In this article, we first give a proof on the Arnold chord conjecture which states that every Reeb flow has at least as many Reeb chords as a smooth function on the Legendre submanifold has critical points on contact manifold. Second, we prove that every Reeb flow has at least as many close Reeb orbits as a smooth round function on the close contact manifold has critical circles on contact manifold. This also implies a proof on the fact that there exists at least number $n$ close Reeb orbits on close $(2n-1)$-dimensional convex hypersurface in $R^{2n}$ conjectured by Ekeland.
Let Σ be a smooth closed oriented manifold of dimension 2n -1. A contact form on Σ is a 1-form such that λ ∧ (dλ) n-1 is a volume form on Σ. Associated to λ there are two important structures. First of all the so-called Reeb vectorfield ẋ = X defined by i X λ ≡ 1, i X dλ ≡ 0; and secondly the contact structure ξ = ξ λ → Σ given by ξ λ = ker(λ) ⊂ T Σ.
By a result of Gray, [7], the contact structure is very stable. In fact, if (λ t ) t∈[0,1] is a smooth arc of contact forms inducing the arc of contact structures (ξ t ) t∈[0,1] , there exists a smooth arc (ψ t ) t∈[0,1] of diffeomorphisms with
here it is important that Σ is compact. From (1.1) and the fact that Ψ 0 = Id it follows immediately that there exists a smooth family of maps
In contrast to the contact structure the dynamics of the Reeb vectorfield changes drastically under small perturbation and in general the flows associated to X t and X s for t = s will not be conjugated. Let (Σ, λ) is a contact manifold with contact form λ of dimension 2n-1, then a Legendre submanifold is a submanifold L of Σ, which is (n -1)dimensional and everywhere tangent to the contact structure ker λ. Then a characteristic chord for (λ, L) is a smooth path
The main results of this paper is following:
Theorem 1.1 Let (Σ, λ) be a contact manifold with contact form λ, X λ its Reeb vector field, L a closed Legendre submanifold. Then there exists at least as many Reeb characteristic chords for (X λ , L) as a smooth function on the Legendre submanifold L has critical points.
Theorem1.1 is asked in [1]. In [9], we have proved there exists at least one Reeb chord by Gromov’s J-holomorphic curves. Partial results is obtained in [12,13].
We recall that a round function f on M is the one whose critical sets consist of smooth circles {C i , i = 1, …, k} (see [3]). One can define Round Morse-Bott function.
Theorem 1.2 Let (Σ, λ) be a contact manifold with contact form λ, X λ its Reeb vector field. Then there exists at least as many close Reeb characteristic orbits for X λ as a smooth round function on the Σ has critical circles.
In [10], we have proved there exists at least one close Reeb orbit by Gromov’s J-holomorphic curves. Theorem1.2 is related to the Arnold-Ginzburg question on magnetic field and partial results was obtained in [1,4,6].
Corollary 1.1 Let (Σ, λ) be a close (2n -1)-dimensional star-shaped hypersurface in R 2n with contact form λ, X λ its Reeb vector field. Then there exists at least number n close Reeb orbits.
This implies that the Ekeland conjecture holds in [5].
The proofs of Theorem1.1-1.2 are the extension of the methods in [11].
2 Proof of Theorem1.1
Proof of Theorem1.1: Let (Σ, λ) be a contact manifold with the contact form λ. By Whitney’s embedding theorem, we first embed Σ in R N , then by considering the cotangent bundles, the symplectizations and contactizations, one can embed (Σ, λ) into (S 2N +1 , λ 0 ) with λ = f λ 0 , f is positive function on Σ. By the contact tubular neighbourhood theorem, the neighbourhood U(Σ) is contactomorphic to the symplectic vector bundle E on Σ with symplectic fibre (R 2N -2n+2 , ω 0 ). By our construction, it is easy to see that there exists Lagrangian sub-bundle L in E with Lagrangian fibre R N -n+1 . So, we can extend the contact form λ on Σ to the neighbourhood U(Σ) as λ such that the Reeb vector fields Xλ|Σ = X λ . We extend f positively to whole S 2N +1 . So the contact form λ on U(Σ) is extended to whole S 2N +1 as f λ 0 .
Let λ s = (sf + 1s)λ 0 be the one parameter family of contact forms on S 2N +1 . Let X λs its Reeb vector field.
Let L be a close Legendre submanifold contained in Σ, i.e., T L ⊂ ξ. Let L be a Legendre submanifold contained in U(Σ) which is fibred on L and T L ⊂ ξ 0 , here ξ 0 is the standard contact structure on (S 2N +1 , λ 0 ).
Let S be a smooth hypersurface in Σ which contains L. We can assume that the Reeb vector fields X λ is transversal to S. Let S be a smooth hypersurface in S 2N +1 which contains L and extends S. We can assume that the Reeb vector fields X λs is transversal to S.
Let η s,t be the Reeb flow generated by the Reeb vector field X λs . Let f s : Ls → R be the arrival time function of the Reeb flow η s,t from the parts Ls of L to S. Then ϕ s = η s,fs (•) : Ls → S is an exact Lagrange embedding for symplectic form dλ s | S.
One observes that ϕ s ( Ls ) ∩ L corresponds to the Reeb chords of X λs . Now we extend the family of functions f s on Ls to the whole L. This extends ϕ s = η s,fs (•) : L → S 2N +1 as an exact isotropic embedding for symplectic form dλ s on S 2N +1 . Then, we obtain φs = η s,fs+t (•) : L×[-ε, ε] → S 2N +1 as an exact Lagrangian embedding for symplectic form d(e t λ s ) on R × S 2N +1 . By Moser’s stability theorem, this defines an exact Lagrangian isotopy φs : L × [-ε, ε] → (R × S 2N +1 , d(e t λ 0 ). It is well known that an exact Lagrangian isotopy is determined by Hamilton isotopy with hamilton function h s on R × S 2N +1 . Let X hs be th
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