Existence of convex hypersurfaces with prescribed Gauss-Kronecker curvature

Existence of Convex Hypersurfaces with Prescribed Gauss-kronecker Curvature

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 ...

On the Existence and Regularity of Hypersurfaces of Prescribed Gauss Curvature with Boundary

In this paper we study the Dirichlet problem for some Monge-Ampère type equations on S, which naturally arise in some geometric problems. The result then is applied to prove the existence of hypersurfaces in R of prescribed Gauss-Kronecker curvature and with fixed boundary.

Hypersurfaces of Prescribed Gauss Curvature in Exterior Domains

We prove an existence theorem for convex hypersurfaces of prescribed Gauß curvature in the complement of a compact set in Euclidean space which are close to a cone.

Entire spacelike hypersurfaces of prescribed Gauss curvature in Minkowski space

which gives an isometric embedding of the hyperbolic space H into R. Hano and Nomizu [11] were probably the first to observe the non-uniqueness of isometric embeddings of H in R by constructing other (geometrically distinct) entire solutions of (1.1)–(1.2) for n 1⁄4 2 (and c1 1) using methods of ordinary di¤erential equations. Using the theory of Monge-Ampère equations, A.-M. Li [12] studied en...

The Existence of Convex Body with Prescribed Curvature Measures