标题： 曲线复形上的定向图 I：双曲性和极值长度 显示英文标题

标题： Oriented graphs on curve complex I: hyperbolic and extremal length

摘要： 我们研究基于闭曲面 $S$ 的曲线复形 $C(S)$ 的有向图，这些有向图由 $C(S)$ 顶点集上的函数诱导。 特别是，我们引入了德恩拟相似变换函数，它们在重复的德恩扭转下行为类似于相似变换。 我们证明，任何两个相同类型的正这样的函数除非成比例，否则会诱导不同的有向图。 这导致了一个新的闭双曲曲面刚性结果——与 $9g-9$ 定理和长度谱刚性不同——仅知道任意两条不相交的简单闭曲线哪一条更长（以双曲或极值长度为标准）就足以确定曲面上的双曲度量。 我们还证明，由具有有限子水平集的函数所诱导的有向图的每个自同构都由 $S$ 的自同胚所诱导。 显示英文摘要

摘要： We investigate oriented graphs based on the curve complex $C(S)$ of a closed surface $S$ and induced by functions on the vertex set of $C(S)$. In particular, we introduce the Dehn quasi-homothetic functions, which behave similarly to homotheties under repeated Dehn twists. We prove that any two positive such functions of the same type induce different oriented graphs unless they are proportional. This leads to a new rigidity result for closed hyperbolic surfaces -- distinct from the $9g-9$ theorem and length spectrum rigidity -- knowing only for any two disjoint simple closed curves which one is longer (in terms of hyperbolic or extremal length) suffices to determine the hyperbolic metric on the surface. We also prove that each automorphism of the oriented graph induced by a function with sublevel sets finite is induced by a self-homeomorphism of $S$.