论文标题
用$ \ mathbb z $ Action在Teichmüller空间的Teichmüller空间建造地球学
Construction of geodesics on Teichmüller spaces of Riemann surfaces with $\mathbb Z$ action
论文作者
论文摘要
Riemann Surface $ r $的Teichmüllerspace $ \ mathrm {teich}（r）$是$ r $的变形空间。在本文中，当$ r $具有$ \ mathbb z $ Action时，我们证明了Beltrami系数极大的条件。作为应用程序，我们讨论了大地测量学的构建。 Earle-kra-krushkaĺ证明了连接$ [0] $和$ [μ] $的地理素的必要条件是$ \ | μ_0\ | _ {\ infty} = | μ_0| （z）$（a.e。$ z $）和``独特的极端''。作为我们结果的副产品，我们表明我们不能排除``唯一的极值''。要显示上述主张，我们在$ \ mathrm {teich}中构造了一个点$ [μ_0] $ μ_0\ | _ {\ infty} = | μ_0| （z）$（a.e。$ z $），并且存在一个大地测量家族$ \ {γ_λ\} _ {λ\ in d} $连接$ [0] $和$ [μ_0] $，带有复杂的分析参数，其中$ d $是$ l^{\ iffty} $的开放式。
Teichmüller space $\mathrm{Teich}(R)$ of a Riemann surface $R$ is a deformation space of $R$. In this paper, we prove a sufficient condition for extremality of the Beltrami coefficients when $R$ has the $\mathbb Z$ action. As an application, we discuss the construction of geodesics. Earle-Kra-Krushkaĺ proved that the necessary and sufficient conditions for the geodesics connecting $[0]$ and $[μ]$ to be unique are $\| μ_0 \|_{\infty} = | μ_0 | ( z )$ (a.e.$z$) and ``unique extremality''. As a byproduct of our results, we show that we cannot exclude ``unique extremality''.To show the above claim, we construct a point $[μ_0]$ in $\mathrm{Teich}(\mathbb C \setminus \mathbb Z)$, satisfying $\| μ_0 \|_{\infty} = | μ_0 | ( z )$ (a.e.$z$) and there exists a family of geodesics $\{ γ_λ\} _{λ\in D}$ connecting $[0]$ and $[μ_0]$ with complex analytic parameter, where $D$ is an open set in $l^{\infty}$.
