## 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

### Keywords:

prescribed mean curvature; convex hypersurfaces
Full Text:

### References:

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