Closed Characteristics on Compact Convex Hypersurfaces in R 2n

Abstract

For any given compact C 2 hypersurface Σ in R 2n bounding a strictly convex set with nonempty interior, in this paper an invariant ̺ n (Σ) is defined and satisfies ̺ n (Σ) ≥ [n/2] + 1, where [a] denotes the greatest integer which is not greater than a ∈ R. The following results are proved in this paper. There always exist at least ̺ n (Σ) geometrically distinct closed characteristics on Σ.

Symmetric Closed Characteristics on Symmetric Compact Convex Hypersurfaces in $$\mathbf{R}^8$$ R 8

“…Note that #J ( ) = 4, we denote by {(τ j , y j )} 1≤ j≤4 all the geometrically distinct closed characteristics on , and by γ j ≡ γ y j the associated symplectic path of (τ j , y j ) on for 1 ≤ j ≤ 4. Then by Lemma 1.3 of [10] (cf. Lemma 15.2.4 of [9]), there exist P j ∈ Sp (8) and M j ∈ Sp(6) such that …”

“…For the existence and multiplicity of geometrically distinct closed characteristics on compact convex hypersurfaces in R 2n , we refer to [3,4,6,7,[9][10][11][12]16,18,19] and references therein. Especially, In [7] of 2002, Liu, Long and Zhu proved that #J ( ) ≥ n, ∀ ∈ SH(2n).…”

“…2 to the Maslov-type index. As Definition 1.1 of [10], we define the following: In the rest of this paper, we fix a ∈ SH(2n). We have the following property, cf.…”

“…Let v ∈ R h be the vector given by (3.11) and A(v) be the set given by Theorem 4.2 of [10] associated to v . For any a = (a 1 , .…”

Seifert Conjecture in the Even Convex Case

“…By Proposition 6.1 of [25] and Lemma 2.8 and Definition 2.5 of [23], we give the following Definition 2.3. For any continuous path γ ∈ P τ (2n), we define the following Maslov-type indices:…”