It is important to understand Lagrangian self-shrinkers with simple geometry since it is the starting point of singularity analysis for the Lagrangian mean curvature flow. One interesting observation is that all known embedded examples in \(\mathbb{R}^4\) become the Clifford Torus. Hence it is natural to ask whether the Clifford Torus is unique as an embedded Lagrangian self-shrinker in \(\mathbb{R}^4\). In this direction, we recently proved that a closed Lagrangian self-shrinker in \(\mathbb{R}^4\) symmetric with respect to a hyperplane is given by the product of two Abresch-Langer curves and obtained a positive answer for the question by assuming reflection symmetry. In this talk, we will focus on the motivation for this work and the reason why reflection symmetry was assumed. Moreover, the idea of proof will also be discussed.

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