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.Comment: 23 pages, 5 figures, Second version. Changes in statement and proof of Corollary

Lagrangian surfaces in complex Euclidean plane via spherical and hyperbolic curves

We present a method to construct a large family of Lagrangian surfaces in complex Euclidean plane 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, Hamiltonian-minimal 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 complex Euclidean plane.Comment: 16 pages To be published in Tohoku Mathematical Journa

A new construction of Lagrangians in the complex Euclidean plane in terms of planar curves

We introduce a new method to construct a large family of Lagrangian surfaces in complex Euclidean plane by means of two planar curves making use of their usual product as complex functions and integrating the Hermitian product of their position and tangent vectors. Among this family, we characterize minimal, constant mean curvature, Hamiltonian stationary, solitons for mean curvature flow and Willmore surfaces in terms of simple properties of the curvatures of the generating curves. As an application, we provide explicitly conformal parametrizations of known and new examples of these classes of Lagrangians in complex Euclidean plane.Comment: 15 pages, 5 figure

Helicoidal minimal surfaces in the 3-sphere: An approach via spherical curves

We prove an existence and uniqueness theorem about spherical helicoidal (in particular, rotational) surfaces with prescribed mean or Gaussian curvature in terms of a continuous function depending on the distance to its axis. As an application in the case of vanishing mean curvature, it is shown that the well-known conjugation between the belicoid and the catenoid in Euclidean three-space extends naturally to the 3-sphere to their spherical versions and determine in a quite explicit way their associated surfaces in the sense of Lawson. As a key tool, we use the notion of spherical angular momentum of the spherical curves that play the role of profile curves of the minimal helicoidal surfaces in the 3-sphere.Comment: 22 pages, 4 figure