Closed characteristics on compact convex hypersurfaces in $\R^{2n}$

Abstract

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