论文标题
相对增长率和接触Banach-Mazur距离
Relative growth rate and contact Banach-Mazur distance
论文作者
论文摘要
在本文中，我们在触点几何设置中定义了Banach-Mazur距离的非线性版本，称为Banach-Mazur距离，并用$ d _ {\ rm cbm} $表示。明确地，我们考虑以下两个设置，要么是在联系歧管$ w \ times s^1 $上，其中$ w $是liouville歧管，要么是封闭的liouville-fillable-fill-fill-follable tocter contact歧管$ m $。在这两种情况下，$ d _ {\ rm cbm} $的输入是不同的。在前一种情况下，输入为（联系）$ w \ times s^1 $的星形域，在后一种情况下，输入是$ m $的1形式的联系人。特别是，在前一种情况下定义的联系Banach-Mazur距离$ D _ {\ rm CBM} $是由Eliashberg和Polterovich最初定义和研究的概念，相对增长率的动机。此外，我们研究了$ d _ {\ rm cbm} $与触点几何和符号几何形状中各种数值测量的关系，例如，触点形状不变，（粗）符号banach-mazur距离。此外，我们以$ d _ {\ rm cbm} $而获得了几种大规模的几何属性。最后，我们提出了模块衍生类别类别（在某些拓扑空间）中的元素之间的定量比较。这是基于或骨的奇异支撑的几种重要特性。
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.
