Four closed characteristics on compact star-shaped hypersurfaces in $\mathbb{R}^{8}$

TL;DR

This paper proves that every non-degenerate, star-shaped hypersurface in 8-dimensional space without certain prime closed characteristics has at least four prime closed characteristics, advancing understanding of geometric properties of such hypersurfaces.

Contribution

The paper establishes a new lower bound of four prime closed characteristics on a class of star-shaped hypersurfaces in 8-dimensional space, under specific non-degeneracy and index conditions.

Findings

At least four prime closed characteristics exist under given conditions

No prime closed characteristic of Maslov-type index -1 on the hypersurface

Results apply to non-degenerate, star-shaped hypersurfaces in 8-space

Abstract

In this paper, we proved that for every non-degenerate $C^3$ compact star-shaped hypersurface $\Sigma$ in $\mathbb{R}^{8}$ which carries no prime closed characteristic of Maslov-type index $-1$, there exist at least four prime closed characteristics on $\Sigma$.