Pseudoholomorphic Curves in Symplectisations and Legendrian Knots

摘要奖：DMS-9971196首席研究员：卡西姆·阿巴斯在研究三维接触流形的Reeb动力学及其病理学意义方面取得了重大进展。过去的研究集中在Reeb矢量场的周期轨道上，而这个项目处理的是一类不同的轨道，即所谓的特征弦。这些是Reeb向量场的轨道，它在两个不同的时间两次击中给定的Legendrian结(或者更一般的，更一般的，给定的Legendrian子流形)。在周期轨道的情况下，接触流形的辛中的主要工具是伪全纯曲线，区域是无边界的穿孔黎曼面，而在特征弦的情况下，区域既有边界又有边界上的穿孔。将施加自由边界条件。这个项目的目的之一是研究何时存在特征弦，这是一个与接触几何中的VI.Arnold猜想有关的问题。该方法是E.Bishopp关于真实曲面在孤立椭圆复切线附近的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的合作，它为三维接触流形中的Legendrian纽结的一个新的不变量(“接触同伦”)提供了分析设置，其中被穿孔的Riemann曲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问题提供一种更系统的方法，因为存在结果在某些情况下将意味着不变量的非平凡。只有当接触流形是三维的时，才能用填充方法来探讨特征弦的存在问题，而接触同伦的构造也应该在对Reeb矢量场的动力学知之甚少的更高维上工作。另一方面，每个封闭的三维流形都有一个接触形式，动力学和拓扑学之间有着密切的联系。本研究项目研究了勒让德结理论和由Reeb矢量场诱导的动力学系统。近年来，纽结理论和现代理论物理中的量子场论之间发现了密切的联系。Reeb类型的动力学系统描述了物理中非常不同的现象，如卫星在重力作用下的运动，或者理想不可压缩流体的粒子在运动处于平衡(稳定)状态且压力几乎恒定的情况下的运动。

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.