The Chord Problem and a new method of filling by pseudoholomorphic curves

Pseudoholomorphic Strips in Symplectisations I: Asymptotic Behavior

This paper is part of a larger program, the investigation of the Chord Problem in three dimensional contact geometry. The main tool will be pseudoholomorphic strips in the symplectisation of a three dimensional contact manifold with two totally real submanifolds $L_0,L_1$ as boundary conditions. The submanifolds $L_0$ and $L_1$ do not intersect transversally. The subject of this paper is to study the asymptotic behavior of such pseudoholomorphic strips.

稳定Morse数作为Reeb弦数的下界

The stable Morse number as a lower bound for the number of Reeb chords

本文研究Legendrian子流形上Reeb弦数的下界问题，证明在spin且Maslov类为零的恰当Lagrangian填充条件下，该下界由填充的稳定Morse数给出，并在一般情形下借助Sharko不变量改进基于Betti数的已知结果。

Quasipositive links and Stein surfaces

We study the generalization of quasipositive links from the three-sphere to arbitrary closed, orientable three-manifolds. Our main result shows that the boundary of any smooth, properly embedded complex curve in a Stein domain is a quasipositive link. This generalizes a result due to Boileau and Orevkov, and it provides the first half of a topological characterization of links in three-manifolds which bound complex curves in a Stein filling. Our arguments replace pseudoholomorphic curve techniques with a study of characteristic and open book foliations on surfaces in three- and four-manifolds.