We present a method to construct a large family of Lagrangian surfaces incomplex Euclidean plane by using Legendre curves in the 3-sphere and in theanti de Sitter 3-space or, equivalently, by using spherical and hyperboliccurves, respectively. Among this family, we characterize minimal, constant meancurvature, Hamiltonian-minimal and Willmore surfaces in terms of simpleproperties of the curvature of the generating curves. As applications, weprovide explicitly conformal parametrizations of known and new examples ofminimal, constant mean curvature, Hamiltonian-minimal and Willmore surfaces incomplex Euclidean plane.
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