John Etnyre
In this paper we develop a method for studying tight contact structures on lens spaces. We then derive uniqueness and non-existence statements for tight contact structures with certain (half) Euler classes on lens spaces. We also prove that any lens space admits only finitely many tight contact structures.
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We exhibit a 3-manifold which admits no tight contact structure.
We consider the problem of realizing tight contact structures on closed orientable three-manifolds. By applying the theorems of Hofer et al., one may deduce tightness from dynamical properties of (Reeb) flows transverse to the contact structure. We detail how two classical constructions, Dehn surgery and branched covering, may be performed on dynamically-constrained links in such a way as to preserve a transverse tight contact structure.
We exhibit tight contact structures on 3-manifolds that do not admit any symplectic fillings.
We develop new techniques in the theory of convex surfaces to prove complete classification results for tight contact structures on lens spaces, solid tori, and T^2 X I. Erratum: In this note we seek to remedy errors which appeared in version 2 and were propagated in subsequent papers.
Classification of invariant tight contact structures on the 3-space, -ball and -sphere
本文研究了三维空间、球体和球面上不变紧致接触结构的分类，发现新的积分扭转导致等价类分裂，并可能为强可逆Legendrian结的分类提供工具。
We prove gluing theorems for tight contact structures. In particular, we rederive (as special cases) gluing theorems due to Colin and Makar-Limanov, and present an algorithm for determining whether a given contact structure on a handlebody is tight. As applications, we construct a tight contact structure on a genus 4 handlebody which becomes overtwisted after Legendrian -1 surgery, and study certain Legendrian surgeries on T^3.
We present a sketch of the proof of the following theorems: (1) Every 3-manifold has only finitely many homotopy classes of 2-plane fields which carry tight contact structures. (2) Every closed atoroidal 3-manifold carries finitely many isotopy classes of tight contact structures.
This paper begins the study of relations between Riemannian geometry and global properties of contact structures on 3-manifolds. In particular we prove an analog of the sphere theorem from Riemannian geometry in the setting of contact geometry. Specifically, if a given three dimensional contact manifold (M,ξ) admits a complete compatible Riemannian metric of positive 4/9-pinched curvature then the underlying contact structure ξis tight; in particular, the contact structure pulled back to the universal cover is the standard contact structure on S^3. We also describe geometric conditions in dime…
Each lens space has a canonical contact structure which lifts to the distribution of complex lines on the three-sphere. In this paper, we show that a symplectic homology cobordism between two lens spaces, which is given with the canonical contact structure on the boundary, must be diffeomorphic to the product of a lens space with the unit interval. As one of the main ingredients in the proof, we also derive in this paper the adjunction and intersection formulae for pseudoholomorphic curves in an almost complex 4-orbifold, extending the relevant work of Gromov and McDuff in the manifold setting…
Geodesic vector fields, induced contact structures and tightness in dimension three
本文简化了Gluck-Harrison定理的证明，证明了在非负混合曲率条件下测地向量场诱导切触结构，并分类了具有等距Reeb流的闭切触3-流形，推广了紧致性结果。
We show that contact homology distinguishes infinitely many tight contact structures on any orientable, toroidal, irreducible 3-manifold. As a consequence of the contact homology computations, on a very large class of toroidal manifolds, all known examples of universally tight contact structures with nonvanishing torsion satisfy the Weinstein conjecture. ----- On montre que l'homologie de contact distingue une infinite de structures de contact tendues sur toute variete toroidale irreductible et orientable de dimension trois. En consequence des calculs d'homologie de contact, sur une tres large…
Kenmotsu Contact Geometry Through the Lens of $\ast-\boldsymbolκ$-Ricci-Bourguignon Almost Solitons
研究 Kenmotsu 流形上 $\ast-\boldsymbol{\kappa}$-Ricci-Bourguignon 几乎孤子的性质，推导标量曲率，构造向量场，并探索扭形向量场的应用，给出五维实例。
Let $M$ be a compact orientable Seifered fibered 3-manifold without a boundary, and $α$ an $S^1$-invariant contact form on $M$. In a suitable adapted Riemannian metric to $α$, we provide a bound for the volume $\text{Vol}(M)$ and the curvature, which implies the universal tightness of the contact structure $ξ=\kerα$.
Sharp systolic inequalities for invariant tight contact forms on principal S1-bundles over S2
本文研究S²上非平凡主S¹丛中S¹不变紧接触形式的收缩问题，证明了依赖Euler类的尖锐不等式，并将其应用于旋转对称Finsler度量、可缩Reeb轨道及Viterbo猜想特例。
Symplectic embeddings of toric domains with boundary a lens space
本研究给出凹透镜空间的嵌入切触复形组合描述，定义带奇点的广义凹环面域，利用ECC计算其ECH容量，并给出可恢复该容量的辛流形填充方式。
Contact structures and Beltrami fields on the torus and the sphere
本文在3-球面和3-环面上构造了新的显式紧和过度扭转接触结构，证明了弱相容度量下接触球面定理不成立，并建立了紧接触结构的几何刚性。
We construct a dynamical quantization for contact manifolds in terms of a flat connection acting on a Hilbert tractor bundle. We show that this contact quantization, which is independent of the choice of contact form, can be obtained by quantizing the Reeb dynamics of an ambient strict contact manifold equivariantly with respect to an R+-action. The contact quantization further determines a certain contact tractor connection whose parallel sections determine a distinguished choice of Reeb dynamics and their quantization. This relationship relies on tractor constructions from parabolic geometri…
We consider a family of tight contact structures on the three-dimensional torus and we compute the relative Contact Homology by using the variational theory of critical points at infinity. We will also show some algebraic equivariant homology reductions.
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