HOLOMORPHIC CURVES, REEB FLOWS AND CONTACT TOPOLOGY

卫星的运动是里布动力系统的众多例子之一。该项目的目的是加深我们对里海水流的了解。特别是Reeb流，包括三维接触型能量表面上的哈密顿流。为了研究Reeb流的行为，我们构造了剖面的整体曲面系统，并研究了庞加莱图的迭代。庞加莱图是通过跟踪流直到它碰到一个表面而得到的。构造截面整体曲面系统的主要工具是在穿孔黎曼曲面上定义并求解非线性Cauchy-Riemann型算子的全纯化曲线。这些曲线也是新的接触流形和辛流形不变量的主要组成部分。

Motion of a satellite is one of many examples of a Reeb dynamical system. The aim of the project is to deepen our understanding of Reeb flows. The Reeb flows, in particular, include Hamiltonian flows on three-dimensional contact type energy surfaces. To study the behaviour of Reeb flows we construct systems of global surfaces of section and study the iterates of the Poincare map, which is obtained by following the flow until it hits a surface. The main tools in constructing systems of global surfaces of section are holomorphic curves in symplectization, which are defined on punctured Riemann surfaces and solve nonlinear Cauchy-Riemann type operator. These curves are also main ingredients of new invariants of contact and symplectic manifolds. These new invariants are now known as Contact Homology and Symplectic Field Theory. In the second part of the project we develop analytical foundations for these theories.