Cubic hypersurfaces admitting an embedding with Gauss map of rank 0

Unirationality of Cubic Hypersurfaces

A remarkable result of [Segre43] says that a smooth cubic surface over Q is unirational iff it has a rational point. [Manin72, II.2] observed that similar arguments work for higher dimensional cubic hypersurfaces satisfying a certain genericity assumption over any infinite field. [CT-S-SD87, 2.3.1] extended the result of Segre to any normal cubic hypersurface (other than cones) over a field of ...

The hypersurfaces with conformal normal Gauss map in H n + 1 and S n + 11 ∗ † ‡

In this paper we introduce the fourth fundamental form for the hypersurfaces in H and the space-like hypersurfaces in S 1 and discuss the conformality of the normal Gauss maps of the hypersurfaces in H and S 1 . Particularly, we discuss the surfaces with conformal normal Gauss maps in H and S 1 and prove a duality property. We give the Weierstrass representation formula for the space-like surfa...

On Cubic Graphs Admitting an Edge-Transitive Solvable Group

Using covering graph techniques, a structural result about connected cubic simple graphs admitting an edge-transitive solvable group of automorphisms is proved. This implies, among other, that every such graph can be obtained from either the 3-dipole Dip3 or the complete graph K4, by a sequence of elementary-abelian covers. Another consequence of the main structural result is that the action of...

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-...

Hypersurfaces of Prescribed Gauss Curvaturein Exterior Domainsfelix