Form Of Hypersurface Research Articles

Horizontal Newton operators and high-order Minkowski formula

In this paper, we study the horizontal Newton transformations, which are nonlinear operators related to the natural splitting of the second fundamental form for hypersurfaces in a complex space form. These operators allow to prove the classical Minkowski formulas in the case of real space forms: unlike the real case, the horizontal ones are not divergence-free. Here, we consider the highest order of nonlinearity and we will show how a Minkowski-type formula can be obtained in this case.

Degenerate hypersurfaces with a two-parametric family of automorphisms

We give a complete classification of Levi-degenerate hypersurfaces of finite type in ℂ2 with two-dimensional symmetry groups. Our analysis is based on the classification of two-dimensional Lie algebras and an explicit description of isotropy groups for such hypersurfaces, which follows from the construction of Chern–Moser type normal forms at points of finite type, developed in [M. Kolář, Normal forms for hypersurfaces of finite type in ℂ2, Math. Res. Lett. 12 (2005), pp. 897–910].

Divisibility of the Second Fundamental Form of Hypersurfaces of Space Forms

We classify the hypersurfaces of space forms for which the cubic form is divisible by the metric (g|C). In other words, when does the symmetric traceless part of the cubic form vanish, where the trace is taken with respect to the first fundamental form?

The hypersurfaces with conformal normal Gauss map in Hn+1 and S1n+1

In this paper, we introduce the fourth fundamental forms for hypersurfaces in Hn+1 and space-like hypersurfaces in S1n+1, and discuss the conformality of the normal Gauss map of the hypersurfaces in Hn+1 and S1n+1. Particularly, we discuss the surfaces with conformal normal Gauss map in H³ and S³1, and prove a duality property. We give a Weierstrass representation formula for space-like surfaces in S³1 with conformal normal Gauss map. We also state the similar results for time-like surfaces in S³1. Some examples of surfaces in S³1 with conformal normal Gauss map are given and a fully nonlinear equation of Monge-Ampère type for the graphs in S³1 with conformal normal Gauss map is derived.