Duke Mathematical Journal

Resonance identity, stability, and multiplicity of closed characteristics on compact convex hypersurfaces

Wei Wang, Xijun Hu, and Yiming Long

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Abstract

There is a long-standing conjecture in Hamiltonian analysis which claims that there exist at least n geometrically distinct closed characteristics on every compact convex hypersurface in R2n with n2. Besides many partial results, this conjecture has been completely solved only for n=2. In this article, we give a confirmed answer to this conjecture for n=3. In order to prove this result, we establish first a new resonance identity for closed characteristics on every compact convex hypersurface Σ in R2n when the number of geometrically distinct closed characteristics on Σ is finite. Then, using this identity and earlier techniques of the index iteration theory, we prove the mentioned multiplicity result for R6. If there are exactly two geometrically distinct closed characteristics on a compact convex hypersuface in R4, we prove that both of them must be irrationally elliptic

Article information

Source
Duke Math. J., Volume 139, Number 3 (2007), 411-462.

Dates
First available in Project Euclid: 24 August 2007

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1187916266

Digital Object Identifier
doi:10.1215/S0012-7094-07-13931-0

Mathematical Reviews number (MathSciNet)
MR2350849

Zentralblatt MATH identifier
1139.58007

Subjects
Primary: 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)
Secondary: 37J45: Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods 34C25: Periodic solutions

Citation

Wang, Wei; Hu, Xijun; Long, Yiming. Resonance identity, stability, and multiplicity of closed characteristics on compact convex hypersurfaces. Duke Math. J. 139 (2007), no. 3, 411--462. doi:10.1215/S0012-7094-07-13931-0. https://projecteuclid.org/euclid.dmj/1187916266


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