Closed Lagrangian self-shrinkers in $\mathbb{R}^4$ symmetric with respect to a hyperplane

TL;DR

This paper classifies closed Lagrangian self-shrinkers in four-dimensional space that are symmetric across a hyperplane, showing they are products of Abresch-Langer curves and identifying the Clifford torus as unique under certain symmetry conditions.

Contribution

It provides a complete classification of symmetric closed Lagrangian self-shrinkers in $\

Findings

Classification of symmetric closed Lagrangian self-shrinkers as products of Abresch-Langer curves

Identification of the Clifford torus as the unique embedded symmetric example

New geometric characterization of the Clifford torus

Abstract

In this paper, we prove that the closed Lagrangian self-shrinkers in $\mathbb{R}^4$ which are symmetric with respect to a hyperplane are given by the products of Abresch-Langer curves. As a corollary, we obtain a new geometric characterization of the Clifford torus as the unique embedded closed Lagrangian self-shrinker symmetric with respect to a hyperplane in $\mathbb{R}^4$.