Resonance identity, stability and multiplicity of closed characteristics on compact convex hypersurfaces

Abstract

There is a long standing conjecture in Hamiltonian analysis which claims that there exist at least $n$ geometrically distinct closed characteristics on every compact convex hypersurface in ${\mathbb{R}}^{2n}$ with $n\ge 2$. Besides many partial results, this conjecture has been only completely solved for $n=2$. In this paper, we give a confirmed answer to this conjecture for $n=3$. In order to prove this result, we establish first a new resonance identity for closed characteristics on every compact convex hypersurface $\Sg$ in ${\mathbb{R}}^{2n}$ when the number of geometrically distinct closed characteristics on $\Sg$ is finite. Then using this identity and earlier techniques of the index iteration theory, we prove the mentioned multiplicity result for ${\mathbb{R}}^6$. If there are exactly two geometrically distinct closed characteristics on a compact convex hypersuface in ${\mathbb{R}}^4$, we prove that both of them must be irrationally elliptic.

48 pages, 1 figure, to appear in Duke Math. J

References in corpus (1)

• Index iteration theory for symplectic paths with applications to nonlinear Hamiltonian systems

Cited by in corpus (1)

• A note on Reeb dynamics on the tight 3-sphere