Proofs On Arnold Chord Conjecture And Weinstein Conjecture In M\times C

A Proof On Arnold Chord Conjecture In Cotangent Bundles

We prove the Arnold chord conjecture on cotangent bundles of open manifold by Gromov's nonlinear Fredholm alternative for $J-$holomorphic curves.

Proof of the Arnold chord conjecture in three dimensions II

In "Proof of the Arnold chord conjecture in three dimensions I", we deduced the Arnold chord conjecture in three dimensions from another result, which asserts that an exact symplectic cobordism between contact three-manifolds induces a map on (filtered) embedded contact homology satisfying certain axioms. The present paper proves the latter result, thus completing the proof of the three-dimensional chord conjecture. We also prove that filtered embedded contact homology does not depend on the choice of almost complex structure used to define it.

Proof of the Arnold chord conjecture in three dimensions I

This paper and its sequel prove that every Legendrian knot in a closed three-manifold with a contact form has a Reeb chord. The present paper deduces this result from another theorem, asserting that an exact symplectic cobordism between contact 3-manifolds induces a map on (filtered) embedded contact homology satisfying certain axioms. The latter theorem will be proved in the sequel using Seiberg-Witten theory.

关于凸接触形式的强阿诺德弦猜想

On the strong Arnold chord conjecture for convex contact forms

本文证明了在严格凸嵌入到$\mathbb{R}^{2n}$中诱导出的接触形式下，强阿诺德弦猜想成立，并给出了当凸性条件不满足时的反例。

通过归一化容量证明强 Arnold 弦猜想

Strong Arnold chord conjecture via normalized capacities

本文利用归一化容量证明，动态凸环面区域附近的所有星形区域边界上的任意 Legendrian 纽结均具有端点不同的 Reeb 弦，并给出高维推广。

On the strict Arnold chord property and coisotropic submanifolds of complex projective space

Let $α$ be a contact form on a manifold $M$, and $L\subseteq M$ a closed Legendrian submanifold. I prove that $L$ intersects some characteristic for $α$ at least twice if all characteristics are closed and of the same period, and $α$ embeds nicely into the product of $\mathbb{R}^{2n}$ and an exact symplectic manifold. As an application of the method of proof, the minimal action of a regular closed coisotropic submanifold of complex projective space is at most $π/2$. This yields an obstruction to presymplectic embeddings, and in particular to Lagrangian embeddings.