Electromagnetic fields from contact forms

Automatic transversality in contact homology II: filtrations and computations

This paper is the sequel to the previous paper [Ne15], which showed that sufficient regularity exists to define cylindrical contact homology in dimension three for nondegenerate dynamically separated contact forms, a subclass of dynamically convex contact forms. The Reeb orbits of these so-called dynamically separated contact forms satisfy a uniform growth condition on their Conley-Zehnder indices with respect to a free homotopy class; see Definition 1.7. {Given a contact form which is dynamically separated up to large action, we demonstrate a filtration by action on the chain complex and show…

关于 Zoll 切触 5-球面

On Zoll Contact 5-Spheres

本文证明了标准切触5-球面上具有Zoll Reeb流的任意切触形式，均严格切触同构于标准Zoll切触形式的伸缩。

Constructions of contact forms on products and piecewise fibered manifolds

We study constructions of contact forms on closed manifolds. A notion of strong symplectic fold structure is defined and we prove that there is a contact form on $M \x X$ provided that $M$ admits such a structure and $X$ is contact. This result is extended to fibrations satisfying certain natural conditions. Some examples and applications are given.

Systolic ratio, index of closed orbits and convexity for tight contact forms on the three-sphere

We construct a dynamically convex contact form on the three-sphere whose systolic ratio is arbitrarily close to 2. This example is related to a conjecture of Viterbo, whose validity would imply that the systolic ratio of a convex contact form does not exceed 1. We also construct a sequence of tight contact forms $α_n$, $n\geq 2$, with systolic ratio arbitrarily close to $n$ and suitable bounds on the mean rotation number of all the closed orbits of the induced Reeb flow.

Contact forms with large systolic ratio in arbitrary dimensions

If a contact form on a (2n+1)-dimensional closed contact manifold admits closed Reeb orbits, then its systolic ration is defined to be the quotient of (n+1)th power of the shortest period of Reeb orbits by the contact volume. We prove that every co-orientable contact structure on any closed contact manifold admits a contact form with arbitrarily large systolic ratio. This statement generalizes the recent result of Abbondandolo et al. in dimension three to higher dimensions. We extend the plug construction of Abbondandolo et. al. to any dimension, by means of generalizing the hamiltonian disc m…

Contact forms with large systolic ratio in dimension three

The systolic ratio of a contact form on a closed three-manifold is the quotient of the square of the shortest period of closed Reeb orbits by the contact volume. We show that every co-orientable contact structure on any closed three-manifold is defined by a contact form with arbitrarily large systolic ratio. This shows that the many existing systolic inequalities in Finsler and Riemannian geometry are not purely contact-topological phenomena.

K接触形式的分类及其子拉普拉斯算子的谱不变量

Classification of K-contact forms and spectral invariants of their sub-Laplacians

本文研究了K接触形式的分类及其子拉普拉斯算子的谱不变量，证明了紧致三维流形上存在此类形式的条件，并与Reeb轨道周期相关联。

三维中S¹不变切触形式的收缩不等式

Systolic inequalities for S1-invariant contact forms in dimension three

该研究在切触几何领域，针对欧拉数非零的Seifert丛，证明了适用于所有底层圆周作用不变切触形式的通用收缩不等式，即用切触体积一致上界闭Reeb轨道的最短周期。

Torsion contact forms in three dimensions have two or infinitely many Reeb orbits

We prove that every nondegenerate contact form on a closed connected three-manifold, such that the associated contact structure has torsion first Chern class, has either two or infinitely many simple Reeb orbits. By previous results it follows that under the above assumptions, there are infinitely many simple Reeb orbits if the three-manifold is not the three-sphere or a lens space. We also show that for non-torsion contact structures, every nondegenerate contact form has at least four simple Reeb orbits.

Contact structures with singularities: from local to global

In this article we introduce and analyze in detail singular contact structures, with an emphasis on $b^m$-contact structures, which are tangent to a given smooth hypersurface $Z$ and satisfy certain transversality conditions. These singular contact structures are determined by the kernel of non-smooth differential forms, called $b^m$-contact forms, having an associated critical hypersurface $Z$. We provide several constructions, prove local normal forms, and study the induced structure on the critical hypersurface. The topology of manifolds endowed with such singular contact forms are related …

Redshift and contact forms

It is shown that the redshift between two Cauchy surfaces in a globally hyperbolic spacetime equals the ratio of the associated contact forms on the space of light rays of that spacetime.

Dirac spectral flow on contact three manifolds II: Thurston--Winkelnkemper contact forms

Given an open book decomposition $(Σ,τ)$ of a three manifold $Y$, Thurston and Winkelnkemper [TW] construct a specific contact form $a$ on $Y$. Given a spin-c Dirac operator $D$ on $Y$, the contact form naturally associates a one parameter family of Dirac operators $D_r = D - \frac{ir}{2}\cl(a)$ for $r\geq0$. When $r>>1$, we prove that the spectrum of $D_r = D_0 - \frac{ir}{2}\cl(a)$ within $[-(r^{1/2})/2, (r^{1/2})/2]$ are almost uniformly distributed. With the result in Part I, it implies that the subleading order term of the spectral flow from $D_0$ to $D_r$ is of order $r (\log r)^{\frac{9…

切触形式的强封闭性与作用选择函子

Strong closing property of contact forms and action selecting functors

受三维Reeb流$C^\infty$封闭引理启发，研究提出切触形式的强封闭性概念，通过幺半函子构建其充分判据，并提出辛椭球边界标准切触形式满足该性质的猜想，为切触拓扑研究提供新方向。

Higher systolic inequalities for 3-dimensional contact manifolds

A contact form is called Besse when the associated Reeb flow is periodic. We prove that Besse contact forms on closed connected 3-manifolds are the local maximizers of suitable higher systolic ratios. Our result extends earlier ones for Zoll contact forms, that is, contact forms whose Reeb flow defines a free circle action.