A uniqueness theorem of complete Lagrangian translator in $$\mathbb C^2$$

Abstract

In this paper we study the complete Lagrangian translators in the complex 2-plane $$\mathbb C^2$$ . As the result, we obtain a uniqueness theorem showing that the plane is the only complete Lagrangian translator in $$\mathbb C^2$$ with constant square norm of the second fundamental form. On the basis of this, we can prove a more general classification theorem for Lagrangian $$\xi $$ -translators in $${\mathbb C}^2$$ . The same idea is also used to give a similar classification of Lagrangian $$\xi $$ -surfaces in $${\mathbb C}^2$$ .