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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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