Convex Hypersurfaces of Prescribed Weingarten Curvatures

Abstract

In this paper we study the existence of closed convex hypersurfaces in the Euclidean space Rn+1 such that a Weingarten curvature, regarded as a function of their unit normal, is equal to a given positive function on the unit sphere Sn. Two classical problems of this type, namely the Christoffel problem [6] and the Minkowski problem [16], which address respectively the cases of harmonic and Gauss curvature, were completely solved in the 1970’s [7,9,17]. Besides the harmonic and Gauss curvatures, the most interesting Weingarten curvatures are probably the mean curvature, or more generally the k-curvatures (1 ≤ k ≤ n)