AI 中文总结
研究在拉格朗日子流形空间基本群上定义的新同态，考虑拉格朗日通量和单值性，通过研究其像的离散性与哈密顿轨道拓扑的关系，对\(\mathbb{R}^4\)、\(\mathbb{C}P^2\)及\(\mathbb{C}P^n\)中的拉格朗日环面进行描述并提升相关结果拓扑。
AI 中文摘要
我们在拉格朗日子流形空间的基本群上定义了一个新的同态，它考虑了拉格朗日通量和单值性。我们研究其像的离散性，并将此离散性与相关拉格朗日量的哈密顿轨道的（\(C^\infty\)）拓扑联系起来。我们完全描述了\(\mathbb{R}^4\)和\(\mathbb{C}P^2\)中拉格朗日环面的这个新同态并给出明确构造。类似构造使我们能研究\(\mathbb{C}P^n\)中克利福德环面的形状不变量。最后，我们将\(\mathbb{R}^4\)中拉格朗日环面的结果从\(C^\infty\)拓扑提升到豪斯多夫拓扑。
英文摘要
We define a new morphism on the fundamental group of the space of Lagrangian submanifolds, which takes into account the Lagrangian flux and monodromy. We study the discreteness of its image and relate this discreteness to the ($C^\infty$) topology of the Hamiltonian orbit of the Lagrangian in question. We completely describe this new morphism for Lagrangian tori in $\mathbb{R}^4$ and $\mathbb{C}P^2$ and give explicit constructions. Similar constructions allow us to study the shape invariant of the Clifford torus in $\mathbb{C}P^n$. Finally, we upgrade our results on Lagrangian tori in $\mathbb{R}^4$ from the $C^\infty$ topology to the Hausdorff one.
Commentsv2: improved Appendix B following private communications with Dominique Rathel-Fournier; 62 pages, 5 figures, 1 table
