Weierstrass Representation For Surfaces Research Articles

The Moutard transformation of two-dimensional Dirac operators and Möbius geometry

We describe the action of inversion on given Weierstrass representations for surfaces and show that the Moutard transformation of two-dimensional Dirac operators maps the potential (the Weierstrass representation) of a surface S to the potential of a surface \(\tilde S\) obtained from S by inversion.

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.

WEIERSTRASS REPRESENTATION FOR SURFACES WITH PRESCRIBED NORMAL GAUSS MAP AND GAUSS CURVATURE IN H3

The author obtains a Weierstrass representation for surfaces with prescribed normal Gauss map and Gauss curvature in H3. A differential equation about the hyperbolic Gauss map is also obtained, which characterizes the relation among the hyperbolic Gauss map, the normal Gauss map and Gauss curvature. The author discusses the harmonicity of the normal Gauss map and the hyperbolic Gauss map from surface with constant Gauss curvature in H3 to S2 with certain altered conformal metric. Finally, the author considers the surface whose normal Gauss map is conformal and derives a completely nonlinear differential equation of second order which graph must satisfy.

On CP 1 and CP 2 Maps and Weierstrass Representations for Surfaces Immersed into Multi-Dimensional Euclidean Spaces

An extension of the classic Enneper–Weierstrass representation for conformally parametrised surfaces in multi-dimensional spaces is presented. This is based on low dimensional CP 1 and CP 2 sigma models which allow the study of the constant mean curvature (CMC) surfaces immersed into Euclidean 3- and 8-dimensional spaces, respectively. Relations of Weierstrass type systems to the equations of these sigma models are established. In particular, it is demonstrated that the generalised Weierstrass representation can admit different CMC-surfaces in ℝ3 which have globally the same Gauss map. A new procedure for constructing CMC-surfaces in ℝn is presented and illustrated in some explicit examples.