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On the existence of convex hypersurfaces with prescribed mean curvature. (English) Zbl 0701.53078

Consider the following problem: Given a function F in \(R^{n+1}\), under what conditions does there exists a smooth closed imbedded hypersurface X in \(R^{n+1}\) with mean curvature H such that (1) \(H(p)=F(p)\), \(p\in X\). A definite answer has been obtained previously [see I. Ya. Bakel’man and B. E. Kantor, Geometry and topology, Leningrad 1974, No.1, 3-10 (1974); L. Caffarelli, L. Nirenberg and J. Spruck, Current topics in partial differential equations, Pap. dedic. S. Mizohata Occas. 60th Birthday, 1-26 (1986; Zbl 0672.35027); and A. Treibergs and S. W. Wei, J. Differ. Geom. 18, 513-521 (1983; Zbl 0529.53043)] for hypersurfaces starshaped with respect to the origin. As pointed out by S. T. Yau, the problem has a variational structure, i.e. the functional \((2)\quad I(X)=(1/n)\int_{X^ 1}- \int_{\tilde X^ F}\) where \(\tilde X\) is the subset bounded by X has (1) as its Euler-Lagrange equation. By studying the critical points of (2) the author establishes results on the existence of convex hypersurface solutions of (1). The proofs are mainly based on the study of the negative gradient flow associated with I.
Reviewer: L.Ornea

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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References:

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