相对增长率和接触Banach-Mazur距离
Relative growth rate and contact Banach-Mazur distance
摘要：
本文在接触几何的基础上定义了一种非线性形式的Banach-Mazur距离，称为接触Banach-Mazur距离，记为$d_{\rm CBM}$ .解释，我们考虑以下两个设置，要么接触流形$W \times S^1$，其中$W$是一个刘维流形，或一个封闭的刘维可填充接触流形$M$。在这两种情况下，$d_{\rm CBM}$的输入不同。在前一种情况下，输入是$W \times S^1$的（接触）星形区域，在后一种情况下，输入是$M$的接触1-形式。特别是，在前一种情况下定义的接触巴拿赫-马祖距离$d_{\rm CBM}$的动机的概念，相对增长率，这是最初定义和研究的Eliashberg和Polterovich。此外，我们还研究了$d_{\rm CBM}$与接触几何和辛几何中的各种数值度量，如接触形状不变量、（粗）辛Banach-Mazur距离等的关系.此外，我们还得到了几个大尺度几何性质的$d_{\rm CBM}$。最后，我们提出了一个定量的比较元素之间的导出范畴层的模（在某些拓扑空间）。这是基于层的奇异支撑的几个重要性质。
英文摘要：
In this paper, we define a non-linear version of Banach-Mazur distance in the contact geometry set-up, called contact Banach-Mazur distance and denoted by $d_{\rm CBM}$. Explicitly, we consider the following two set-ups, either on a contact manifold $W \times S^1$ where $W$ is a Liouville manifold, or a closed Liouville-fillable contact manifold $M$. The inputs of $d_{\rm CBM}$ are different in these two cases. In the former case the inputs are (contact) star-shaped domains of $W \times S^1$, and in the latter case the inputs are contact 1-forms of $M$. In particular, the contact Banach-Mazur distance $d_{\rm CBM}$ defined in the former case is motivated by the concept, relative growth rate, which was originally defined and studied by Eliashberg and Polterovich. In addition, we investigate the relations of $d_{\rm CBM}$ to various numerical measurements in contact geometry and symplectic geometry, for instance, contact shape invariant, (coarse) symplectic Banach-Mazur distance. Moreover, we obtain several large-scale geometric properties in terms of $d_{\rm CBM}$. Finally, we propose a quantitative comparison between elements in the derived categories of sheaves of modules (over certain topological spaces). This is based on several important properties of the singular support of sheaves.
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