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

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

2007-01-22

arxiv.org/abs/math/0701608v4

Mathematics

Symplectic Geometry

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

Scientific paper

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 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 $\R^{2n}$ 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 $\R^6$. 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.

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