发表机构
Swarthmore College; Universidad Nacional de Colombia, Sede Manizales, Facultad de Ciencias Exactas y Naturales, Departamento de Matemáticas y Estadística（斯沃斯莫尔学院; 哥伦比亚国立大学马尼萨莱斯校区自然科学理学院数学与统计系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为无限型驯服平移曲面引入基于奇点数据的分层理论，证明Loch Ness monster上有限锥角奇点个数为有限是实现障碍，且所有其他奇点数据均可实现，从而给出该曲面分层的完整刻画。
AI 中文摘要
紧致平移曲面的经典理论通过根据相关阿贝尔微分的零点阶数对模空间进行分层来组织。在无限拓扑类型的驯服平移曲面的背景下，尚未系统地发展出类似的分层概念。在本文中，我们引入了一种基于奇点数据的驯服平移曲面的分层理论。对于每个非紧致驯服平移曲面，我们关联一个序列，该序列记录了每个可能锥角的有限角锥奇点的基数以及无限角奇点集合的基数。我们称这个序列为曲面的奇点数据。我们研究了Loch Ness monster上奇点数据的实现问题。我们的第一个主要结果表明，在Loch Ness monster上不存在仅具有有限个有限锥角奇点的驯服平移结构。然后我们证明这是唯一的障碍：每个未被先前结果排除的奇点数据都可以通过Loch Ness monster上的驯服平移结构实现。因此，我们获得了该曲面上驯服平移结构的分层根据其奇点数据的完整刻画。这些结果提供了非紧致驯服平移曲面分层的第一个实现定理，并揭示了奇点数据与端点空间之间的强相互作用。
英文摘要
The classical theory of compact translation surfaces is organized through the stratification of moduli spaces according to the orders of the zeros of the associated Abelian differentials. In the setting of tame translation surfaces of infinite topological type, no analogous notion of strata has been systematically developed. In this article, we introduce a stratification theory for tame translation surfaces based on their singularity data. To each non-compact tame translation surface, we associate a sequence recording the cardinalities of finite-angle cone singularities of each possible cone angle together with the cardinality of the set of infinite-angle singularities. We call this sequence the singularity data of the surface. We study the realization problem for singularity data on the Loch Ness monster. Our first main result shows that there is no tame translation structure on the Loch Ness monster having only a finite number of finite cone angle singularities. We then prove that this is the only obstruction: every singularity data not excluded by the previous result is realized by a tame translation structure on the Loch Ness monster. As a consequence, we obtain a complete characterization of the strata of tame translation structures on this surface in terms of their singularity data. These results provide the first realization theorem for strata of non-compact tame translation surfaces and reveal a strong interaction between singularity data and the ends space.
Comments21 pages, 8 figures
