紧星型超曲面上闭特征问题的若干新进展——献给陈化教授 70 寿辰

数学物理学报 ›› 2026, Vol. 46 ›› Issue (2): 493-502.

紧星型超曲面上闭特征问题的若干新进展——献给陈化教授 70 寿辰

刘会()

武汉大学数学与统计学院武汉 430072

收稿日期:2025-12-02 修回日期:2025-12-28 出版日期:2026-04-26 发布日期:2026-04-27
作者简介:刘会，Email：huiliu00031514@whu.edu.cn
基金资助:
国家自然科学基金(12371195)

Some New Progress for the Problems About Closed Characteristics on Compact Star-Shaped Hypersurfaces

Hui Liu()

School of Mathematics and Statistics, Wuhan University, Wuhan 430072

Received:2025-12-02 Revised:2025-12-28 Online:2026-04-26 Published:2026-04-27
Supported by:
NSFC(12371195)

摘要/Abstract

摘要：

该文主要介绍 ${\mathbb{R}}^{2n}$ 中紧星型超曲面上闭特征的多重性猜想和 Hofer-Wysocki-Zehnder 猜想, 以及近期的一些相关研究进展, 并进一步介绍一般切触流形上对应的闭轨道问题, 阐述其中的研究方法, 涉及变分方法和 Morse 理论、动力系统、辛几何等.

关键词: 紧星型超曲面, 闭特征, 多重性, 切触流形

Abstract:

This paper mainly introduces the multiplicity conjecture and Hofer-Wysocki-Zehnder conjecture for closed characteristics on compact star-shaped hypersurfaces in ${\mathbb{R}}^{2n}$, and some recent related progresses on them, furthermore we introduce the corresponding closed orbit problems for contact manifold and explain the research methods therein, which involve Variational Method and Morse Theory、Dynamical System、Symplectic Geometry, and so on.

Key words: compact star-shaped hypersurface, closed characteristic, multiplicity, contact manifold

中图分类号:

O175.23

刘会. 紧星型超曲面上闭特征问题的若干新进展——献给陈化教授 70 寿辰[J]. 数学物理学报, 2026, 46(2): 493-502.

Hui Liu. Some New Progress for the Problems About Closed Characteristics on Compact Star-Shaped Hypersurfaces[J]. Acta mathematica scientia,Series A, 2026, 46(2): 493-502.

参考文献 48

[1]	álvarez Paiva J C. Some problems on Finsler geometry. Handbook of Differential Geometry. 2006, 2: 1--33
[2]	Bangert V. On the existence of closed geodesics on two-spheres. Internat J Math, 1993, 4: 1-10
[3]	Bangert V, Klingenberg W. Homology generated by iterated closed geodesics. Topology. 1983, 22: 379-388
[4]	Berestycki H, Lasry J M, Mancini G, Ruf B. Existence of multiple periodic orbits on starshaped Hamiltonian systems. Comm Pure Appl Math, 1985, 38: 253-289
[5]	Bourgeois F, Cieliebak K, Ekholm T. A note on Reeb dynamics on the tight 3-sphere. J Mod Dyn, 2007, 1: 597-613
[6]	Burns K, Matveev S. Open problems and questions about geodesics. Ergodic Theory Dynam Systems, 2021, 41(3): 641-684
[7]	Cineli E, Ginzburg V, Gürel B. Closed orbits of dynamically convex reeb flows: Towards the HZ- and multiplicity conjectures. arXiv:2410.13093
[8]	Colin V, Dehornoy P, Rechtman A. On the existence of supporting broken book decompositions for contact forms in dimension $3$. Invent Math, 2023, 231: 1489-1539
[9]	Cristofaro-Gardiner D, Hryniewicz U, Hutchings M, Liu H. Contact three-manifolds with exactly two simple Reeb orbits. Geom Topol, 2023, 27: 3801-3831
[10]	Cristofaro-Gardiner D, Hryniewicz U, Hutchings M, Liu H. Proof of Hofer-Wysocki-Zehnder's two or infinity conjecture. 2026. DOI:10.1090/jams/1072
[11]	Cristofaro-Gardiner D, Hutchings M. From one Reeb orbit to two. J Diff Geom, 2016, 102: 25-36
[12]	Cristofaro-Gardiner D, Hutchings M, Pomerleano D. Torsion contact forms in three dimensions have two or infinitely many Reeb orbits. Geom Topol, 2019, 23: 3601-3645
[13]	Duan H, Liu H. Multiplicity and ellipticity of closed characteristics on compact star-shaped hypersurfaces in $\mathbb{R}^{2n}$. Calc Var and PDEs, 2017, 56(3): Art 65
[14]	Duan H, Liu H, Ren W. A dichotomy result for closed characteristics on compact star-shaped hypersurfaces in $\mathbb{R}^{2n}$. Math Z, 2022, 302: 743-757
[15]	Duan H, Long Y, Wang W. The enhanced common index jump theorem for symplectic paths and non-hyperbolic closed geodesics on Finsler manifolds. Calc Var and PDEs, 2016, 55: Art 145
[16]	Duan H, Liu H, Long Y, Wang W. Generalized common index jump theorem with applications to closed characteristics on star-shaped hypersurfaces and beyond. J Funct Anal, 2024, 286(7): Art 110352
[17]	Ekeland I. Une théorie de Morse pour les systémes hamiltoniens convexes. Ann IHP Anal non Linéaire, 1984, 1: 19-78
[18]	Ekeland I. Convexity Methods in Hamiltonian Mechanics. Berlin: Springer-Verlag, 1990
[19]	Ekeland I, Hofer H. Convex Hamiltonian energy surfaces and their periodic trajectories. Comm Math Phys, 1987, 113: 419-469
[20]	Ekeland I, Lassoued L. Multiplicité des trajectoires fermées de systéme hamiltoniens convexes. Ann IHP Anal Nonlinéaire, 1987, 4: 307-335
[21]	Franks J. Geodesics on $S^2$ and periodic points of annulus homeomorphisms. Invent Math, 1992, 108: 403-418
[22]	Ginzburg V, Goren Y. Iterated index and the mean Euler characteristic. J Topol Anal, 2015, 7: 453-481
[23]	Ginzburg V, Gürel B Z. Lusternik-Schnirelmann theory and closed Reeb orbits. Math Z, 2020, 295: 515-582
[24]	Ginzburg V, Hein D, Hryniewicz U, Macarini L. Closed Reeb orbits on the sphere and symplectically degenerate maxima. Acta Math Vietnam, 2013, 38: 55-78
[25]	Grayson M A. Shortening embedded curves. Ann of Math, 1989, 129(1): 71-111
[26]	Gürel B Z. Perfect Reeb flows and action-index relations. Geom Dedicata, 2015, 174: 105-120
[27]	Hu X, Long Y. Closed characteristics on non-degenerate star-shaped hypersurfaces in $\mathbb{R}^{2n}$. Sci China Ser A, 2002, 45: 1038-1052
[28]	Hofer H, Wysocki K, Zehnder E. The dynamics on three-dimensional strictly convex energy surfaces. Ann of Math, 1998, 148: 197-289
[29]	Hofer H, Wysocki K, Zehnder E. Finite energy foliations of tight three-spheres and Hamiltonian dynamics. Ann Math, 2003, 157: 125-255
[30]	Hutchings M, Taubes C H. The Weinstein conjecture for stable Hamiltonian structures. Geom Topol, 2009, 13: 901-941
[31]	Liu C, Long Y, Zhu C. Multiplicity of closed characteristics on symmetric convex hypersurfaces in $\mathbb{R}^{2n}$. Math Ann, 2002, 323: 201-215
[32]	Liu H, Long Y. The existence of two closed characteristics on every compact star-shaped hypersurface in $\mathbb{ R}^4$. Acta Math Sin, 2016, 32: 40-53
[33]	Liu H, Long Y. Resonance identities and stability of symmetric closed characteristics on symmetric compact star-shaped hypersurfaces. Calc Var and PDEs, 2015, 54: 3753-3787
[34]	Liu H, Long Y, Wang W. Resonance identities for closed charactersitics on compact star-shaped hypersurfaces in $\mathbb{R}^{2n}$. J Funct Anal, 2014, 266: 5598-5638
[35]	Long Y. Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics. Adv Math, 2000, 154: 76-131
[36]	Long Y. Index Theory for Symplectic Paths with Applications. Basel: Birkhäuser, 2002
[37]	Long Y. Multiplicity and stability of closed geodesics on Finsler $2$-spheres. J Eur Math Soc, 2006, 8(2): 341-353
[38]	Long Y, Zhu C. Closed characteristics on compact convex hypersurfaces in $\mathbb{R}^{2n}$. Ann Math, 2002, 155: 317-368
[39]	Lusternik L, Schnirelmann L. Sur le probléme de trois g'eod'esiques ferm'ees sur les surfaces de genre 0. C R Acad Sci Paris, 1929, 189: 269-271
[40]	Rabinowitz P. Periodic solutions of Hamiltonian systems. Comm Pure Appl Math, 1978, 31: 157-184
[41]	Senior D B, Hryniewicz U L, Salomão P A S. On the relation between action and linking. J Mod Dyn, 2021, 17: 319-336
[42]	Szulkin A. Morse theory and existence of periodic solutions of convex Hamiltonian systems. Bull Soc Math France, 1988, 116: 171-197
[43]	Taubes C H. The Seiberg-Witten equations and the Weinstein conjecture. Geom Topol, 2007, 11: 2117-2202
[44]	Viterbo C. Equivariant Morse theory for starshaped Hamiltonian systems. Trans Amer Math Soc, 1989, 311: 621-655
[45]	Wang W. Existence of closed characteristics on compact convex hypersurfaces in $\mathbb{R}^{2n}$. Calc Var and PDEs, 2016, 55(1): 1-25
[46]	Wang W. Closed characteristics on compact convex hypersurfaces in $\mathbb{R}^8$. Adv Math, 2016, 297: 93-148
[47]	Wang W, Hu X, Long Y. Resonance identity, stability and multiplicity of closed characteristics on compact convex hypersurfaces. Duke Math J, 2007, 139(3): 411-462
[48]	Weinstein A. Periodic orbits for convex Hamiltonian systems. Ann Math, 1978, 108(3): 507-518