Finite-energy pseudoholomorphic planes with multiple asymptotic limits

Siefring, Richard

doi:10.1007/s00208-016-1478-y

Cited by 14 publications

(12 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A fairly standard argument [, § 7] yields the following correspondence between cylinders

u \in {scriptM}^{J_{ε}} false(γ_{p}^{k}, γ_{q}^{k} false)

and gradient flow lines between the critical points

p

and

q

. Theorem then implies that these cylinders correspond to Floer trajectories between the 1‐periodic orbits of

H^{ε}

, which are the constant loops at the critical points

p

and

q

.…”

Section: Fun With Filtrationsmentioning

confidence: 99%

“…Proof When

H

is small but not identically zero, the projection of the curve to

normalΣ

is no longer holomorphic as in the proof of [, Theorem 3.1]. However, one can appeal to the asymptotic behavior of holomorphic curves, along with intersection theory and the relationship between the Conley–Zehnder indices and extremal winding numbers as in the proof of [, Proposition 4.11, Theorem C.10].…”

Section: Fun With Filtrationsmentioning

confidence: 99%

See 1 more Smart Citation

Automatic transversality in contact homology II: filtrations and computations

Nelson

2019

Proc. London Math. Soc.

View full text Add to dashboard Cite

This paper is the sequel to the previous paper [Nelson, Abh. Math. Semin. Univ. Hambg. 85 (2015) , 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. Given a contact form which is dynamically separated up to large action, we demonstrate a filtration by action on the chain complex and show how to obtain the desired cylindrical contact homology by taking direct limits. We give a direct proof of invariance of cylindrical contact homology within the class of dynamically separated contact forms, and elucidate the independence of the filtered cylindrical contact homology with respect to the choice of the dynamically separated contact form and almost complex structure. We also show that these regularity results are compatible with geometric methods of computing cylindrical contact homology of prequantization bundles, proving a conjecture of Eliashberg [Symplectic field theory and its applications, International Congress of Mathematicians I (European Mathematical Society, Zürich, 2007) 217-246] in dimension three.Definition 1.6. Let (M, λ) be a contact 3-manifold with c 1 (ker λ) = 0 such that all the Reeb orbits of R λ are contractible. Then λ is said to be dynamically separated whenever the following conditions hold.(I) If γ is a closed simple Reeb orbit then 3 μ CZ (γ) 5. (II) If γ k is the k-fold cover of a simple orbit γ then μ CZ (γ k ) = μ CZ (γ k−1 ) + 4. * (Y, λ, J).Corollary 1.18. CH EGH * is an invariant of closed contact manifolds (Y, ξ) for which there exists a pair (λ, J) where λ is a nondegenerate hypertight contact form with ker(λ) = ξ, and J is an admissible λ-compatible almost complex structure.Again, we will upgrade these results to hold for dynamically convex contact forms in dimension three in a forthcoming paper by Hutchings and Nelson. In contrast to [30] and a forthcoming paper by Hutchings and Nelson, this paper is concerned with the more restricted class of dynamically separated contact forms which allows us to directly obtain regularity for S 1 -independent pseudoholomorphic cylinders in cobordisms. Filtered cylindrical contact homologyThe action of a Reeb orbit γ is given by A(γ) := γ λ.. Since J is a λ-compatible almost complex structure on the symplectization it follows [40, Lemma 2.18] that the cylindrical contact homology differential(s) decreases the action, for example, ifThus, given any real number L it makes sense to define the filtered cylindrical contact homology, denoted by CH EGH,L * (M, λ, J), to be the homology of the subcomplex C EGH,L * (M, λ, J) of the chain complex spanned by generators of action less than L. The invariance of these filtered cylindrical contact homology groups is more subtle than in the unfiltered case, as they typically depend ...

show abstract

“…A fairly standard argument [, § 7] yields the following correspondence between cylinders

u \in {scriptM}^{J_{ε}} false(γ_{p}^{k}, γ_{q}^{k} false)

and gradient flow lines between the critical points

p

and

q

. Theorem then implies that these cylinders correspond to Floer trajectories between the 1‐periodic orbits of

H^{ε}

, which are the constant loops at the critical points

p

and

q

.…”

Section: Fun With Filtrationsmentioning

confidence: 99%

“…Proof When

H

is small but not identically zero, the projection of the curve to

normalΣ

Section: Fun With Filtrationsmentioning

confidence: 99%

Automatic transversality in contact homology II: filtrations and computations

Nelson

2019

Proc. London Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Furthermore, if the curve converges exponentially fast, at a negative (respectively, positive) puncture the rate of convergence is governed by the smallest positive (largest negative) eigenvalue of the corresponding asymptotic operator [68]. [69]. Similarly, a gradient trajectory of a smooth function that is not Morse-Bott can fail to converge to a single critical point, and may contain sequences converging to different critical points.…”

Section: Morse-bott Symplectic Homologymentioning

confidence: 99%

“…Remark There are several constructions of examples where the limit is not unique in the degenerate case. In particular, Siefring has carefully constructed such examples in the case of pseudoholomorphic curves in symplectizations . Similarly, a gradient trajectory of a smooth function that is not Morse–Bott can fail to converge to a single critical point, and may contain sequences converging to different critical points.…”

Section: Floer Moduli Spaces Before and After Splittingmentioning

confidence: 99%

Symplectic homology of complements of smooth divisors

Diogo

Lisi

2019

Journal of Topology

View full text Add to dashboard Cite

If (X, ω) is a closed symplectic manifold, and Σ is a smooth symplectic submanifold Poincaré dual to a positive multiple of ω, then X \ Σ can be completed to a Liouville manifold (W, dλ).Under monotonicity assumptions on X and on Σ, we construct a chain complex whose homology computes the symplectic homology of W . We show that the differential is given in terms of Morse contributions, Gromov-Witten invariants of X relative to Σ and Gromov-Witten invariants of Σ.We use a Morse-Bott model for symplectic homology. Our proof involves comparing Floer cylinders with punctures to pseudoholomorphic curves in the symplectization of the unit normal bundle to Σ. Contents

show abstract

“…The last statement of the preceding theorem tells us that if an asymptotic limit P of u is non-degenerate, then Ω consists of a single element P , up to reparametrization. In [55], Siefring provides explicit examples of finite energy planes with the image of the ω-limit set being diffeomorphic to the two-torus.…”

Section: Pseudoholomorphic Curves In Symplectizationsmentioning

confidence: 99%

Symmetric periodic orbits and invariant disk-like global surfaces of section on the three-sphere

Kim¹

2020

Preprint

View full text Add to dashboard Cite

We study Reeb dynamics on the three-sphere equipped with a tight contact form and an anti-contact involution. We prove the existence of a symmetric periodic orbit and provide necessary and sufficient conditions for it to bound an invariant disk-like global surface of section. We also study the same questions under the presence of additional symmetry and obtain similar results in this case. The proofs make use of pseudoholomorphic curves in symplectizations. As applications, we study Birkhoff's conjecture on disk-like global surfaces of section in the planar circular restricted three-body problem and the existence of symmetric closed Finsler geodesics on the two-sphere. We also present applications to some classical Hamiltonian systems.

show abstract

Finite-energy pseudoholomorphic planes with multiple asymptotic limits

Cited by 14 publications

References 15 publications

Automatic transversality in contact homology II: filtrations and computations

Automatic transversality in contact homology II: filtrations and computations

Symplectic homology of complements of smooth divisors

Symmetric periodic orbits and invariant disk-like global surfaces of section on the three-sphere

Contact Info

Product

Resources

About