Symmetric Closed Characteristics on Symmetric Compact Convex Hypersurfaces in $$\mathbf{R}^8$$
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Symmetric Closed Characteristics on Symmetric Compact Convex Hypersurfaces in R8 Hui Liu · Yiming Long · Wei Wang · Ping’an Zhang
Received: 7 December 2014 / Accepted: 11 February 2015 / Published online: 13 March 2015 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag Berlin Heidelberg 2015
Abstract Let be a C 3 compact symmetric convex hypersurface in R8 . We prove that when carries exactly four geometrically distinct closed characteristics, then all of them must be symmetric. Due to the example of weakly non-resonant ellipsoids, our result is sharp. Keywords Compact convex hypersurfaces · Symmetric closed characteristics · Hamiltonian systems · Morse theory · Index iteration theory Mathematics Subject Classification
58E05 · 37J45 · 37C75
H. Liu Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, Anhui, People’s Republic of China e-mail: [email protected] Y. Long (B) Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, People’s Republic of China e-mail: [email protected] W. Wang Key Laboratory of Pure and Applied Mathematics, School of Mathematical Science, Peking University, Beijing 100871, People’s Republic of China e-mail: [email protected];[email protected] P. Zhang School of Mathematics and Statistics, Xidian University, Xi’an 710071, Shaanxi, People’s Republic of China e-mail: [email protected]
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1 Introduction and Main Results In this paper, let be a fixed C 3 compact convex hypersurface in R2n , i.e., is the boundary of a compact and strictly convex region U in R2n . We denote the set of all such hypersurfaces by H(2n). Without loss of generality, we suppose U contains the origin. We denote the set of all compact convex hypersurfaces which are symmetric with respect to the origin by SH(2n), i.e., = − for ∈ SH(2n). We consider closed characteristics (τ, y) on , which are solutions of the following problem
y˙ = J N (y), y(τ ) = y(0),
(1.1)
0 −In , In is the identity matrix in Rn , τ > 0 and N (y) is the where J = In 0 outward normal vector of at y normalized by the condition N (y) · y = 1. Here a · b denotes the standard inner product of a, b ∈ R2n . A closed characteristic (τ, y) is prime if τ is the minimal period of y. Two closed characteristics (τ, y) and (σ, z) are geometrically distinct if y(R) = z(R). We denote by J () and J() the set of all closed characteristics (τ, y) on with τ being the minimal period of y and the set of all geometrically distinct ones, respectively. Note that J () = {θ · y | θ ∈ S 1 , y is prime}, while J() = J ()/S 1 , where the natural S 1 -action is defined by θ · y(t) = y(t + τ θ ), ∀θ ∈ S 1 , t ∈ R. A closed characteristic (τ, y) on ∈ SH(2n) is symmetric if y(R) = −y(R). For the existence and multiplicity of geometrically distinct closed characteristics on compact convex hypersurfaces in R2n , we re
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