Existence of convex hypersurfaces with prescribed Gauss-Kronecker curvature

@inproceedings{Wang1996ExistenceOC,
  title={Existence of convex hypersurfaces with prescribed Gauss-Kronecker curvature},
  author={Xu-Jia Wang},
  year={1996}
}
Let f(x) be a given positive function in Rn+1. In this paper we consider the existence of convex, closed hypersurfaces X so that its GaussKronecker curvature at x ∈ X is equal to f(x). This problem has variational structure and the existence of stable solutions has been discussed by Tso (J. Diff. Geom. 34 (1991), 389–410). Using the Mountain Pass Lemma and the Gauss curvature flow we prove the existence of unstable solutions to the problem. 

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