The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form

Hui Liu; Yiming Long; Yuming Xiao

doi:10.3934/dcds.2018165

Article Contents

2018, Volume 38, Issue 8: 3803-3829. Doi: 10.3934/dcds.2018165

This issue Previous Article Impulsive control of conservative periodic equations and systems: Variational approach Next Article Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros

The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form

a.
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, China
b.
Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China
c.
School of Mathematics, Sichuan University, Chengdu 610064, China
d.
Beijing Advanced Innovation Center for Imaging Technology, Capital Normal University, Beijing 100048, China

1 Partially supported by NSFC (Nos. 11401555, 11771341), Anhui Provincial Natural Science Foundation (No. 1608085QA01)
2 Partially supported by NSFC (Nos. 11131004, 11671215 and 11790271), MCME and LPMC of MOE of China, Nankai University and BAICIT of Capital Normal University
3 Corresponding author: Yuming Xiao. Supported by the Sichuan Science and Technology Program (No. 2018JY0140)

Received: June 30, 2017

Revised: January 31, 2018

Published: May 2018

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Let $M = S^n/ Γ$ and $h$ be a nontrivial element of finite order $p$ in $π_1(M)$, where the integer $n≥2$, $Γ$ is a finite group which acts freely and isometrically on the $n$-sphere and therefore $M$ is diffeomorphic to a compact space form. In this paper, we establish first the resonance identity for non-contractible homologically visible minimal closed geodesics of the class $[h]$ on every Finsler compact space form $(M, F)$ when there exist only finitely many distinct non-contractible closed geodesics of the class $[h]$ on $(M, F)$. Then as an application of this resonance identity, we prove the existence of at least two distinct non-contractible closed geodesics of the class $[h]$ on $(M, F)$ with a bumpy Finsler metric, which improves a result of Taimanov in [39] by removing some additional conditions. Also our results extend the resonance identity and multiplicity results on $\mathbb{R}P^n$ in [25] to general compact space forms.

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Mathematics Subject Classification: 53C22, 58E05, 58E10.

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References

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