论文标题
通过Fenchel-Nielsen参数的Riemann表面的类型问题
The type problem for Riemann surfaces via Fenchel-Nielsen parameters
论文作者
论文摘要
如果Riemann Surface $ X $据说为\ emph {抛物线类型}，如果它支持绿色的功能。同等地，$ x $的单位切线上的测量流是ergodic的。鉴于Riemann表面$ x $的任意拓扑类型和$ x $的双曲线裤子分解，我们在分解的Fenchel-Nielsen参数方面获得了$ x $的寄生额的足够条件。特别是，我们启动了扭曲参数对抛物线的影响的研究。我们工作中的关键要素是\ textit {non Standard Half-Collar}关于双曲线测量的概念。我们表明，这种半领的模量比标准半领的模量大得多，因为核心测量的双曲线长度倾向于无穷大。此外，与标准项圈的情况不同，通过粘合两个非标准半奖币获得的环的模量取决于扭曲参数。在许多情况下，我们的结果很明显。例如，对于零扭式长笛表面以及具有长度凹入序列的半扭式长笛表面，我们的结果就长度参数提供了抛物性的完整表征。因此，在这些情况下，抛物线性等同于完整性。还研究了其他拓扑类型的应用，例如具有无限属和一端的表面（又称无限的尼斯尼斯怪物），梯形表面，紧凑型表面的abelian覆盖物。
A Riemann surface $X$ is said to be of \emph{parabolic type} if it supports a Green's function. Equivalently, the geodesic flow on the unit tangent of $X$ is ergodic. Given a Riemann surface $X$ of arbitrary topological type and a hyperbolic pants decomposition of $X$ we obtain sufficient conditions for parabolicity of $X$ in terms of the Fenchel-Nielsen parameters of the decomposition. In particular, we initiate the study of the effect of twist parameters on parabolicity. A key ingredient in our work is the notion of \textit{non standard half-collar} about a hyperbolic geodesic. We show that the modulus of such a half-collar is much larger than the modulus of a standard half-collar as the hyperbolic length of the core geodesic tends to infinity. Moreover, the modulus of the annulus obtained by gluing two non standard half-collars depends on the twist parameter, unlike in the case of standard collars. Our results are sharp in many cases. For instance, for zero-twist flute surfaces as well as half-twist flute surfaces with concave sequences of lengths our results provide a complete characterization of parabolicity in terms of the length parameters. It follows that parabolicity is equivalent to completeness in these cases. Applications to other topological types such as surfaces with infinite genus and one end (a.k.a. the infinite Loch-Ness monster), the ladder surface, Abelian covers of compact surfaces are also studied.
