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15 September 2007 Resonance identity, stability, and multiplicity of closed characteristics on compact convex hypersurfaces
Wei Wang, Xijun Hu, Yiming Long
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Duke Math. J. 139(3): 411-462 (15 September 2007). DOI: 10.1215/S0012-7094-07-13931-0

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

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Wei Wang. Xijun Hu. Yiming Long. "Resonance identity, stability, and multiplicity of closed characteristics on compact convex hypersurfaces." Duke Math. J. 139 (3) 411 - 462, 15 September 2007. https://doi.org/10.1215/S0012-7094-07-13931-0

Information

Published: 15 September 2007
First available in Project Euclid: 24 August 2007

zbMATH: 1139.58007
MathSciNet: MR2350849
Digital Object Identifier: 10.1215/S0012-7094-07-13931-0

Subjects:
Primary: 58E05
Secondary: 34C25, 37J45

Rights: Copyright © 2007 Duke University Press

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Vol.139 • No. 3 • 15 September 2007
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