In this paper,the authors give a characterization theorem for the standard tori S^(1)(a)×S^(1)(b),a,b>0,as the compact Lagrangianξ-submanifolds in the two-dimensional complex Euclidean space C^(2),and obtain the best version of a former rigidity theorem for compact Lagrangianξ-submanifold in C^(2).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 C^(2).
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