An index inequality for embedded pseudoholomorphic curves in symplectizations

Gluing pseudoholomorphic curves along branched covered cylinders I

This paper and its sequel prove a generalization of the usual gluing theorem for two index 1 pseudoholomorphic curves u_+ and u_- in the symplectization of a contact 3-manifold. We assume that for each embedded Reeb orbit gamma, the total multiplicity of the negative ends of u_+ at covers of gamma agrees with the total multiplicity of the positive ends of u_- at covers of gamma. However, unlike in the usual gluing story, here the individual multiplicities are allowed to differ. In this situation, one can often glue u_+ and u_- to an index 2 curve by inserting genus zero branched covers of R-in…

Compactness for Embedded Pseudoholomorphic Curves in 3-manifolds

We prove a compactness theorem for holomorphic curves in 4-dimensional symplectizations that have embedded projections to the underlying 3-manifold. It strengthens the cylindrical case of the SFT compactness theorem by using intersection theory to show that degenerations of such sequences never give rise to multiple covers or nodes, so transversality is easily achieved. This has application to the theory of stable finite energy foliations, and also suggests a new approach to defining SFT-type invariants for contact 3-manifolds, or more generally, 3-manifolds with stable Hamiltonian structures.

Intersection theory of punctured pseudoholomorphic curves

We study the intersection theory of punctured pseudoholomorphic curves in $4$-dimensional symplectic cobordisms. We first study the local intersection properties of such curves at the punctures. We then use this to develop topological controls on the intersection number of two curves. We also prove an adjunction formula which gives a topological condition that will guarantee a curve in a given homotopy class is embedded, extending previous work of Hutchings. We then turn our attention to curves in the symplectization $\mathbb{R}\times M$ of a $3$-manifold $M$ admitting a stable Hamiltonian str…

Asymptotic Behaviour for $\mathcal{H}-$holomorphic Cylinders of Small Area

$\mathcal{H}-$holomorphic curves are solutions of a specific modification of the pseudoholomorphic curve equation in symplectizations involving a harmonic $1-$form as perturbation term. In this paper we study the asymptotics of $\mathcal{H}-$holomorphic curves defined on a sequence of degenerating cylinders.

Holomorphic curves in the presence of holomorphic hypersurface foliations

We prove a result which establishes restrictions on the pseudoholomorphic curves which can exist in a stable Hamiltonian manifold in the presence of certain $\mathbb{R}$-invariant foliations of the symplectization by holomorphic hypersurfaces. This result has applications in the first author's work on algebraic torsion in higher dimensional contact manifolds.

Topological contact dynamics I: symplectization and applications of the energy-capacity inequality

We introduce topological contact dynamics of a smooth manifold carrying a cooriented contact structure, generalizing previous work in the case of a symplectic structure [MO07] or a contact form [BS12]. A topological contact isotopy is not generated by a vector field; nevertheless, the group identities, the transformation law, and classical uniqueness results in the smooth case extend to topological contact isotopies and homeomorphisms, giving rise to an extension of smooth contact dynamics to topological dynamics. Our approach is via symplectization of a contact manifold, and our main tools ar…

Embedded contact knot homology and a surgery formula

Embedded contact knot homology (ECK) is a variation on Embedded contact homology (ECH), defined with respect to an open book decomposition compatible with a contact structure on some 3-manifold, M. The knot in question is given by the (null-homologous) binding of the open book and the chain complex is defined in terms of closed orbits of the Reeb vector field and certain pseudoholomorphic curves in the symplectization of the knot complement. In this thesis we first generalize this construction to the case of rational open book decompositions, allowing us to define ECK for rationally null-homol…

利用 ECH 研究某些实射影三维空间上的椭圆 Reeb 轨道

Elliptic Reeb orbit on some real projective three-spaces via ECH

本文借助 ECH 的 $U$ 映射所诱导的伪全纯曲线，证明 $\mathbb{R}P^3$ 上某些切触形式存在椭圆 Reeb 轨道，并刻画、计算了相关 ECH 谱。

The embedded contact homology index revisited

Let Y be a closed oriented 3-manifold with a contact form such that all Reeb orbits are nondegenerate. The embedded contact homology (ECH) index associates an integer to each relative 2-dimensional homology class of surfaces whose boundary is the difference between two unions of Reeb orbits. This integer determines the relative grading on ECH; the ECH differential counts holomorphic curves in the symplectization of Y whose relative homology classes have ECH index 1. A known index inequality implies that such curves are (mostly) embedded and satisfy some additional constraints. In this paper we…

Quasiregular curves

We extend the notion of a pseudoholomorphic vector of Iwaniec, Verchota, and Vogel to mappings between Riemannian manifolds. Since this class of mappings contains both quasiregular mappings and (pseudo)holomorphic curves, we call them quasiregular curves. Let $n\le m$ and let $M$ be an oriented Riemannian $n$-manifold, $N$ a Riemannian $m$-manifold, and $ω\in Ω^n(N)$ a smooth closed non-vanishing $n$-form on $N$. A continuous Sobolev map $f\colon M \to N$ in $W^{1,n}_{\mathrm{loc}}(M,N)$ is a $K$-quasiregular $ω$-curve for $K\ge 1$ if $f$ satisfies the distortion inequality $(\lVertω\rVert\cir…