Abstract

Hypersurfaces of prescribed weighted mean curvature, or F-mean curvature, are introduced as critical immersions of anisotropic surface energies, thus generalizing minimal surfaces and surfaces of prescribed mean curvature. We first prove enclosure theorems in Rⁿ⁺¹ for such surfaces in cylindrical boundary configurations. Then we derive a general second variation formula for the anisotropic surface energies generalizing corresponding formulas of do Carmo for minimal surfaces, and Sauvigny for prescribed mean curvature surfaces. Finally we prove that stable surfaces of prescribed F-mean curvature in R³ can be represented as graphs over a planar strictly convex domain Ω, if the given boundary contour in R³ is a graph over ∂Ω.

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