Families of contact structures on 3-manifolds; surgery descriptions and open book decompostions of contact manifolds

本计画研究三维流形上的某些接触结构族，即所谓的接触圆与接触球。目的是了解这些结构与表面上复杂结构的泰希米勒理论，三维流形上特殊流动的动力学以及物理学中产生的超凯勒度量的构造（如Gibbons-Hawking度量）的关系。最终目标是建立联系圈，作为回答这些领域出现的问题的工具。具体目标是分类和理解三维流形上的横向共形流的几何，研究由接触圆引起的高斯-博内定理的推广，并研究高阶和orbifolds的自旋结构的推广，以及Teichmüller空间的相关覆盖。该项目的一部分与接触手术3流形的目的是找到明确的手术介绍接触3流形和应用这些介绍接触拓扑结构中出现的问题。该项目的第三部分涉及高维流形上接触结构的存在和分类。特别强调的是存在领域的这种结构是兼容的有限群行动领域。

This project studies certain families of contact structures, so-called contact circles and contact spheres, on 3-dimensional manifolds. The aim is to understand the relation of these structures to the Teichmüller theory of complex structures on surfaces, the dynamics of special flows on 3-manifolds, and constructions of hyperkähler metrics arising in physics such as the Gibbons-Hawking ansatz. The ultimate goal is to develop contact circles as a tool for answering questions arising in those areas. Specific aims are to classifiy and understand the geometry of transversely conformal flows on 3-manifolds, to study a generalisation of the Gauß-Bonnet theorem arising from contact circles, and to investigate a generalisation of spin structures to higher orders and orbifolds, and related coverings of Teichmüller space. The part of the project concerned with contact surgery on 3-manifolds aims to find explicit surgery presentations for contact 3-manifolds and applications of these presentations to questions arising in contact topology. A third strand of the project is concerned with the existence and classification of contact structures on higher-dimensional manifolds. Particular stress is laid on the existence on spheres of such structures that are compatible with finite group actions on that sphere.