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 $R^{2n}$ with $n\ge 2$. Besides many partial results, this conjecture has been completely solved only for $n=2$. In this article, 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 $\Sigma$ in $R^{2n}$ when the number of geometrically distinct closed characteristics on $\Sigma$ is finite. Then, using this identity and earlier techniques of the index iteration theory, we prove the mentioned multiplicity result for $R^6$. If there are exactly two geometrically distinct closed characteristics on a compact convex hypersuface in $R^4$, we prove that both of them must be irrationally elliptic