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arXiv 1810.02421math.GT
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泛Teichmüller空间中Teichmüller变形的收敛性

Convergence of Teichmüller deformations in the universal Teichmüller space

Hideki Miyachi, Dragomir Šarić

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14 pages, 1 figure
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摘要：

设$φ:\mathbb{D}\to\mathbb{C}$是泛Teichmüller空间$T(\mathbb{D})$中单位圆盘$\mathbb{D}$上的可积全纯函数，$D_φ:\mathbb{D}\to T(\mathbb{D})$是Teichmüller圆盘.对于正的$t$，已知$D_φ(t)\to [μ_φ]\in PML_b(\mathbb{D})$为$t\to 1$，其中$μ_φ$是表示$T(\mathbb{D})$的Thurston边界上的点的有界测量叠层。我们扩展这个结果表明，$D_φ\colon \mathbb{D}\to T(\mathbb{D})$延伸作为一个连续的地图从封闭的磁盘$\overline{\mathbb{D}}$的Thurston bordification。此外，我们还证明了$λ\to e^{iθ}$时$D_φ(λ)$的收敛速度与$e^{iθ}\in\partial\mathbb{D}$方法的类型无关。

英文摘要：

Let $φ:\mathbb{D}\to\mathbb{C}$ be an integrable holomorphic function on the unit disk $\mathbb{D}$ and $D_φ:\mathbb{D}\to T(\mathbb{D})$ the Teichmüller disk in the universal Teichmüller space $T(\mathbb{D})$. For a positive $t$ it is known that $D_φ(t)\to [μ_φ]\in PML_b(\mathbb{D})$ as $t\to 1$, where $μ_φ$ is a bounded measured lamination representing a point on the Thurston boundary of $T(\mathbb{D})$. We extend this result by showing that $D_φ\colon \mathbb{D}\to T(\mathbb{D})$ extends as a continuous map from the closed disk $\overline{\mathbb{D}}$ to the Thurston bordification. In addition, we prove that the rate of convergence of $D_φ(λ)$ when $λ\to e^{iθ}$ is independent of the type of the approach to $e^{iθ}\in\partial\mathbb{D}$.

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