Hamiltonian Dynamics and Pseudoholomorphic Curves

经典力学中研究的一个系统的例子是小卫星在行星引力影响下的运动。从历史上看，辛几何是从研究这样的情况中发展起来的，在这种情况下，系统（相空间）的所有可能位置和动量的集合被认为具有一个特殊空间的结构，称为辛流形。物理定律，如能量守恒，然后产生了系统的模型，作为对应于不同能级的相空间子流形上的动力系统。现代辛方法已经在辛流形的全局性质和一类特殊能级上的动力学之间建立了深刻的联系，即接触型超曲面，但迄今为止已经证明对一般能级的使用相当有限。这个项目的目的是纠正这一不足，并使用全局辛技术，通过分析一个广义的曲线类，称为野生pseudoholomorphic曲线，研究任意能级上的动力学。本计画主要研究几何分析、辛拓扑与动力系统之介面。其关键思想是使用一个新定义的一类无限Hofer能量伪全纯曲线，即野生曲线，研究Hamilton流在指定的能量表面上的辛流形的任意尺寸和某些体积保持流的三个流形一般。更具体地说，我们的目标是研究这些曲线和它们的属性，并使用它们来建立非极小的范围广泛的体积保持流在三维。另一个目的是研究一般的渐近极限，这样的野生曲线，并确定是否或不辛场理论承认扩展到辛流形与一般光滑边界。这项研究的成功完成有可能解决保体积的Gottschalk猜想，广泛扩展辛场论，并可能产生关键的新见解低维微分拓扑。

An example of a system studied in classical mechanics is the motion of a small satellite under the gravitational influence of planets. Historically, symplectic geometry grew out of studying situations such as this, in which the collection of all possible positions and momenta of the system (phase space) was seen to have the structure of a special space known as a symplectic manifold. Physical laws, like conservation of energy, then gave rise to models of the systems as dynamical systems on submanifolds of phase space corresponding to different energy levels. Modern symplectic methods have established deep connections between global properties of symplectic manifolds and the associated dynamics on a special class of energy levels, namely contact-type hypersurfaces, but have thus far proved to be of rather limited use for a general energy level. This project aims to rectify this deficiency and use global symplectic techniques to study dynamics on arbitrary energy levels by analyzing a generalized class of curves, known as feral pseudoholomorphic curves. This project concerns research at the interface of geometric analysis, symplectic topology, and dynamical systems. The key idea is the use of a certain newly defined class of infinite Hofer-energy pseudoholomorphic curves, namely feral curves, to study Hamiltonian flows on prescribed energy surfaces in symplectic manifolds of arbitrary dimension and certain volume-preserving flows on three-manifolds in general. More specifically, the aim is to study these curves and their properties and to use them to establish non-minimality of a wide range of volume preserving flows in dimension three. A further aim is to study the generic asymptotic limit of such feral curves and to determine whether or not symplectic field theory admits an extension to symplectic manifolds with generic smooth boundary. Successful completion of this research has the potential to solve the volume-preserving Gottschalk conjecture, broadly extend symplectic field theory, and possibly yield key new insights on low dimensional differential topology.