Fredholm theory for pseudoholomorphic curves with brake symmetry

Zhou, Beijia; Zhu, Chaofeng

doi:10.1007/s11464-021-0935-4

Cited by 3 publications

(3 citation statements)

References 22 publications

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“…Instead, it can be computed using the linear gluing operation and the Riemann-Roch theorem, see [53]. We also refer to [58] in which Zhou studies moduli spaces of real holomorphic curves.…”

Section: 1mentioning

confidence: 99%

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

Kim¹

2020

Preprint

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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.

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“…Instead, it can be computed using the linear gluing operation and the Riemann-Roch theorem, see [53]. We also refer to [58] in which Zhou studies moduli spaces of real holomorphic curves.…”

Section: 1mentioning

confidence: 99%

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

Kim¹

2020

Preprint

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“…Proof. If k " 1, then u is a plane and by [17,Main Theorem] ind R puq " ´1 2 `µ1 pβq ě 1 by the dynamically convex condition. with one positive puncture, which converges to the brake orbit β.…”

Section: Appendix a Applicationmentioning

confidence: 99%

“…In [17] the author and C.Zhu calculated that the virtual dimension of the moduli space of Real pseudoholomorphic curves, which is given by…”

mentioning

confidence: 99%

Iteration Formulae for Brake Orbit and Index Inequalities for Real Pseudoholomorphic Curves

Zhou¹

2020

Preprint

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

Kim

2022

Trans. Amer. Math. Soc.

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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

Fredholm theory for pseudoholomorphic curves with brake symmetry

Cited by 3 publications

References 22 publications

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

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

Iteration Formulae for Brake Orbit and Index Inequalities for Real Pseudoholomorphic Curves

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

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