LetX be a compact, strictly convex C-hypersurface in the (n+1)-dimensional Euclidean space R. The Gauss map ofX maps the hypersurface one-to-one and onto the unit n-sphere S. One may parametrize X by the inverse of the Gauss map. Consequently, the Gauss curvature can be regarded as a function on S. The classical Minkowski problem asks conversely when a positive function K on S is the Gauss curv...

In this paper, by the studying of the Gauss map, Laplacian operator, curvatures of surfaces in R 1 and Bour’s theorem, we are going to identify surfaces of revolution with pointwise 1-type Gauss map property in 3−dimensional Minkowski space. Introduction The classification of submanifolds in Euclidean and Non-Euclidean spaces is one of the interesting topics in differential geometry and in this...

We give an estimate of the Gauss curvature for minimal surfaces in Rm whose Gauss map omits more than m(m + 1)/2 hyperplanes in P(C).

Let M be a connected oriented surface and let G'2 be the Grassmannian of oriented 2-planes in Euclidean (2 + c)-space. E2 + l. Smooth maps t: M -» (7f are studied to determine whether or not they are Gauss maps. Both local and global results are obtained. If í is a Gauss map of an immersion X: M -» E2 + 1, we study the extent to which / uniquely determines X under certain circumstances. Let X: ...

We consider an interval map which is a generalization of the well-known Gauss transformation. In particular, we prove a result concerning the asymptotic behavior of the distribution functions of this map. 1. Introduction. In 1800, Gauss studied the following problem. In modern notation, it reads as follows. Write x ∈ [0, 1) as a regular continued fraction

We give the best possible upper bound on the number of exceptional values and totally ramified value number of the hyperbolic Gauss map for pseudo-algebraic Bryant surfaces and some partial results on the Osserman problem for algebraic Bryant surfaces. Moreover, we study the value distribution of the hyperbolic Gauss map for complete constant mean curvature one faces in de Sitter three-space.

Many topics in integrable surface geometry may be unified by application of the highly developed theory of harmonic maps of surfaces into (pseudo-)Riemannian symmetric spaces. On the one hand, such harmonic maps comprise an integrable system with spectral deformations, algebro-geometric solutions and dressing actions of loop groups generated by Bäcklund transforms [5], [6], [14], [21], [24]. On...

We give an effective estimate for the totally ramified value number of the hyperbolic Gauss maps of complete flat fronts in the hyperbolic three-space. As a corollary, we give the upper bound of the number of exceptional values of them for some topological cases. Moreover, we obtain some new examples for this class.