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

TL;DR

This paper provides new characterizations of the Clifford torus as a Lagrangian self-shrinker, proving its uniqueness under certain curvature and embedding conditions in complex Euclidean space.

Contribution

It establishes the Clifford torus as the unique compact orientable Lagrangian self-shrinker with specific curvature bounds and embedding properties, confirming a conjecture.

Findings

Clifford torus is the unique compact orientable Lagrangian self-shrinker with |A|^2 ≤ 2.

It is the only such self-shrinker with nonnegative or nonpositive Gauss curvature.

Confirmed the Castro-Lerma conjecture regarding the Clifford torus.

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 $|A|^2\leq 2$, 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 $\mathbb{C}^2$.