The rigidity theorem for complete Lagrangian self-shrinkers

TL;DR

This paper proves a rigidity theorem for 2D complete Lagrangian self-shrinkers with constant mean curvature squared norm in 4D Euclidean space, extending to Lagrangian ξ-submanifolds.

Contribution

It establishes a new rigidity result for Lagrangian self-shrinkers with constant mean curvature norm, using novel techniques applicable to ξ-submanifolds.

Findings

Rigidity theorem for 2D Lagrangian self-shrinkers with constant |H|^2

Extension of results to Lagrangian ξ-submanifolds

Methodology applicable to related geometric submanifolds

Abstract

In this paper, we obtain a rigidity result of $2$-dimensional complete lagrangian self-shrinkers with constant squared norm $|\vec{H}|^{2}$ of the mean curvature vector in the Euclidean space $\mathbb{R}^{4}$. The same idea is also used to give a similar result of Lagrangian $\xi$-submanifolds in $\mathbb{R}^{4}$.