Symmetric closed characteristics on symmetric compact convex hypersurfaces in $\mathbf{R}^8$: Special cases

TL;DR

This paper investigates symmetric closed characteristics on symmetric convex hypersurfaces in 8-dimensional space, proving that under certain conditions with exactly four such characteristics, all must exhibit symmetry.

Contribution

It establishes that for specific cases with four closed characteristics, all are symmetric, advancing understanding of symmetric convex hypersurfaces in higher dimensions.

Findings

All four closed characteristics are symmetric in the special cases considered.

The result applies to symmetric convex hypersurfaces in $ extbf{R}^8$ with exactly four closed characteristics.

Provides conditions under which symmetry of characteristics is guaranteed.

Abstract

Let $\Sigma$ be a $C^3$ compact symmetric convex hypersurface in $\mathbf{R}^{8}$. For some special cases, we prove that when $\Sigma$ carries exactly four geometrically distinct closed characteristics, then all of them must be symmetric.