A Characterization of the Standard Tori in ℂ² as Compact Lagrangian ξ-Submanifolds

Abstract

In this paper, the authors give a characterization theorem for the standard tori $\mathbb{S}^1(a) \times \mathbb{S}^1(b)$, a, b > 0, as the compact Lagrangian ξ-submanifolds in the two-dimensional complex Euclidean space ℂ², and obtain the best version of a former rigidity theorem for compact Lagrangian ξ-submanifold in ℂ². Furthermore, their argument in this paper also proves a new rigidity theorem which is a direct generalization of a rigidity theorem by Li and Wang for Lagrangian self-shrinkers in ℂ².

Keywords

ξ-Submanifold / the Second fundamental form / Mean curvature vector / Standard tori

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