On the number of P-invariant closed characteristics on partially symmetric compact convex hypersurfaces in ${\mathbf{R}}^{2n}$

Abstract

In this paper, let $n\ge 2$ be an integer, $P=diag(-I_{n-\kappa },I_{\kappa },-I_{n-\kappa },I_{\kappa })$ for some integer $\kappa \in [0,n)$, and $\Sigma \subset {\mathbf{R}}^{2n}$ 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 $\frac{R}{r}<\sqrt{2}$, then there exist at least $n-\kappa$ geometrically distinct P-symmetric closed characteristics on $\Sigma$, as a consequence, $\Sigma$ carry at least $n$ geometrically distinct P-invariant closed characteristics.

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Cited by in corpus (2)

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