Weak Symplectic Fillings and Holomorphic Curves
Weak Symplectic Fillings and Holomorphic Curves
复制标题
DOI：
10.24033/asens.2155
复制
发表时间：
2010-03
影响因子：
1.9
通讯作者：
K. Niederkruger;C. Wendl
中科院分区：
文献类型：
--
作者：
K. Niederkruger;C. Wendl
我们证明了接触3流形弱辛填充的几个结果，包括：(1)任意平面接触流形的每一个弱填充都可以变形为Stein填充的膨胀。(2)具有完全分离平面扭转的接触流形不是弱可填充的——这给出了许多没有吉鲁扭转的接触流形没有弱填充的新例子。(3)接触流形沿辛前拉格朗日环面拼接时保留了弱可填充性，给出了许多没有吉鲁扭转但弱但不强可填充的接触流形的新例子。利用全纯曲线，通过两种平行方法建立了弱填充的障碍物。在第一种方法中，我们推广了原始的Gromov-Eliashberg“Bishop盘”论证，研究了边界在“锚定过扭环”上的全纯环空的Bishop族的Giroux扭转的特殊情况。第二种方法使用穿孔全纯曲线，并基于观察到每个弱填充都可以在环邻域中变形，从而在边界上诱导稳定的哈密顿结构。这也使得辛场论技术的应用成为可能，我们在一个测试案例中证明了弱可填充和强可填充之间的区别可以转化为接触同调，就像扭曲系数和非扭曲系数之间的区别一样。
We prove several results on weak symplectic fillings of contact 3-manifolds, including: (1) Every weak filling of any planar contact manifold can be deformed to a blow up of a Stein filling. (2) Contact manifolds that have fully separating planar torsion are not weakly fillable - this gives many new examples of contact manifolds without Giroux torsion that have no weak fillings. (3) Weak fillability is preserved under splicing of contact manifolds along symplectic pre-Lagrangian tori - this gives many new examples of contact manifolds without Giroux torsion that are weakly but not strongly fillable. We establish the obstructions to weak fillings via two parallel approaches using holomorphic curves. In the first approach, we generalize the original Gromov-Eliashberg "Bishop disk" argument to study the special case of Giroux torsion via a Bishop family of holomorphic annuli with boundary on an "anchored overtwisted annulus". The second approach uses punctured holomorphic curves, and is based on the observation that every weak filling can be deformed in a collar neighborhood so as to induce a stable Hamiltonian structure on the boundary. This also makes it possible to apply the techniques of Symplectic Field Theory, which we demonstrate in a test case by showing that the distinction between weakly and strongly fillable translates into contact homology as the distinction between twisted and untwisted coefficients.