哈塞尔度量和边界深度
Hofer's metrics and boundary depth
摘要：
本文证明了：如果（M，ω）是一个闭辛流形，其中存在一个非平凡的Hamilton向量场，且该向量场的所有可收缩闭轨道都是常数，则（M，ω）的Hamilton同态群上的Hynst度量具有无穷大直径，并且确实存在无穷维拟等距嵌入赋范向量空间.类似的结论也适用于各种拉格朗日子流形空间上的哈塞尔度规，包括当M满足上述动力学条件时，M × M中对角线的哈塞尔度规。为了证明这一点，我们使用称为边界深度的弗洛尔理论量的属性，它以编码鲁棒辛拓扑信息的方式测量弗洛尔复体上边界算子的非平凡性。
英文摘要：
We show that if (M,ω) is a closed symplectic manifold which admits a nontrivial Hamiltonian vector field all of whose contractible closed orbits are constant, then Hofer's metric on the group of Hamiltonian diffeomorphisms of (M,ω) has infinite diameter, and indeed admits infinite-dimensional quasi-isometrically embedded normed vector spaces. A similar conclusion applies to Hofer's metric on various spaces of Lagrangian submanifolds, including those Hamiltonian-isotopic to the diagonal in M x M when M satisfies the above dynamical condition. To prove this, we use the properties of a Floer-theoretic quantity called the boundary depth, which measures the nontriviality of the boundary operator on the Floer complex in a way that encodes robust symplectic-topological information.
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