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Non-hyperbolic closed characteristics on non-degenerate star-shaped hypersurfaces in ℝ2n

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Abstract

In this paper, we prove that for every index perfect non-degenerate compact star-shaped hypersurface Σ ⊂ R2n, there exist at least n non-hyperbolic closed characteristics with even Maslovtype indices on Σ when n is even. When n is odd, there exist at least n closed characteristics with odd Maslov-type indices on Σ and at least (n−1) of them are non-hyperbolic. Here we call a compact star-shaped hypersurface Σ ⊂ R2n index perfect if it carries only finitely many geometrically distinct prime closed characteristics, and every prime closed characteristic (τ, y) on Σ possesses positive mean index and whose Maslov-type index i(y,m) of its m-th iterate satisfies i(y,m) ≠ −1 when n is even, and i(y,m) ∉ {−2,−1, 0} when n is odd for all m ∈ N.

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Acknowledgements

The authors would like to sincerely thank the anonymous referees for their careful reading of the manuscript and valuable comments.

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Authors and Affiliations

  1. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, P. R. China

    Hua Gui Duan

  2. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, P. R. China

    Hui Liu

  3. Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, 300071, P. R. China

    Yi Ming Long

  4. School of Mathematical Sciences and LMAM, Peking University, Beijing, 100871, P. R. China

    Wei Wang

Authors
  1. Hua Gui Duan
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  2. Hui Liu
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  3. Yi Ming Long
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  4. Wei Wang
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Correspondence to Hua Gui Duan.

Additional information

The first author is supported by NSFC (Grant Nos. 11671215, 11131004 and 11471169) and LPMC of MOE of China; the second author is supported by NSFC (Grant No. 11401555) and Anhui Provincial Natural Science Foundation (Grant No. 1608085QA01); the third author is supported by NSFC (Nos. 11131004 and 11671215), MCME, LPMC of MOE of China, Nankai University and BAICIT of Capital Normal University; the fourth author is supported by NSFC (Grant Nos. 11222105 and 11431001)

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Duan, H.G., Liu, H., Long, Y.M. et al. Non-hyperbolic closed characteristics on non-degenerate star-shaped hypersurfaces in ℝ2n . Acta. Math. Sin.-English Ser. 34, 1–18 (2018). https://doi.org/10.1007/s10114-016-6019-9

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  • DOI: https://doi.org/10.1007/s10114-016-6019-9

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