3D convex contact forms and the Ruelle invariant

Abstract

A. We give a construction of embedded contact homology (ECH) for a contact 3-manifold with convex sutured boundary and a pair of Legendrians Λ `and Λ ´contained in B satisfying an exactness condition. The chain complex is generated by certain configurations of closed Reeb orbits of and Reeb chords of Λ `to Λ ´. The main ingredients include ‚ a general Legendrian adjunction formula for curves in R ˆ with boundary on R ˆΛ.‚ a relative writhe bound for curves in contact 3-manifolds asymptotic to Reeb chords.‚ a Legendrian ECH index and an accompanying ECH index inequality.The (action filtered) Legendrian ECH of any pair p , Λq of a closed contact 3-manifold and a Legendrian link Λ can also be defined using this machinery after passing to a sutured link complement. This work builds on ideas present in Colin-Ghiggini-Honda's proof of the equivalence of Heegaard-Floer homology and ECH.The independence of our construction of choices of almost complex structure and contact form should require a new flavor of monopole Floer homology. It is beyond the scope of this paper.

Remarks on symplectic capacities of p-products

“…(iv) Another notable class of domains in R 2n , which includes convex domains, is the class DC 2n of dynamically convex domains (see [25]). Contrary to previous assumptions, and in response to a longstanding open question, it has recently been demonstrated (first in R 4 [12], and then in every dimension [13]) that there exist dynamically convex domains in R 2n that are not symplectically convex. The maximal systolic constant for this class Sys(DC 2n ) can be defined in a similar manner as above, where one replaces the Ekeland-Hofer-Zehnder capacity with the minimal action among closed characteritcs on the boundary.…”

On dynamically convex contact manifolds and filtered symplectic homology

“…(2) By the work of Hofer-Wysocki-Zehnder [23] (see also [16,Theorem 4.10]), geometrically convex domains in ℝ 2𝑛 are dynamically convex, but the converse is not true as discovered in [10,14].…”

“…It requires that $$\operatorname{CZ}(\gamma) > 3-n$$ for any contractible Reeb orbits $$\gamma$$, which morally means that the contact homology is trivial in negative degrees. This notion is also referred to as index‐positivity of contact manifolds as in [12, section 9.5]. (2)By the work of Hofer–Wysocki–Zehnder [23] (see also [16, Theorem 4.10]), geometrically convex domains in $${\mathbb {R}}^{2n}$$ are dynamically convex, but the converse is not true as discovered in [10, 14]. …”