1008.3545Aug 20, 2010Formerly, "The Plateau problem for general curvature functions". Significantly revised for the sake of improved clarity, incorporating notational changes made to arXiv:1002.2982, modifying emphasis on results so as to better reflect the main interests of experts in the field, and tidying up results and removing inelegant technical conditions which have proven themselves to be unnecessary

The Plateau problem for convex curvature functions

Graham Smith

Differential Geometry

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Abstract

We present a novel and comprehensive approach to the study of the parametric Plateau problem for locally strictly convex (LSC) hypersurfaces of prescribed curvature for general convex curvature functions inside general Riemannian manifolds. We prove existence of solutions to the Plateau problem with outer barrier for LSC hypersurfaces of constant or prescribed curvature for general curvature functions inside general Hadamard manifolds modulo a single scalar condition. In particular, convex curvature functions of bounded type are fully treated.

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