Construction of Lagrangian surfaces in complex Euclidean plane with Legendre curves

Abstract

An important problem in the theory of Lagrangian submanifolds is to find non-trivial examples of Lagrangian submanifolds in complex Euclidean spaces with some given special geometric properties. In this article, we provide a new method to construct Lagrangian surfaces in the complex Euclidean plane C² by using Legendre curves in S³(1) $\subset$ C². We also investigate intrinsic and extrinsic geometric properties of the Lagrangian surfaces in C² obtained by applying our construction method. As an application we provide some new families of Hamiltonian minimal Lagrangian surfaces in C² via our construction method.