三维流形上的切触结构

三维流形上的切触结构的研究是目前低维拓扑中的活跃课题之一.本项目研究三维流形上的切触结构, 包括切触三维流形的Stein填充，切触三维流形中的 Legendrian纽结，以及切触三维流形与open book分解. 具体来说，本项目研究双曲三维流形上的Stein可填充的切触结构的存在性，overtwisted切触三维流形中的non-loose Legendrian纽结的存在性， 以及open book分解的分数的Dehn twist系数的估计，切触三维流形的支撑亏格的计算. 这些问题的研究和解决将加深人们对切触三维流形的拓扑，几何，特别是它们的各种不变量，的理解.

Recently, the study of contact structures on 3-dimensional manifolds becomes one of the active topics in low dimensional topology. In this project, we study the contact structures on 3-manifolds, including the Stein fillings of contact 3-manifolds, Legendrian knots in contact 3-manifolds, and open book decompositions. Precisely, we study the existence of Stein fillable contact structures on hyperbolic 3-dimensional manifolds, the existence of non-loose Legendrian knots in overtwisted contact 3-manifolds, the estimation of fractional Dehn twist coefficients of the open book decompositions, and the determination of support genus of contact 3-manifolds. The study of these problems will shed new light on the understanding of the topology, geometry, and invariants of contact 3-manifolds.