Stable P-symmetric closed characteristics on partially symmetric compact convex hypersurfaces

TL;DR

This paper proves the existence of multiple P-symmetric closed characteristics on certain partially symmetric convex hypersurfaces, with at least two having a specified number of Floquet multipliers on the unit circle.

Contribution

It establishes the existence of multiple P-symmetric closed characteristics on partially symmetric convex hypersurfaces under a pinching condition, extending previous results in symplectic geometry.

Findings

At least two P-symmetric closed characteristics exist under the given conditions.

These characteristics have at least 2n-4κ Floquet multipliers on the unit circle.

The result applies to hypersurfaces satisfying a specific pinching condition.

Abstract

In this paper, let $n\geq2$ be an integer, $P=diag(-I_{n-\kappa},I_\kappa,-I_{n-\kappa},I_\kappa)$ for some integer $\kappa\in[0, n-1)$, and $\Sigma \subset {\bf R}^{2n}$ be a partially symmetric compact convex hypersurface, i.e., $x\in \Sigma$ implies $Px\in\Sigma$. We prove that if $\Sigma$ is $(r,R)$-pinched with $\frac{R}{r}<\sqrt{\frac{5}{3}}$, then $\Sigma$ carries at least two geometrically distinct P-symmetric closed characteristics which possess at least $2n-4\kappa$ Floquet multipliers on the unit circle of the complex plane.