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

TL;DR

This paper presents a novel method for constructing a wide variety of Lagrangian surfaces in the complex Euclidean plane using pairs of planar curves, enabling explicit parametrizations of important geometric classes.

Contribution

It introduces a new construction technique based on planar curves and characterizes key geometric properties of resulting Lagrangians, including minimality and constant mean curvature.

Findings

Explicit conformal parametrizations of known Lagrangians.

Characterization of minimal and CMC Lagrangians via curve curvatures.

Construction of new examples of Lagrangian surfaces.

Abstract

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.