MAXIMAL SPACE-LIKE HYPERSURFACES IN H14(-1) WITH ZERO 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 ...

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

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...

A pr 2 00 3 Immersions with fractal set of points of zero Gauss - Kronecker curvature

We construct, for any " good " Cantor set F of S n−1 , an immersion of the sphere S n with set of points of zero Gauss-Kronecker curvature equal to F ×D 1 , where D 1 is the 1-dimensional disk. In particular these examples show that the theorem of Matheus-Oliveira strictly extends two results by do Carmo-Elbert and Barbosa-Fukuoka-Mercuri.

Motion of Hypersurfaces by Gauss Curvature

We consider n-dimensional convex Euclidean hypersurfaces moving with normal velocity proportional to a positive power α of the Gauss curvature. We prove that hypersurfaces contract to points in finite time, and for α ∈ (1/(n + 2], 1/n] we also prove that in the limit the solutions evolve purely by homothetic contraction to the final point. We prove existence and uniqueness of solutions for non-...