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

### Abstract

For any given compact $C^2$ hypersurface $\Sigma$ in $\mathbf{R}^{2n}$ bounding a strictly convex set with nonempty interior, in this paper an invariant $\rho_n(\Sigma)$ is defined and satisfies $\rho_n(\Sigma) \ge [n/2]+1$, where $[a]$ denotes the greatest integer which is not greater than $a\in \mathbf{R}$. The following results are proved in this paper. 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 $\rho_n(\Sigma) \ge n$. If the total number of geometrically distinct closed characteristics on $\Sigma$ is finite, there exists at least an elliptic one among them, 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.

Yiming Long

Chaofeng Zhu