Pseudoholomorphic Curves in Symplectisations and Legendrian Knots

[摘要]主持人：Casim abba在三维接触流形的三维动力学及其形态学意义的研究方面取得了重大进展。过去的研究集中在Reeb向量场的周期性轨道上，而这个项目处理的是另一类轨迹，即所谓的特征和弦。这些是Reeb向量场的轨道，它们在两个不同的时间（或者更一般地说，给定的legendian子流形）两次击中给定的legendian结。在周期轨道的情况下，主要工具是接触流形的复合中的伪全纯曲线，其区域是一个没有边界的穿孔黎曼曲面，而在特征弦的情况下，该区域将有边界和边界上的穿孔。将施加一个自由边界条件。本项目的目的之一是研究特征和弦存在的情况，这是一个与V.I.阿诺德接触几何猜想有关的问题。主要研究者开发的方法是E. bishop关于孤立椭圆复切线附近实曲面局部填充的经典结果的新版本。该研究项目的另一个方面是与Helmut Hofer合作，为三维接触流形中的一个新的Legendrian结不变量（“接触学”）提供了分析设置，其中穿孔黎曼表面上的伪全纯曲线起着关键作用。这也将为特征和弦的存在性问题提供一个更系统的方法，因为存在性结果将意味着在某些情况下不变量的非平凡性。只有在接触流形是三维的情况下，才能用填充法求解特征弦的存在性问题，而接触流形的构造也应该适用于对Reeb矢量场的动力学知之甚少的高维情况。另一方面，每一个封闭的三维流形都有接触形式，动力学和拓扑学之间有着密切的联系。本研究项目涉及由Reeb矢量场诱导的Legendrian结和动力系统的理论。近年来，人们发现，作为现代理论物理的一部分，结理论与量子场论之间存在着密切的关系。Reeb类型的动力系统描述了物理学中各种各样的现象，比如在重力作用下卫星的运动，或者理想不可压缩流体中粒子的运动，如果运动处于不平衡（稳定）状态，并且压力几乎是恒定的。

AbstractAward: DMS-9971196Principal Investigator: Casim AbbasSignificant progress has been made in the study of theReeb-dynamics of a three dimensional contact manifold and itstopological implications. Past research focused on periodicorbits of the Reeb vectorfield, while this project deals with adifferent class of trajectories, the so-called characteristicchords. These are orbits of the Reeb vectorfield that hit a givenLegendrian knot twice at two different times (or more general, agiven Legendrian submanifold). In the case of periodic orbits themain tools were pseudoholomorphic curves in the symplectisationof the contact manifold with domain being a punctured Riemannsurface without boundary, while in the case of characteristicchords the domain will have boundary and also punctures on theboundary. A free boundary condition will be imposed. One aim ofthis project is to investigate when characteristic chords exist,a question which is related to a conjecture of V.I. Arnold incontact geometry. The method developed by the principalinvestigator is a new version of a classical result by E. Bishopabout local fillings of a real surface near an isolated ellipticcomplex tangency. The other aspect of the research project isjoint work with Helmut Hofer which is providing the analyticsetting for a new invariant of Legendrian knots ("ContactHomology") in a three dimensional contact manifold, wherepseudoholomorphic curves on punctured Riemann surfaces play thekey role. This would also provide a more systematic approach tothe existence question for characteristic chords, since existenceresults would imply nontriviality for the invariants in certaincases. Approaching the existence question of charactericticchords by filling methods is only possible if the contactmanifold is three dimensional while the construction of ContactHomology should also work in higher dimensions where very littleis known about the dynamics of the Reeb vectorfield. On theother hand, every closed three dimensional manifold admits acontact form, with between dynamics and topology closely related.This research project deals with the theory of Legendrian knotsand dynamical systems induced by a Reeb vectorfield. In recentyears close relationships have been found between knot theory andquantum field theory, a part of modern theoretical physics.Dynamical systems of Reeb type describe a very diverse spectrumof phenomena in physics, such as the motion of a satellite in thepresence of gravitational forces, or the motion of the particlesof an ideal incompressible fluid if the motion is in anequilibrium (steady) state and the pressure is almost constant.