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On surfaces of prescribed \(F\)-mean curvature. (English) Zbl 1058.53003

Generalizing minimal surfaces and surfaces of prescribed mean curvature, the authors introduce hypersurfaces of prescribed \(F\)-mean curvature as critical immersions of anisotropic surface energies. After proving some enclosure results, the authors derive a general second variation formula for the anisotropic surface energies, generalizing the corresponding well known formulas for minimal surfaces and surfaces of prescribed mean curvature. As an application, the authors prove that stable surfaces, of prescribed \(F\)-mean curvature, in the Euclidean space \(\mathbb E^{3}\) can be represented as graphs over a strictly convex domain \(\Omega \) in the plane, if the given contour in \(\mathbb E^{3}\) is a graph over the boundary \(\partial \Omega \).

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
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