The Schwarzian derivative and measured laminations on Riemann surfaces
The Schwarzian derivative and measured laminations on Riemann surfaces
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DOI：
10.1215/s0012-7094-07-14021-3
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发表时间：
2005-10
影响因子：
2.5
通讯作者：
D. Dumas
中科院分区：
文献类型：
--
作者：
D. Dumas
双曲黎曼曲面上的全纯二次微分具有相关的测量叶状结构，可以将其拉直以产生测量的测地层状结构。另一方面，二次微分可以被视为 1 结构的 Schwarzian 导数，人们可以使用嫁接将另一个测量的测地层压自然地与它关联起来。我们比较二次微分和测得的测地层压之间的这两种关系，每个关系都会产生同胚 ML(S)！ Q(X) 对于紧凑表面 S 上的每个共形结构 X。我们证明这些图几乎是相同的，区别在于乘法因子 2 和比图本身低阶的误差项（我们明确绑定）。作为一个应用，我们证明具有 Fuchsian 完整性的 1 结构的 Schwarzian 导数接近于 2�-积分 Jenkins-Strebel 微分。我们还使用 Schwarzian 导数和嫁接坐标研究了 1 结构空间的紧化；我们证明了它们之间的自然映射延伸到 Teichmuller 空间上每个纤维的边界，并且我们描述了这种延伸。
A holomorphic quadratic differential on a hyperbolic Rie- mann surface has an associated measured foliation, which can be straight- ened to yield a measured geodesic lamination. On the other hand, a quadratic differential can be considered as the Schwarzian derivative of a 1 structure, to which one can naturally associate another measured geodesic lamination using grafting. We compare these two relationships between quadratic differentials and measured geodesic laminations, each of which yields a homeomor- phism ML(S) ! Q(X) for each conformal structure X on a compact surface S. We show that these maps are nearly the same, differing by a multiplicative factor of 2 and an error term of lower order than the maps themselves (which we bound explicitly). As an application we show that the Schwarzian derivative of a 1 structure with Fuchsian holonomy is close to a 2�-integral Jenkins- Strebel differential. We also study compactifications of the space of 1 structures using the Schwarzian derivative and grafting coordinates; we show that the natural map between these extends to the boundary of each fiber over Teichmuller space, and we describe this extension.