Classification of Lagrangian translators and Lagrangian self-expanders in $\mathbb{C}^{2}$

TL;DR

This paper classifies 2D complete Lagrangian translators and self-expanders in ^2 with constant mean curvature norm, using a new maximum principle, and extends results to Lagrangian ^2 -translators.

Contribution

It provides new classification results for Lagrangian translators and self-expanders in ^2 using an innovative maximum principle approach.

Findings

Classified 2D complete Lagrangian translators in ^2.

Classified Lagrangian self-expanders with constant  norm of mean curvature.

Extended classification to Lagrangian ^2 -translators.

Abstract

In this paper, we obtain several classification results of $2$-dimensional complete Lagrangian translators and lagrangian self-expanders with constant squared norm $|\vec{H}|^{2}$ of the mean curvature vector in $\mathbb{C}^{2}$ by using a new Omori-Yau type maximum principle which was proved by Chen and Qiu \cite{CQ}. The same idea is also used to give a similar result of Lagrangian $\xi$-translators in $\mathbb{C}^{2}$.