Existence of closed characteristics on compact convex hypersurfaces in $\R^{2n}

TL;DR

This paper proves the existence of multiple closed characteristics on compact convex hypersurfaces in even-dimensional Euclidean space, including non-hyperbolic ones under certain conditions.

Contribution

It establishes lower bounds on the number of geometrically distinct closed characteristics, advancing the understanding of their existence on convex hypersurfaces.

Findings

At least [ (n+1)/2 ] + 1 closed characteristics exist.

At least [ n/2 ] + 1 non-hyperbolic closed characteristics exist if finite.

Provides new lower bounds for closed characteristics in symplectic geometry.

Abstract

In this paper, we prove there exist at least $[\frac{n+1}{2}]+1$ geometrically distinct closed characteristics on every compact convex hypersurface $\Sg$ in $\R^{2n}$. Moreover, there exist at least $[\frac{n}{2}]+1$ geometrically distinct non-hyperbolic closed characteristics on $\Sg$ in $\R^{2n}$ provided the number of geometrically distinct closed characteristics on $\Sg$ is finite.