We construct a family of small analytic discs attached to Levi non-degenerate hypersurfaces in Cn+1, which is globally biholomorphically invariant. We then apply this technique to study unique determination problems along Levi non-degenerate hypersurfaces that are merely of class C4. This method gives 2-jet determination results for germs of biholomorphisms, CR diffeomorphisms, as well as in th...

A well known result of Forstnerić [18] states that most real-analytic strictly pseudoconvex hypersurfaces in complex space are not holomorphically embeddable into spheres of higher dimension. A more recent result by Forstnerić [19] states even more: most real-analytic hypersurfaces do not admit a holomorphic embedding even into a merely algebraic hypersurface of higher dimension, in particular,...

In this paper we consider a variational problem for spacelike hypersurfaces in the (n + 1)-dimensional Lorentz-Minkowski space L, whose critical points are hypersurfaces supported in a spacelike hyperplane determined by two facts: the mean curvature is a linear function of the distance to and the hypersurface makes a constant angle with along its boundary. We prove that the hypersurface is rota...

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...

In this paper we describe all rotation H-hypersurfaces in Hn×R and use them as barriers to prove existence and characterization of certain vertical H-graphs and to give symmetry and uniqueness results for compact H-hypersurfaces whose boundary is one or two parallel submanifolds in slices. We also describe examples of translation H-hypersurfaces in Hn × R. In particular, for H < n−1 n , we obta...

Abstract. We determine conditions on q for the nonexistence of deep holes of the standard Reed–Solomon code of dimension k over Fq generated by polynomials of degree k+d. Our conditions rely on the existence of q–rational points with nonzero, pairwise–distinct coordinates of a certain family of hypersurfaces defined over Fq. We show that the hypersurfaces under consideration are invariant under...

We propose a ‘geometric Chevalley-Warning’ conjecture, that is a motivic extension of the Chevalley-Warning theorem in number theory. It is equivalent to a particular case of a recent conjecture of F. Brown and O.Schnetz. In this paper, we show the conjecture is true for linear hyperplane arrangements, quadratic and singular cubic hypersurfaces of any dimension, and cubic surfaces in P. The las...