Pseudoholomorphic Curves in Symplectisations and Legendrian Knots

辛化中的伪全纯曲线和勒让结

• 批准号: 9971196
• 负责人: Casim Abbas
• 金额: $ 7.52万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 1999-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971196&HistoricalAwards=false
• 关键词: Pseudoholomorphic; Curves; Symplectisations; Legendrian; Knots

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.

摘要奖项：DMS-9971196 首席研究员：Casim Abbas 在三维接触流形的 Reeb 动力学及其停止学意义的研究中取得了重大进展。 过去的研究主要集中在里布矢量场的周期轨道上，而这个项目则处理不同类别的轨迹，即所谓的特征弦。这些是里布矢量场的轨道，在两个不同时间两次撞击给定的勒让德结（或更一般地，给定的勒让德子流形）。在周期轨道的情况下，主要工具是接触流形辛化中的伪全纯曲线，其域是无边界的穿孔黎曼曲面，而在特征弦的情况下，域将具有边界并且在边界上也有穿孔。将施加自由边界条件。该项目的一个目的是调查何时存在特征和弦，这个问题与 V.I. 的猜想有关。阿诺德接触几何。首席研究员开发的方法是 E. Bishop 经典结果的新版本，该结果涉及孤立椭圆复形切线附近真实曲面的局部填充。 该研究项目的另一个方面是与 Helmut Hofer 的联合工作，该项目为三维接触流形中的勒让德结（“接触同调”）的新不变量提供分析设置，其中穿孔黎曼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性问题提供更系统的方法，因为存在性结果意味着在某些情况下不变量的非平凡性。只有当接触流形是三维时，才可能通过填充方法来解决特征和弦的存在问题，而接触同调的构造也应该在更高的维度上工作，而在更高的维度上，人们对里布矢量场的动力学知之甚少。 另一方面，每个封闭的三维流形都承认一个接触形式，动力学和拓扑学之间密切相关。本研究项目涉及勒让结理论和由Reeb矢量场导出的动力系统。 近年来，结理论和量子场论（现代理论物理学的一部分）之间发现了密切的关系。里布型动力系统描述了物理学中非常多样化的现象，例如卫星在重力存在下的运动，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。