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Existence and convexity for hyperspheres of prescribed mean curvature. (English) Zbl 0599.53044
The author is concerned with the problem of determining embedded hyperspheres \(Y: S^ n\to {\mathbb{R}}^{n+1}\) whose mean curvature H is prescribed as a function of position and normal vectors, \(H=\Phi (Y,N)\). It is assumed that the hypersurface Y is starlike with respect to the origin and that the function \(\Phi\) is homogeneous of degree -1 in the first variable. Under various conditions the author proves the existence of such a hypersurface Y satisfying \(H=\lambda \Phi (Y,N)\) for some eigenvalue parameter \(\lambda\), both Y and \(\lambda\) are uniquely determined. Furthermore, the paper contains a stability result for solutions close to a sphere. Moreover, in the case when \(\Phi\) does not depend on the normal N a sufficient (but not very explicit) criterion on \(\Phi\) for the convexity of Y is given.
Reviewer: F.Tomi

MSC:
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
35J60 Nonlinear elliptic equations
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