HOLOMORPHIC CURVES, REEB FLOWS AND CONTACT TOPOLOGY

全纯曲线、Reeb流和接触拓扑。卫星的运动是Reeb动力系统的许多例子之一。该项目的目的是加深我们对Reeb流动的理解。尤其是Reeb流，包括三维接触型能面上的哈密顿流。为了研究Reeb流的行为，我们构造了截面的全局曲面系统，并研究了Poincare映射的迭代，Poincare映射是通过跟踪流直到到达曲面而获得的。构造整体截面系的主要工具是辛化的全纯曲线，它们定义在穿孔黎曼曲面上，用于求解非线性Cauchy-Riemann型算子。这些曲线也是接触和辛流形的新不变量的主要成分。这些新不变量现在被称为接触同调和辛场理论。在项目的第二部分中，我们为这些理论建立了分析基础。

HOLOMORPHIC CURVES, REEB FLOWS AND CONTACT TOPOLOGY. 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.\r\nThese 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.