Lagrangian tori in a symplectic vector space and global symplectomorphisms
Lagrangian tori in a symplectic vector space and global symplectomorphisms
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辛向量空间中的拉格朗日环面和全局辛同态
DOI：
10.1007/pl00004278
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发表时间：
1996
影响因子：
0.8
通讯作者：
Yu. V. Chekanov
中科院分区：
文献类型：
--
作者：
Yu. V. Chekanov
近年来，环面拉格朗日嵌入辛向量空间的研究取得了一些进展。拉格朗日浸没空间的一些连通分量已被证明不包含嵌入[G1，P2，V2]，并且所有嵌入的空间的一些连通分量已被证明不包含拉格朗日嵌入[L]。然而，嵌入式拉格朗日环面空间连通分量的分类问题还远未得到解决。似乎这种环面的所有已知例子都是拉格朗日嵌入类中基本汤姆（根据定义，2 中圆的乘积）的环面同位素。本文考虑以下问题：给定两个嵌入拉格朗日环L、L'，是否存在全局辛同态9使得9(L)=L'。针对本文构造的基本环面和某些单调环面的情况给出了解决方案。还讨论了主要定理的一些后果。
In recent years some progress has been made in the study of Lagrangian embeddings of a torus into a symplectic vector space. Some connected components of the space of Lagrangian immersions have been shown to contain no embeddings [G1, P2, V2], and some connected components of the space of of all embeddings have been shown to contain no Lagrangian embeddings [L]. However, the classification problem for connected components of the space of embedded Lagrangian tori is far from being solved. It seems that all known examples of such tori are tori isotopic to an elementary toms (which is, by definition, a product of circles in~ 2) in the class of Lagrangian embeddings. In the present paper, the following question is considered: given two embedded Lagrangian tori L, L', whether or not there exists a global symplectomorphism 9 such that 9 (L)= L'. The solution is given for the case of elementary tori and certain monotone tori which are constructed in the paper. Some consequences of the main theorems are also discussed.