Closed characteristics on compact convex hypersurfaces in \(\mathbb{R}^{2n}\). (English) Zbl 1028.53003

Long, Yiming; Zhu, Chaofeng

Ann. Math. (2) 155, No. 2, 317-368 (2002).

For any given compact \(C^2\) hypersurface \(\Sigma\) in \(\mathbb{R}^{2n}\) bounding a strictly convex set with nonempty interior, in this paper an invariant \(\rho_n(\Sigma)\) is defined satisfying \(\rho_n(\Sigma)\geq [n/2]+1\), where \([a]\) denotes the greatest integer which is not greater than \(a\in \mathbb{R}\). The following results are proved. There always exist at least \(\rho_n(\Sigma)\) geometrically distinct closed characteristics on \(\Sigma\). If all the geometrically distinct closed characteristics on \(\Sigma\) are nondegenerate, then \(\rho_n(\Sigma)\geq n\). If the total number of geometrically distinct closed characteristics on \(\Sigma\) is finite, there exists among them at least one elliptic one, and there exist at least \(\rho_n(\Sigma)-1\) of them possessing irrational mean indices. If this total number is at most \(2\rho_n(\Sigma)-2\), there exist at least two elliptic ones among them.

Reviewer: A.A.Borisenko (Khar’kov)

MSC:

53A07	Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
52A20	Convex sets in \(n\) dimensions (including convex hypersurfaces)
53C45	Global surface theory (convex surfaces à la A. D. Aleksandrov)

Keywords:

convex hypersurface; Maslov-type index; Hamiltonian system