On<i>J</i>-holomorphic curves in almost complex manifolds with asymptotically cylindrical ends

Bao, Erkao

doi:10.2140/pjm.2015.278.291

Cited by 5 publications

(17 citation statements)

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“…Under these hypotheses, we establish the following C ∞ uniform exponential convergence of w to a closed Reeb orbit z of period T = T . This result was already known in [HWZ3,Bou,Ba] in the context of pseudo-holomorphic curves u = (w, a) in symplectizations. However we emphasize that our proof presented here, which uses the three-interval framework, is different from the ones [HWZ3,Bou,Ba] even in the symplectization case.…”

Section: Introductionsupporting

confidence: 60%

Analysis of Contact Cauchy–riemann Maps ii: Canonical Neighborhoods and Exponential Convergence for the Morse–bott Case

Wang²

2017

Nagoya Math. J.

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This is a sequel to the papers [OW1], [OW2]. In [OW1], the authors introduced a canonical affine connection on M associated to the contact triad (M, λ, J). In [OW2], they used the connection to establish a priori W k,pcoercive estimates for maps w :Σ → M satisfying ∂ π w = 0, d(w * λ • j) = 0 without involving symplectization. We call such a pair (w, j) a contact instanton. In this paper, we first prove a canonical neighborhood theorem of the locus Q foliated by closed Reeb orbits of a Morse-Bott contact form. Then using a general framework of the three-interval method, we establish exponential decay estimates for contact instantons (w, j) of the triad (M, λ, J), with λ a Morse-Bott contact form and J a CR-almost complex structure adapted to Q, under the condition that the asymptotic charge of (w, j) at the associated puncture vanishes. We also apply the three-interval method to the symplectization case and provide an alternative approach via tensorial calculations to exponential decay estimates in the Morse-Bott case for the pseudoholomorphic curves on the symplectization of contact manifolds. This was previously established by Bourgeois [Bou] (resp. by Bao [Ba]), by using special coordinates, for the cylindrical (resp. for the asymptotically cylindrical) ends. The exponential decay result for the Morse-Bott case is an essential ingredient in the set-up of the moduli space of pseudoholomorphic curves which plays a central role in contact homology and symplectic field theory (SFT).

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Section: Introductionsupporting

confidence: 60%

Analysis of Contact Cauchy–riemann Maps ii: Canonical Neighborhoods and Exponential Convergence for the Morse–bott Case

Wang²

2017

Nagoya Math. J.

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“…Proof. This follows immediately from the proof of Theorem 2 in [2]. Namely, from the assumption, we could derive that the convergence in (7) and (8) is exponentially fast.…”

Section: Definitionmentioning

confidence: 69%

“…In this paper, we assume that J is Morse-Bott. The Morse-Bott condition is the condition assumed in [2] to guarantee Theorem 6, Lemma 7 and Theorem 9. For the application in section 4, it is easy to check that this requirement is satisfied.…”

Section: Definitionmentioning

confidence: 99%

“…The following theorem is one of the most important theorems in [10,11,5,4] for the case when J is cylindrical, and it is proved in the asymptotically cylindrical setting in [2].…”

Section: Definitionmentioning

confidence: 99%

“…Notice that this definition is equivalent to the definition given in [2]. In [2] for J being asymptotically cylindrical, besides (ACC1)-(ACC5) we require that there exists a 2-form ω on W − such that…”

Section: Introductionmentioning

confidence: 99%

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On Hofer energy of $J$-holomorphic curves for asymptotically cylindrical $J$

Bao

2016

Journal of Symplectic Geometry

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In this paper, we provide a bound for the generalized Hofer energy of punctured J-holomorphic curves in almost complex manifolds with asymptotically cylindrical ends. As an application, we prove a version of Gromov's Monotonicity Theorem with multiplicity. Namely, for a closed symplectic manifold (M, ω ′ ) with a compatible almost complex structure J and a ball B in M, there exists a constant > 0, such that any J-holomorphic curveũ passing through the center of B for k times (counted with multiplicity) with boundary mapped to ∂B has symplectic area´ũ −1 (B)ũ * ω ′ > k , where the constant depends only on (M, ω ′ , J) and the radius of B. As a consequence, the number of times that any closed J-holomorphic curve in M passes through a point is bounded by a constant depending only on (M, ω ′ , J) 1 and the symplectic area ofũ. Here J is any ω ′ −compatible smooth almost complex structure on M . In particular, we do not require J to be integrable.1 Following the notation in [5] we save ω for something else.1 energy bound can be obtained by specifying the behavior the J-holomorphic curves at infinity and bounding their symplectic areas (see [10,5]). In [2] the notion of Hofer energy and Property (A) are further generalized to include Jholomorphic curves in "almost complex manifolds with asymptotically cylindrical ends". Here "asymptotically cylindrical" means that the difference between the almost complex structure J and a translation invariant one is exponentially small. In this paper, we prove Property (B) in this setting. Property (A) and property (B) together imply the expected useful compactness results in Symplectic Field Theory.One of the main advantages of this generalization is that the asymptotically cylindrical J arises naturally. As an application, we prove a version of Gromov's Monotonicity Theorem with multiplicity 2 , namely for a closed symplectic manifold (M, ω ′ ) with a compatible almost complex structure J and a ball B in M, there exists a constant > 0, such that any J-holomorphic curveũ passing through the center of B k times (counted with multiplicity) with the boundary mapped to ∂B has symplectic area´ũ −1 (B)ũ * ω ′ > k , where the constant depends only on (M, ω ′ , J) and the radius of B. The inequality k < 1 ´ũ −1 (B)ũ * ω ′ is closely related to a question asked in [6], where they study J-holomorphic curves with boundaries lying inside two clean intersecting Lagrangian submanifolds, and prove that the number of "boundary switches" at the intersecting loci is uniformly bounded by the Hofer Energy. Their proof in an essential way relies on the additional requirement that the almost complex structure J is integrable near the intersecting loci. They ask to what extent their results are still true without assuming the integrability of J. In this paper, we provide a simple proof for the closed version of their result for arbitrary J. Namely, the J-holomorphic curves we consider in this paper have no boundaries. In this case, "boundary switches" just means that the J-holomorphic curve passes a fixed...

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Analysis of contact Cauchy-Riemann maps II: canonical neighborhoods and exponential convergence for the Morse-Bott case

Oh¹,

Wang²

2013

Preprint

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OnJ-holomorphic curves in almost complex manifolds with asymptotically cylindrical ends

Cited by 5 publications

References 10 publications

Analysis of Contact Cauchy–riemann Maps ii: Canonical Neighborhoods and Exponential Convergence for the Morse–bott Case

Analysis of Contact Cauchy–riemann Maps ii: Canonical Neighborhoods and Exponential Convergence for the Morse–bott Case

On Hofer energy of $J$-holomorphic curves for asymptotically cylindrical $J$

Analysis of contact Cauchy-Riemann maps II: canonical neighborhoods and exponential convergence for the Morse-Bott case

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