New characterizations of the Clifford torus as a Lagrangian self-shrinker

Abstract

In this paper, we obtain several new characterizations of the Clifford torus as a Lagrangian self-shrinker. We first show that the Clifford torus ${\mathbb{S}}^1(1)\times {\mathbb{S}}^1(1)$ is the unique compact orientable Lagrangian self-shrinker in ${\mathbb{C}}^2$ with $|A{|}^2\le 2$, which gives an affirmative answer to Castro-Lerma's conjecture. We also prove that the Clifford torus is the unique compact orientable embedded Lagrangian self-shrinker with nonnegative or nonpositive Gauss curvature in ${\mathbb{C}}^2$.

arXiv comment16 pages, accepted by The Journal of Geometric Analysis