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Elliptic and non-hyperbolic closed characteristics on compact convex P-cyclic symmetric hypersurfaces in \(\mathbf{R}^{2n}\)

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Abstract

Let \(\Sigma \) be a compact convex hypersurface in \(\mathbf{R}^{2n}\) which is P-cyclic symmetric, i.e., \(x\in \Sigma \) implies \(Px\in \Sigma \) with P being a \(2n\times 2n\) symplectic orthogonal matrix and \(P^k=I_{2n}\), where \(n, k\ge 2\), \(ker(P-I_{2n})=0\). In this paper, we first generalize Ekeland index theory for periodic solutions of convex Hamiltonian system to a index theory with P boundary value condition and study its relationship with Maslov P-index theory, then we use index theory to prove the existence of elliptic and non-hyperbolic closed characteristics on compact convex P-cyclic symmetric hypersurfaces in \(\mathbf{R}^{2n}\) for a broad class of symplectic orthogonal matrix P.

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Acknowledgements

The authors would like to sincerely thank the anonymous referee for his/her careful reading of the manuscript and valuable comments.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, Hubei, People’s Republic of China

    Hui Liu & Chongzhi Wang

  2. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, People’s Republic of China

    Duanzhi Zhang

Authors
  1. Hui Liu
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  2. Chongzhi Wang
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  3. Duanzhi Zhang
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Correspondence to Duanzhi Zhang.

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Communicated by P. Rabinowitz.

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H. Liu: Partially supported by NSFC (Nos. 11401555, 11771341).

C. Wang: Partially supported by NSFC (No. 11771341).

D. Zhang: Partially supported by NSFC (Nos. 11790271, 11771341, 11422103) and LPMC of Nankai University.

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Liu, H., Wang, C. & Zhang, D. Elliptic and non-hyperbolic closed characteristics on compact convex P-cyclic symmetric hypersurfaces in \(\mathbf{R}^{2n}\). Calc. Var. 59, 24 (2020). https://doi.org/10.1007/s00526-019-1681-2

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