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The understanding of Type I singularities for the mean curvature flow is closely related with the classification of self-shrinkers. Since there do not exist Lagrangian self-shrinking spheres, the Clifford torus is the most regular example of a compact self-shrinker for Lagrangian mean curvature flow in complex Euclidean plane. In this paper, we provide four rigidity results for the Clifford torus in the class of compact self-shrinkers for Lagrangian mean curvature flow.

The Clifford Torus as a Self-Shrinker for the Lagrangian Mean Curvature Flow