Gauss Map Of Surfaces Research Articles

Gauss Maps of the Surfaces in the Three-Dimensional Heisenberg Group

In this paper we study the Gauss map of surfaces in three-dimensional Heisenberg group using the Gans model of the hyperbolic plane. We establish a relationship between the tension fields of the Gauss maps and the mean curvatures of the surfaces in $\mathcal{H}_{3}$.

Weierstrass representation for surfaces in the three-dimensional Heisenberg group

The authors define the Gauss map of surfaces in the three-dimensional Heisenberg group and give a representation formula for surfaces of prescribed mean curvature. Furthermore, a second order partial differential equation for the Gauss map is obtained, and it is shown that this equation is the complete integrability condition of the representation.

On twistor Gauss maps of surfaces in 4-spheres

In this paper we study surfaces in S4 and their twistor Gauss maps. Some necessary and sufficient conditions that the twistor Gauss map is harmonic are given. We find many examples of nonisotropic harmonic maps from a surface to ℂP3.

TIME-INDEPENDENT SOLUTIONS TO THE TWO-DIMENSIONAL NONLINEAR O(3) SIGMA MODEL AND SURFACES OF CONSTANT MEAN CURVATURE

It is shown that time-independent solutions to the (2+1)-dimensional nonlinear O(3) sigma model may be placed in correspondence with surfaces of constant mean curvature in three-dimensional Euclidean space. The tools required to establish this correspondence are provided by the classical differential geometry of surfaces. A constant-mean-curvature surface induces a solution to the O(3) model through the identification of the Gauss map, or normal vector, of the surface with the field vector of the sigma model. Some explicit solutions, including the solitons and antisolitons discovered by Belavin and Polyakov, and a more general solution due to Purkait and Ray, are considered and the surfaces giving rise to them are found explicitly. It is seen, for example, that the Belavin-Polyakov solutions are induced by the Gauss maps of surfaces which are conformal to their spherical images, i.e. spheres and minimal surfaces, and that the Purkait-Ray solution corresponds to the family of constant-mean-curvature helicoids first studied by do Carmo and Dajczer in 1982. A generalization of this method to include time dependence may shed new light on the role of the Hopf invariant in this model.

The Gauss Map of Surfaces in R3 and R4

The Gauss Map of Surfaces in R³ and R⁴

The Gauss map for surfaces in 4-space

The Gauss map for surfaces in 4-space

On the volume of the generalized Gauss map

Let f :M~ N"+P be an immersion of the n-manifold M into Euclidean (n + p)-space. The classical Gauss map for surfaces in 3-space has two natural generalizations to this situation. In the obvious way we can assign (a) to every unit normal vector of f a point in the unit sphere S "+p- 1, or (b) to every point of M its p-dimensional normal space through tile origin of ~"+P. Both the sphere S "+p- 1 and the Grassmannian Gv(~ "+p) have canonical Riemannian metrics, and we can talk about the (n + p - 1)-dimensional or the n-dimensional volumes of the Gauss maps (a) and (b). In the first case this (suitably normalized) volume is the well-known total absolute curvature r(f) off. The Gauss map (b) has also been studied extensively (see [8] for a survey and a list of references). In the present note we shall affirmatively answer a conjecture of Osserman [8] stating that the (again suitably normalized) volume a(f) of the map (b) satisfies z(f) 0 denote the volume of the unit k-sphere, and define