AI 中文总结
本文构造了从六边形图到增广Teichmüller空间的拟等距映射，并证明其宽度至多$\sqrt{g+n}\log(g+n)$，同时给出边界层的显式投影。
AI 中文摘要
我们给出了从三角剖分的翻转图 $\mathscr{F}_{g,n}$ 到配备Teichmüller度量的Teichmüller空间$\epsilon$-厚部分的一个显式拟等距映射，通过将每个理想三角剖分映射到沿该三角剖分的剪切坐标全为零的双曲曲面。扩展此构造，我们得到了从六边形图 $\mathscr{H}_{g,n}$ 到配备Weil-Petersson度量的增广Teichmüller空间 $\overline{\operatorname{Teich}}_{g,n}$ 的拟等距映射 $Q$，并估计了其宽度，即 $Q(\mathscr{H}_{g,n})$ 与 $\overline{\operatorname{Teich}}_{g,n}$ 之间的Hausdorff距离。通过界定沿嫁接射线的Weil-Petersson距离和沿剪切变形的Teichmüller距离，我们证明宽度至多为 $\sqrt{g+n}\log(g+n)$。我们还提供了从 $\overline{\operatorname{Teich}}_{g,n}$ 中任意点到边界层的厚部分的显式投影，同时保持对Weil-Petersson距离和剪切坐标的同时控制。
英文摘要
We give an explicit quasi-isometry from the flip graph of triangulations $\mathscr{F}_{g,n}$ to the $ε$-thick part of Teichmüller space equipped with the Teichmüller metric, by mapping each ideal triangulation to the hyperbolic surface whose shearing coordinates along that triangulation are all equal to zero. Extending this construction, we obtain a quasi-isometry $Q$ from the hexagon graph $\mathscr{H}_{g,n}$ to the augmented Teichmüller space $\overline{\operatorname{Teich}}_{g,n}$ equipped with the Weil-Petersson metric, and we estimate its width, that is, the Hausdorff distance between $Q(\mathscr{H}_{g,n})$ and $\overline{\operatorname{Teich}}_{g,n}$. By bounding the Weil-Petersson distance along grafting rays and the Teichmüller distance along shearing deformations, we show that the width is at most $\sqrt{g+n}\log(g+n)$. We also provide an explicit projection from any point in $\overline{\operatorname{Teich}}_{g,n}$ to the thick part of boundary strata, maintaining simultaneous control over the Weil-Petersson distance and the shearing coordinates.
Comments75 pages, 10 figures
