使用伪全纯曲线的哈密顿偏微分方程的时间周期解

Pseudoholomorphic discs and symplectic structures in Hilbert space

We develop the theory of $J$-holomorphic discs in Hilbert spaces with almost complex structures. As an aplication, we prove a version of Gromov's symplectic non-squeezing theorem for Hilbert spaces. It can be applied to short-time symplectic flows of a wide class of Hamiltonian PDEs.

Periodic approximations of irrational pseudo-rotations using pseudoholomorphic curves

We prove that every $C^\infty$-smooth, area preserving diffeomorphism of the closed 2-disk having not more than one periodic point is the uniform limit of periodic $C^\infty$-smooth diffeomorphisms. In particular every smooth irrational pseudo-rotation can be $C^0$-approximated by integrable systems. This partially answers a long standing question of A. Katok regarding zero entropy Hamiltonian systems in low dimensions. Our approach uses pseudoholomorphic curve techniques from symplectic geometry.

伪全纯曲线的极大极小能量与三维 Reeb 流

Max-min energy of pseudoholomorphic curves and periodic Reeb flows in dimension $3$

本文利用伪全纯曲线的极大极小能量谱不变量，刻画了三维 Reeb 流中 Zoll 接触形式，并推广至 Lens 空间及 Besse 情形。

An index inequality for embedded pseudoholomorphic curves in symplectizations

Let $Σ$ be a surface with a symplectic form, let $ϕ$ be a symplectomorphism of $Σ$, and let $Y$ be the mapping torus of $ϕ$. We show that the dimensions of moduli spaces of embedded pseudoholomorphic curves in $\R\times Y$, with cylindrical ends asymptotic to periodic orbits of $ϕ$ or multiple covers thereof, are bounded from above by an additive relative index. We deduce some compactness results for these moduli spaces. This paper establishes some of the foundations for a program with Michael Thaddeus, to understand the Seiberg-Witten Floer homology of $Y$ in terms of such pseudoholomorphic c…

The periodic Floer homology of a Dehn twist

The periodic Floer homology of a surface symplectomorphism, defined by the first author and M. Thaddeus, is the homology of a chain complex which is generated by certain unions of periodic orbits, and whose differential counts certain embedded pseudoholomorphic curves in R cross the mapping torus. It is conjectured to recover the Seiberg-Witten Floer homology of the mapping torus for most spin-c structures, and is related to a variant of contact homology. In this paper we compute the periodic Floer homology of some Dehn twists.

Feral Curves and Minimal Sets

Here we prove that for each Hamiltonian function $H\in \mathcal{C}^\infty(\mathbb{R}^4, \mathbb{R})$ defined on the standard symplectic $(\mathbb{R}^4, ω_0)$, for which $M:=H^{-1}(0)$ is a non-empty compact regular energy level, the Hamiltonian flow on $M$ is not minimal. That is, we prove there exists a closed invariant subset of the Hamiltonian flow in $M$ that is neither $\emptyset$ nor all of $M$. This answers the four dimensional case of a twenty year old question of Michel Herman, part of which can be regarded as a special case of the Gottschalk Conjecture. Our principal technique is the…