The Gauss Map of Minimal Surfaces in the Heisenberg Group
نویسنده
چکیده
We study the Gauss map of minimal surfaces in the Heisenberg group Nil3 endowed with a left-invariant Riemannian metric. We prove that the Gauss map of a nowhere vertical minimal surface is harmonic into the hyperbolic plane H. Conversely, any nowhere antiholomorphic harmonic map into H is the Gauss map of a nowhere vertical minimal surface. Finally, we study the image of the Gauss map of complete nowhere vertical minimal surfaces.
منابع مشابه
L_1 operator and Gauss map of quadric surfaces
The quadrics are all surfaces that can be expressed as a second degree polynomialin x, y and z. We study the Gauss map G of quadric surfaces in the 3-dimensional Euclidean space R^3 with respect to the so called L_1 operator ( Cheng-Yau operator □) acting on the smooth functions defined on the surfaces. For any smooth functions f defined on the surfaces, L_f=tr(P_1o hessf), where P_1 is t...
متن کاملTranslation invariant surfaces in the 3-dimensional Heisenberg group
In this paper, we study translation invariant surfaces in the 3-dimensional Heisenberg group $rm Nil_3$. In particular, we completely classify translation invariant surfaces in $rm Nil_3$ whose position vector $x$ satisfies the equation $Delta x = Ax$, where $Delta$ is the Laplacian operator of the surface and $A$ is a $3 times 3$-real matrix.
متن کاملDigital cohomology groups of certain minimal surfaces
In this study, we compute simplicial cohomology groups with different coefficients of a connected sum of certain minimal simple surfaces by using the universal coefficient theorem for cohomology groups. The method used in this paper is a different way to compute digital cohomology groups of minimal simple surfaces. We also prove some theorems related to degree properties of a map on digital sph...
متن کاملAN ESTIMATE FOR THE GAUSS CURVATURE OF MINIMAL SURFACES IN Rm WHOSE GAUSS MAP OMITS A SET OF HYPERPLANES
We give an estimate of the Gauss curvature for minimal surfaces in R m whose Gauss map omits more than m(m + 1)=2 hyperplanes in P m?1 (C).
متن کاملAn Estimate for the Gauss Curvature of Minimal Surfaces in R Whose Gauss Map Omits a Set of Hyperplanes
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).
متن کامل