Precise Iteration Formulae of the Maslov-type Index Theory and Ellipticity of Closed Characteristics

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

“…Motivated by these results and conjectures, we confirm the symmetry conjecture for the case n = 4 when #J ( ) = 4: Theorem 1.1 For every ∈ SH (8) satisfying #J ( ) = 4, all the closed characteristics on are symmetric.…”

“…In Sect. 3, we prove the main result using equivariant Morse theory, index iteration theory developed by Long and his coworkers, especially the common index jump theorem of Long and Zhu, and a commutative property for closed characteristics in the common index jump intervals discovered by Wang in [16] which played a crucial role in his proof of the existence of four geometrically distinct closed characteristics on every compact convex hypersurface in R 8 . Note that this commutative property was also used in [17].…”

“…In [14] of 2009, Wang proved that if there exist exactly n closed characteristics on some ∈ SH(2n), then all of them are symmetric when n = 2, 3. Recently, Wang [16] proved that #J ( ) ≥ 4 for every ∈ H (8). But in both [7,16] it is left open on whether the closed characteristics found are symmetric.…”

“…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 …”

Selected new aspects of the calculus of variations in the large

“…In [L-Z], they proved that any such hypersurface carries at least [n/2] + 1 closed trajectories, while in [L-L-Z], they show that there are at least n if, in addition, one assumes symmetry of the surface around the origin. Unfortunately, we cannot go into the proofs of these beautiful results here and the reader is referred to [E2], [Lo1] and [Lo2] for that story. Instead, we present a very simple example to illustrate the use of Theorem 5.1.…”