$X\in F$$ and x is the radial projection of X on $$S^ n$$. Earlier the reviewer has shown that the above problem has a unique solution of class $$C^{m+1,\alpha}$$, for any $$\alpha\in (0,1)$$, if the following conditions are satisfied: $$K>0$$; $$K\in C^ m(R^{n+1}\setminus \{0\})$$, $$m\geq 3$$; there exist two numbers $$R_ 1$$ and $$R_ 2$$, $$0<R_ 1\leq 1\leq R_ 2$$, such that $$K(X)>| X|^{-n}$$ when $$| X| <R_ 1$$ and $$K(X)<| X|^{-n}$$ when $$| X| >R_ 2$$; and, finally, $$(\partial /\partial \rho)(\rho^ n K(x,\rho))\leq 0$$, $$\rho \in [R_ 1,R_ 2].$$
In this paper in the first theorem the author shows that the strict inequalities in the decay conditions on K can be replaced by nonstrict ones and the last condition can be omitted. In such form these conditions are sufficient for existence; however no assertion about uniqueness can be made. In the second theorem it is shown that for a positive function $$\Phi$$ (x), $$x\in S^ n$$ one can find a convex hypersurface (as a graph over $$S^ n)$$ such that its Gauss curvature is proportional to $$\Phi$$ (x) $$| Y(x)|^{-n}$$, where Y(x) is the position vector of the hypersurface.
Reviewer: V.Oliker

##### MSC:
 53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov) 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related $$n$$-spaces 35J60 Nonlinear elliptic equations 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 47J25 Iterative procedures involving nonlinear operators
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