Lagrangian Surfaces in Complex Euclidean Plane via Spherical and Hyperbolic Curves
نویسندگان
چکیده
We present a method to construct a large family of Lagrangian surfaces in complex Euclidean plane C by using Legendre curves in the 3-sphere and in the anti de Sitter 3-space or, equivalently, by using spherical and hyperbolic curves, respectively. Among this family, we characterize minimal, constant mean curvature, Hamiltonianminimal and Willmore surfaces in terms of simple properties of the curvature of the generating curves. As applications, we provide explicitly conformal parametrizations of known and new examples of minimal, constant mean curvature, Hamiltonian-minimal and Willmore surfaces in C.
منابع مشابه
Construction of Hamiltonian-minimal Lagrangian Submanifolds in Complex Euclidean Space
We describe several families of Lagrangian submanifolds in the complex Euclidean space which are H-minimal, i.e. critical points of the volume functional restricted to Hamiltonian variations. We make use of various constructions involving planar, spherical and hyperbolic curves, as well as Legendrian submanifolds of the odd-dimensional unit sphere.
متن کاملLagrangian surfaces with circular ellipse of curvature in complex space forms
We classify the Lagrangian orientable surfaces in complex space forms with the property that the ellipse of curvature is always a circle. As a consequence, we obtain new characterizations of the Clifford torus in the complex projective plane and of the Whitney spheres in the complex projective, complex Euclidean and complex hyperbolic planes. MSC 2000: 53C42, 53C40.
متن کاملSpherical Arc-length as a Global Conformal Parameter for Analytic Curves in the Riemann Sphere
We prove that for every analytic curve in the complex plane C, Euclidean and spherical arc-lengths are global conformal parameters. We also prove that for any analytic curve in the hyperbolic plane, hyperbolic arc-length is also a global parameter. We generalize some of these results to the case of analytic curves in R and C and we discuss the situation of curves in the Riemann sphere C ∪ {∞}. ...
متن کاملGeometric properties of hyperbolic geodesics
In the unit disk D hyperbolic geodesic rays emanating from the origin and hyperbolic disks centered at the origin exhibit simple geometric properties. The goal is to determine whether analogs of these geometric properties remain valid for hyperbolic geodesic rays and hyperbolic disks in a simply connected region Ω. According to whether the simply connected region Ω is a subset of the unit disk ...
متن کاملHamiltonian-minimal Lagrangian submanifolds in complex space forms
Using Legendrian immersions and, in particular, Legendre curves in odd dimensional spheres and anti De Sitter spaces, we provide a method of construction of new examples of Hamiltonian-minimal Lagrangian submanifolds in complex projective and hyperbolic spaces, including explicit one parameter families of embeddings of quotients of certain product manifolds. In addition, new examples of minimal...
متن کامل