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 R2n with n≥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 Σ in R2n when the number of geometrically distinct closed characteristics on Σ is finite. Then, using this identity and earlier techniques of the index iteration theory, we prove the mentioned multiplicity result for R6. If there are exactly two geometrically distinct closed characteristics on a compact convex hypersuface in R4, we prove that both of them must be irrationally elliptic
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