Fredholm Theory for Pseudoholomorphic Curves with Brake Symmetry

Zhou, Beijia; Zhu, Chaofeng

doi:10.48550/arxiv.2011.07598

2020

DOI: 10.48550/arxiv.2011.07598

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Fredholm Theory for Pseudoholomorphic Curves with Brake Symmetry

Beijia Zhou¹,

Chaofeng Zhu²

Abstract: In this paper we study pseudoholomorphic curves with brake symmetry in symplectization of a closed contact manifold. We introduce the pseudoholomorphic curve with brake symmetry and the corresponding moduli space. Then we get the virtual dimension of the moduli space.

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2022

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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.

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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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show abstract

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Fredholm Theory for Pseudoholomorphic Curves with Brake Symmetry

Abstract: In this paper we study pseudoholomorphic curves with brake symmetry in symplectization of a closed contact manifold. We introduce the pseudoholomorphic curve with brake symmetry and the corresponding moduli space. Then we get the virtual dimension of the moduli space.

Cited by 1 publication

References 21 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

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