aa r X i v : . [ m a t h . S G ] A ug On the degenerated Arnold-Givental conjecture
Guangcun Lu ∗ Department of Mathematics,Beijing Normal University, China([email protected])First version, May 30, 2008Revised version, August 6, 2018
Abstract
We present another view dealing with the Arnold-Givental conjecture on a realsymplectic manifold (
M, ω, τ ) with nonempty and compact real part L = Fix( τ ). Forgiven Λ ∈ (0 , + ∞ ] and m ∈ N ∪ { } we show the equivalence of the following twoclaims: (i) ♯ ( L ∩ φ H ( L )) ≥ m for any Hamiltonian function H ∈ C ∞ ([0 , × M )with Hofer’s norm k H k < Λ; (ii) ♯ P ( H, τ ) ≥ m for every H ∈ C ∞ ( R / Z × M )satisfying H ( t, x ) = H ( − t, τ ( x )) ∀ ( t, x ) ∈ R × M and with Hofer’s norm k H k < P ( H, τ ) is the set of all 1-periodic solutions of ˙ x ( t ) = X H ( t, x ( t )) satisfying x ( − t ) = τ ( x ( t )) ∀ t ∈ R (which are also called brake orbits sometimes). Supposethat ( M, ω ) is geometrical bounded for some J ∈ J ( M, ω ) with τ ∗ J = − J and hasa rationality index r ω > r ω = + ∞ . Using Hofer’s method we prove that ifthe Hamiltonian H in (ii) above has Hofer’s norm k H k < r ω then ♯ ( L ∩ φ H ( L )) ≥ ♯ P ( H, τ ) ≥ Cuplength F ( L ) for F = Z , and further for F = Z if L is orientable,where P ( H, τ ) consists of all contractible solutions in P ( H, τ ). A real symplectic manifold is a triple ( M, ω, τ ) consisting of a symplectic manifold(
M, ω ) and an anti-symplectic involution τ on ( M, ω ), i.e. τ ∗ ω = − ω and τ = id M .The Marsden-Weinstein quotients of real Hamiltonian systems provide a great dealof examples of such manifolds. Let J ( M, ω ) denote the space of all ω -compatiblesmooth almost complex structures on M , and R J ( M, ω ) = { J ∈ J ( M, ω ) | J ◦ dτ = − dτ ◦ J } , ∗ The first author is partially sponsored by the NNSF 10671017 and 11271044 of China and the Programfor New Century Excellent Talents of the Education Ministry of China. hat is, J ∈ R J ( M, ω ) if and only if τ is anti-holomorphic with respect to J . Withthe standard trick of S´evennec (see [McSa1, p.64]) one can prove that R J ( M, ω ) isa separable Frech´et submanifold of J ( M, ω ) which is nonempty and contractible (cf.[Wel, Prop. 1.1]). The fixed point set L := Fix( τ ) of τ is called the real part of M . Since τ is an isometry of the natural Riemann metric g J = ω ◦ ( id M × J ) for any J ∈ R J ( M, ω ), L is either empty or a Lagrange submanifold (cf. [Vi, p.4]). Arnold-Givental conjecture ([Gi]): Let (
M, ω, τ ) be a real symplectic manifold ofdimension 2 n , and L = Fix( τ ) be a nonempty compact submanifold without bound-ary. Then for every Hamiltonian diffeomorphism φ on ( M, ω ), it holds that ♯ (cid:0) L ∩ φ ( L ) (cid:1) ≥ n X k =0 b k ( L, Z ) or n X k =0 b k ( L, Z ) (1.1)provided that L and φ ( L ) intersect transversally, and that ♯ (cid:0) L ∩ φ ( L ) (cid:1) ≥ Cuplength Z ( L ) or Cuplength Z ( L ) (1.2)generally. Hereafter the F -cuplength of a paracompact topological space X over anintegral domain F , Cuplength F ( X ), is defined the supremum of natural numbers k such that there exist cohomology classes α , · · · , α k − in H ∗ ( X, F ) of positive degreesatisfying α ∪ · · · ∪ α k − = 0.This conjecture is a special case of Arnold’s conjecture on Lagrangian intersections([Ar1, Ar2]). If M is closed, the estimate (1.1) in Z -coefficients follows from Floer[Fl1] if π ( M, L ) = 0, [Oh] if L is a real form of compact Hermitian spaces withsome assumptions on the Maslov index, [Laz] if L is the strongly negative monotone,and [FuOOO, Theorem H] if L is the semipositive, and [Fr] if L is in Marsden-Weinstein quotients. The estimate (1.2) in Z -coefficients follows from Floer andHofer [Fl2, Ho2], and Liu [Liu] if ( M, ω ) has positive rationality index r ω and φ maybe generated by H ∈ C ∞ ([0 , × M ) with Hofer’s norm k H k < r ω /
2. The estimates in(1.1) and (1.2) were obtained for (
M, L ) = ( C P n , R P n ) [ChJi, Gi]. (The author [Lu2]also generalized the arguments in [ChJi] to the case of weighted complex projectivespaces, which are symplectic orbifolds).Arnold-Givental conjecture contains Arnold conjecture for the symplectic fixedpoints ([Ar1, Ar2]), which stated that for every Hamiltonian diffeomorphism φ on aclosed symplectic manifold ( M, ω ) the following estimates hold true, ♯ Fix( φ ) ≥ Cuplength F ( M ) , (1.3) ♯ Fix( φ ) ≥ dim M X k =0 b k ( M ; F ) (1.4)if each x ∈ Fix( φ ) is nondegenerate in the sense that the tangent map dφ ( x ) : T x M → T x M has no eigenvalue 1. After Floer [Fl3] first invented Floer homologies to prove(1.4) for monotone ( M, ω ) and F = Z , Fukaya-Ono [FuO] and Liu-Tian [LiuT] de-veloped Floer homologies to affirm it for any closed symplectic manifold ( M, ω ) and F = Q . n this paper we consider a smooth time dependent Hamiltonian function H : R × M → R , ( t, x ) H ( t, x ) = H t ( x ) satisfying H t ( x ) = H t +1 ( x ) and H ( t, x ) = H ( − t, τ ( x )) ∀ ( t, x ) ∈ R × M. (1.5)Such a Hamiltonian function H is said to be 1 -periodic in time and symmetric .Let X H t be defined by ω ( X H t , · ) = − dH t ( · ). Then X H t = X H t +1 and X H − t ( x ) = − dτ ( τ ( x )) X H t ( τ ( x )) ∀ ( t, x ) ∈ R × M. (1.6)For x ∈ M let x : R → M be the solution of˙ x ( t ) = X H t ( x ( t )) (1.7)through x at t = 0. Then both y ( t ) := x ( − t ) and z ( t ) := τ ( x ( t )) are solutions of˙ x ( t ) = dτ ( τ ( x ( t )) X H t ( τ ( x ( t ))) . So y = z if and only if x = y (0) = z (0) = τ ( x (0)) = τ ( x ). We are interested inthose 1-periodic solutions x of the equation (1.7) which satisfy x ( − t ) = τ ( x ( t )) ∀ t ∈ R . (1.8)Clearly, such a solution x satisfies: x (0) , x (1 / ∈ L and x (1 − t ) = τ ( x ( t )) ∀ t . A loop x : S = R / Z → M satisfying (1.8) is called a τ -reversible . ( τ -reversible 1-periodicsolutions are also called brake orbits in literature.) Denote by P ( H, τ ) (resp . P ( H, τ ) )the set of all τ -reversible 1-periodic solutions (resp. contractible τ -reversible 1-periodic solutions) of (1.7). Clearly, P ( H, τ ) must be empty if L = ∅ . Let φ Ht : M → M be the Hamiltonian diffeomorphisms defined by ddt φ Ht = X H t ◦ φ Ht , φ H = id M . From (1.6) it easily follows that φ Ht ◦ τ = τ ◦ φ H − t ∀ t ∈ R . Moreover, it always holdsthat φ Ht +1 = φ Ht ◦ φ H ∀ t ∈ R . So we get that φ H ◦ τ = τ ◦ ( φ H ) − . (1.9)One also easily checks that the elements of P ( H, τ ) are one-to-one correspondencewith points in L ∩ Fix( φ H ). So we have ♯ ( L ∩ Fix( φ H )) = ♯ P ( H, τ ) ≥ ♯ P ( H, τ ) . (1.10)Recall Hofer’s norm of a Hamiltonian function H ∈ C ∞ ([0 , × M ) is defined by k H k = Z [sup x H t ( x ) − inf x H t ( x )] dt. Our first result is heorem 1.1 Let ( M, ω, τ ) be a real symplectic manifold of dimension n , and thefixed point set L = Fix( τ ) be nonempty. Let Λ ∈ (0 , + ∞ ] and m ∈ N ∪ { } . Then thefollowing two claims are equivalent. (i) Every Hamiltonian diffeomorphism φ on M generated by a Hamiltonian function H ∈ C ∞ ([0 , × M ) with k H k < Λ , satisfies ♯ ( L ∩ φ ( L )) ≥ m. (ii) Every 1-periodic in time and symmetric H ∈ C ∞ ( R / Z × M ) with k H k < ,satisfies ♯ P ( H, τ ) ≥ m. Remark 1.2
The proof of “(i)= ⇒ (ii)” in the proof of Theorem 1.1 actually shows P ( H, τ ) = (cid:8) x ( t ) = φ Ht ( x ) (cid:12)(cid:12) x ∈ L ∩ ( φ H / ) − ( L ) (cid:9) and so ♯ P ( H, τ ) = ♯ ( L ∩ φ H ( L )) . So using the results obtained for the Arnold conjecture on Lagrangian intersectionsone may get the estimates of the lower bound of ♯ P ( H, τ ) under certain assumptions.For example, it follows from Theorem 1.1 and [FuOOO, Theorem H] that if M isclosed, L is semipositive, and L ⋔ φ H / ( L ) then ♯ P ( H, τ ) ≥ P rank H ∗ ( L ; Z ).Recall that a symplectic manifold ( M, ω ) without boundary is said to be geo-metrically bounded if there exist a geometrically bounded Riemannian metric µ on M (i.e., its sectional curvature is bounded above by some constant K > i ( M, µ ) >
0) and a ω -compatible almost complex structure J suchthat such that ω ( X, J X ) ≥ α k X k µ and | ω ( X, Y ) | ≤ β k X k µ k Y k µ ∀ X, Y ∈ T M for some positive constants α and β (cf. [Gr], [AuLaPo], [CGK], [Lu1]). For areal symplectic manifold ( M, ω, τ ) without boundary, if the almost complex structure J above can be chosen in R J ( M, ω ) we say (
M, ω, τ ) to be real geometricallybounded (with respect to (
J, µ )).The rationality index of a symplectic manifold (
M, ω ) is defined by r ω = r ( M, ω ) := inf (cid:8) h ω, A i (cid:12)(cid:12) A ∈ π ( M ) , h ω, A i > (cid:9) ∈ [0 , + ∞ ] , where we use the convention that the infimum over the empty set is equal to + ∞ .Since { ω ( A ) | A ∈ π ( M ) } is a subgroup of ( R , +), it is easily checked that r ω is a finitepositive number if and only if ω (cid:0) π ( M ) (cid:1) = r ω Z . For J ∈ J ( M, ω ) let m ( M, ω, J ) ∈ [0 , + ∞ ] denote the infimum of the area of all nonconstant J -holomorphic spheres in M , where as usual we understand m ( M, ω, J ) = ∞ if no nonconstant J -holomorphicsphere exists. Clearly, r ω ≤ m ( M, ω, J ) ∀ J ∈ J ( M, ω ). As showed by (ii)-(iii) ofExample 4.2, there exist closed symplectic manifolds (
M, ω ) such that0 < r ω < sup J ∈J ( M,ω ) m ( M, ω, J ) = + ∞ . f M is compact, it directly follows from the Gromov compactness theorem that m ( M, ω, J ) > . If (
M, ω, J ) is only geometrically bounded as above, this may be derived from themonotonicity principle ([Sik, Prop.4.3.1(ii)]):
For r = min { i ( M, µ ) , π/ √ K } , a com-pact Riemann surface with boundary S and a J -holomorphic map f : S → M , assumethat there exists a µ -metric ball B ( x, r ) with r ≤ r and with x ∈ f ( S ) such that f ( ∂S ) ⊂ ∂B ( x, r ) , then πα β r ≤ Area µ ( f ( S )) ≤ α Z S f ∗ ω. In fact, put δ = min { πα β r , πα β ( i ( M, µ )) / } . It follows that R Σ u ∗ ω ≥ δ for everynonconstant J -holomorphic map u from a closed Riemann surface Σ to M . (See theproof of [FuO, Lemma 8.1] below Lemma 8.10 therein).Based on Hofer’method in [Ho2] we can get our second result. Theorem 1.3
Let ( M, ω, τ ) be a real geometrical bounded symplectic manifold withrespect to J ∈ R J ( M, ω ) and a Riemannian metric µ , and L = Fix( τ ) be a nonemptycompact submanifold without boundary. Let H ∈ C ∞ ( R / Z × M ) be a symmetricHamiltonian function. If r ω > and k H k < r ω , then ♯ ( L ∩ Fix( φ H )) ≥ ♯ P ( H, τ ) ≥ Cuplength Z ( L ) , (1.11) and ♯ P ( H, τ ) ≥ Cuplength Z ( L ) if L is orientable. Note that r ω ∈ (0 , + ∞ ) (resp. = + ∞ ) implies ω ( π ( M, L )) = r ω Z (resp. = 0).As a direct consequence of (1.10) and Theorems 1.1, 1.3 we get Theorem 1.4
Let ( M, ω, τ, J, L ) be as in Theorem 1.3. If r ω > , then everyHamiltonian diffeomorphism φ on M generated by a Hamiltonian function H ∈ C ∞ ([0 , × M ) with k H k < r ω / , satisfies the estimates ♯ ( L ∩ φ ( L )) ≥ Cuplength Z ( L ) , (1.12) and ♯ ( L ∩ φ ( L )) ≥ Cuplength Z ( L ) if L is orientable. Remark 1.5 If M is closed, (1.12) is a special case of the main result in [Liu] provedwith Floer homology; the latter and Theorems 1.1 can only lead to ♯ P ( H, τ ) ≥ Cuplength Z ( L ), which is weaker than the second inequality in (1.11). Actually, themain result in [Liu] can also be proved by refining Hofer’ arguments in [Ho2] asdone in this paper. Hofer’method does not involve Floer and Morse homologies (andthus complicated transversality arguments). Recently, Albers and Hein [AH] gave anabstract result based on Morse cohomology. As in the proof of [AH, Theorem 5.1], itmay lead to (1.11), but no better result. he twisted product ( c M , b ω ) = ( M × M, ω × ( − ω )) of a symplectic manifold ( M, ω )and itself with anti-symplectic involution given by τ : M × M → M × M, ( x, y ) ( y, x ) , is a real symplectic manifold with Fix( τ ) = △ M . For any J ∈ J ( M, ω ) it is easilychecked that J × ( − J ) ∈ R J ( M × M, ω × ( − ω )) and m ( M × M, ω × ( − ω ) , J × ( − J )) = 2 m ( M, ω, J ) . (1.13)If H ∈ C ∞ ( R / Z × M ), then b H : R × M × M → R , ( t, x, y ) H t ( x ) + H − t ( y ) , is 1-periodic in time and symmetric. Note that X b H t ( x, y ) = ( X H t ( x ) , − X H − t ( y )) bythe definition of X H t above (1.6). One easily proves that z = ( x, y ) : R / Z → R belongsto P ( b H , τ ) (resp. P ( b H, τ )) if and only if x ∈ P ( H ) (resp. x ∈ P ( H )) and y ( t ) = x ( − t ) ∀ t ∈ R . Here P ( H ) (resp. P ( H )) always denote the set of 1-periodic solutions(resp. contractible 1-periodic solutions) of the equation ˙ x = X H ( t, x ). Moreover, k b H k = Z [sup ( x,y ) H t ( x, y ) − inf ( x,y ) H t ( x, y )] dt = 2 k H k and r b ω = r ω are clear. Using this and (1.13) we derive from Theorem 1.3: Theorem 1.6
Let ( M, ω ) be a closed symplectic manifold, and H ∈ C ∞ ( R / Z × M ) satisfy k H k < r ω . Then ♯ Fix( φ H ) ≥ Cuplength Z ( M ) and ♯ Fix( φ H ) ≥ Cuplength Z ( M ) . The first inequality was proved in [Sch, Theorem 1.1] by Floer homology method.It is a generalization of the result in [Fl2, Ho2]. Without the assumption “ k H k < r ω ”,Le and Ono [LeO] got the estimates (1.3) for F = Z if ( M, ω ) is negative monotoneand has minimal Chern number N ≥ dim M/ N , ( T ∗ N, ω can = − dλ can ), is a real symplecticmanifold with the anti-symplectic involution given by τ : T ∗ N → T ∗ N, ( q, p ) ( q, − p ) , where q ∈ N and p ∈ T ∗ q N . Recall that the Liouville 1-form λ can on T ∗ N is definedby λ can ( ξ ) = p ( T π ∗ ξ ) ∀ ξ ∈ T p T ∗ N , where π ∗ : T ∗ N → N is the natural projection.The fixed point set Fix( τ ) is the zero section 0 N which can be identified with N .Assume now that N is closed. As in [CGK, Lu1] we can prove that ( T ∗ N, ω can , τ )is geometrically bounded for some J ∈ R J ( T ∗ N, ω can ) and some metric G on T ∗ N .Applying Theorem 1.3 to ( T ∗ N, ω can , τ ) we immediately obtain: orollary 1.7 Let N be a closed manifold, and H ∈ C ∞ ( R / Z × T ∗ N ) satisfy H ( − t, q, p ) = H ( t, q, − p ) for all t ∈ R and ( q, p ) ∈ T ∗ N . Then ♯ (0 N ∩ Fix( φ H )) ≥ ♯ P ( H, τ ) ≥ Cuplength Z ( N ) , and ♯ P ( H, τ ) ≥ Cuplength Z ( N ) if N is orientable. This and Theorem 1.1 immediately lead to
Corollary 1.8 ([Ho1, LaSi])
Let N be a closed manifold. Then any Hamilto-nian diffeomorphism φ on ( T ∗ N, ω can ) generated by a Hamiltonian function H ∈ C ∞ ([0 , × T ∗ N ) satisfies estimates: ♯ ( N ∩ φ ( N )) ≥ Cuplength Z ( N ) , and ♯ ( N ∩ φ ( N )) ≥ Cuplength Z ( N ) if N is orientable. The arrangements of the paper as follows. In Section 2.1 we first prove Theo-rem 1.1. Then in Section 2.2 we complete the proof of Theorem 1.3 by improvingthe arguments in [HoZe, § Z -degree for Fredholm section having Fredholm index zeroas in [HoZe, § Acknowledgements : The results of this paper were reported in the workshop onFloer Theory and Symplectic Dynamics at CRM of University of Montreal, May 19-23, 2008. I would like to thank the organizers for their invitation, and CRM forhospitality. (i) = ⇒ (ii) : Let φ t be the Hamiltonian flow generated by H . Define Q : [0 , × M → R by Q ( t, x ) = H ( t/ , x ), and denote by ϕ t the flow of X Q . It is easily proved that φ = ϕ and k Q k = 12 k H k < Λ . (2.1)It follows from (i) that ♯ ( L ∩ φ ( L )) ≥ m. For any x ∈ L ∩ φ − ( L ), x ( t ) := φ t ( x ) satisfies ˙ x ( t ) = X H t ( x ( t )) ∀ t and x ( ) = φ ( x ) ∈ L . Since H t = H − t ◦ τ , for ≤ t ≤ x ( t ) = X H t ( x ( t )) = − dτ ( τ ( x ( t ))) X H − t ( τ ( x ( t ))) or equivalently ddt τ ( x ( t )) = − X H − t ( τ ( x ( t ))) . It follows that y ( t ) := τ ( x (1 − t )) on [0 , ] satisfies ˙ y ( t ) = X H t ( y ( t )). Note that x ( ) ∈ L implies y ( ) = τ ( x ( )) = x ( ), i.e., φ ( y (0)) = φ ( x ). Hence y (0) = x nd thus τ ( x (1 − t )) = y ( t ) = x ( t ) ∀ ≤ t ≤ . This implies x (1 − t ) = τ ( x ( t )) ∀ t ∈ [0 , x (1) = τ ( x ) = x = x (0). Moreover, since H = H ,one has ˙ x (1) = ˙ x (0). Hence x is a 1-periodic solution of ˙ x ( t ) = X H t ( x ( t )) satisfying x (1 − t ) = τ ( x ( t )) ∀ t , that is, x ∈ P ( H, τ ). It is also clear that two different x , x ∗ ∈ L ∩ φ − ( L ) give two different elements in P ( H, τ ), x ( t ) = φ t ( x ) and x ∗ ( t ) = φ ∗ t ( x ).Conversely, each x ∈ P ( H, τ ) determines a point x (0) ∈ L ∩ φ − ( L ) uniquely. Sowe get P ( H, τ ) = { x ( t ) = φ t ( x ) | x ∈ L ∩ φ − ( L ) } (2.2)which implies ♯ P ( H, τ ) = ♯ ( L ∩ φ ( L )) ≥ m . (ii) = ⇒ (i) : By the assumption there exists a Hamiltonian H ∈ C ∞ ([0 , × M )with k H k < Λ, such that its Hamiltonian flow φ t satisfies φ = φ . The proof will befinished along the line of proof of [BiPoSa, Propsition 2.1.3]. Take a small δ > k H k + 2 δ < λ : [0 , → [0 ,
1] such that fora given small 0 < ǫ ≪ / λ ( t ) = 0 for t ∈ [0 , ǫ ] ,λ ( t ) = 0 for t ∈ [1 − ǫ, ,λ ′ ( t ) > t ∈ ( ǫ, − ǫ ) . (2.3)Clearly, R λ ′ ( t ) dt = 1. Take a time independent compactly supported function F : M → R which is τ -invariant, such that k F k C < δ/
4. Let f t be the Hamiltonianflow generated by F . Then the Hamiltonian isotopy ϕ t := f t − λ ( t ) ◦ φ λ ( t ) is generatedby the Hamiltonian function H t := F + λ ′ ( t )( H λ ( t ) − F ) ◦ f λ ( t ) − t . The function H t equals F near t = 0 and t = 1 and hence defines a smooth Hamilto-nian on S × M . Moreover, ϕ = φ . Denote by A H ( t ) = sup x ∈ M H t ( x ) − inf x ∈ M H t ( x ) ∀ t ∈ [0 , . Then k H k = R A H ( t ) dt , and it is easily computed that A H ( t ) : = sup x ∈ M H t ( x ) − inf x ∈ M H t ( x ) ≤ λ ′ ( t ) (cid:0) sup x ∈ M H λ ( t ) ( x ) − inf x ∈ M H λ ( t ) ( x ) (cid:1) + 2 k F k C + 2 λ ′ ( t ) k F k C . From this and (2.3) we arrive at k H k = Z A H ( t ) dt ≤ Z λ ′ ( t ) A H ( λ ( t )) dt + 4 k F k C = Z − ǫǫ A H ( λ ( t )) dλ ( t ) + 4 k F k C = Z A H ( t ) dt + 4 k F k C = k H k + 4 k F k C . et us define a smooth Hamiltonian G : [0 , × M → R by G t ( x ) = ( H t ( x ) if 0 ≤ t ≤ / , H − t ) ( τ x ) if 1 / ≤ t ≤ . It is easy to see that G t = F near t = 0 , / ,
1, and G − t ( x ) = G t ( τ x ) for any( t, x ) ∈ [0 , × M . Extend G to R × M t , still denoted by G , weeasily see that G satisfies k G k = 2 k H k < k H k + 2 δ <
2Λ and (1.5), i.e., G t +1 = G t and G − t ( x ) = G t ( τ x ) ∀ ( t, x ) ∈ R × M. It follows that X G t ( x ) = ( X H t ( x ) if 0 ≤ t ≤ / , − dτ ( τ x ) X H − t ) ( τ x ) if 1 / ≤ t ≤ ψ t of X G and the flow ϕ t of X H satisfy ψ t/ ( x ) = ϕ t ( x ) for ( t, x ) ∈ [0 , × M. Specially, we have ψ / = ϕ = φ . Now for any y ∈ P ( G, τ ), the map x : [0 , → M defined by x ( t ) = y ( t/
2) satisfies ˙ x ( t ) = X H t ( x ( t )). Note that both x (0) = y (0) = y (1) and x (1) = y (1 /
2) belong to L = Fix( τ ). Hence x (1) = ϕ ( x (0)) ∈ L ∩ ϕ ( L ) = L ∩ φ ( L ) = L ∩ φ ( L ) since ϕ = φ = φ .Moreover, for two different y , y ∈ P ( G, τ ) we have y ( t ) = y ( t ) for some t ∈ [0 , / t ∈ [0 ,
1] let x i ( t ) = y i ( t/ i = 1 ,
2. Both satisfy ˙ x ( t ) = X H t ( x ( t )).Since x (2 t ) = x (2 t ), that is, ϕ t ( x (0)) = ϕ t ( x (0)), we obtain x (0) = x (0)and thus x (1) = x (1).In summary, we have proved ♯ ( L ∩ φ ( L )) ≥ ♯ P ( G, τ ). Applying Theorem 1.1(ii)to G we have also that ♯ P ( G, τ ) ≥ Cuplength Z ( L ). The desired claim is proved. ✷ Let (
M, ω, τ ) be real geometrical bounded for J ∈ R J ( M, ω ) and a Riemannianmetric µ on M . By the assumptions of Theorem 1.3 there exists a compact subset K ⊂ M such thatsupp( H t ) ⊂ K ∀ t ∈ R , L ⊂ K and [ x ∈P ( H,τ ) x ( R ) ⊂ K. (2.4)From now on, we assume ( M, g J ) ⊂ ( R N , h· , ·i ) by the Nash embedding theorem.Consider the standard Riemannian sphere ( S = C ∪ {∞} , j ) and the submanifold ofthe Banach manifolds W ,p ( S , M ) for a fixed p > B = { w ∈ W ,p ( S , M ) | w is contractible } . Let E J → S × M be the vector bundle, whose fiber over ( z, m ) ∈ S × M con-sists of all linear maps φ : T z S → T m M such that J ( m ) φ = − φ ◦ j . Due to the nclusion W ,p ( S , M ) ֒ → C ( S , M ), for given w ∈ W ,p ( S , M ), we can denote by¯ w : S → S × M the “graph map” ¯ w ( z ) = ( z, w ( z )) and write ¯ w ∗ E J → S for thepull back bundle. There exists a natural Banach space bundle E → B whose fiber E w = L p ( ¯ w ∗ E J ) at w ∈ B consists of all L p sections of the vector bundle ¯ w ∗ E J → S .The nonlinear Cauchy-Riemannian operator ¯ ∂ J ,¯ ∂ J ( w ) = dw + J ◦ dw ◦ j, can be considered as a smooth section of the bundle E → B .Denote by Z T = [ − T, T ] × S for T >
1. Take a smooth function γ : R → [0 , γ ( s ) = 1 for s ≤ − γ ( s ) = 0 for s ≥
0, and γ ′ ( s ) ≤ s ∈ R .Define γ T ( s ) = , s ∈ [ − T + 1 , T − γ ( s − T ) , s ≥ T − γ ( − s − T ) , s ≤ − T + 1. (2.5)Then γ ′ T ( s ) ≤ s ≥ T −
1, and γ ′ T ( s ) ≥ s ≤ − T + 1. Denote by ∇ theLevi-Civita connection with respect to the metric h· , ·i = g J ( · , · ). By the definition of X H t above (1.6), ∇ H t = − J X H t . For ( z, m ) ∈ ( S \ { , ∞} ) × M let us define h TJ ( z, m ) (cid:16) ξ ∂∂x | z + η ∂∂y | z (cid:17) = ξ (cid:16) γ T ( s ) e − πs cos(2 πt )2 π ∇ H t ( m ) − γ T ( s ) e − πs sin(2 πt )2 π J ( m ) ∇ H t ( m ) (cid:17) − η (cid:16) γ T ( s ) e − πs sin(2 πt )2 π ∇ H t ( m )+ γ T ( s ) e − πs cos(2 πt )2 π J ( m ) ∇ H t ( m ) (cid:17) for ξ, η ∈ R and z = e π ( s + it ) ∈ C . It is easily checked that h TJ ( z, m ) ◦ j = − J ◦ h TJ ( z, m ), i.e., h TJ ( z, m ) ∈ ( E J ) ( z,m ) . Note that0 < | z | = e πs ≤ e − π ( T +1) ⇐⇒ s ∈ ( −∞ , − T − ⇒ γ T ( s ) = 0 , ∞ > | z | = e πs ≥ e π ( T +1) ⇐⇒ s ∈ [ T + 1 , + ∞ ) ⇒ γ T ( s ) = 0 . Hence we can define h TJ (0 , m ) = 0 , h TJ ( ∞ , m ) = 0 and get a smooth family of sections h TJ : S × M → E J , T >
1. These give rise to a smooth family of sections of theBanach bundle
B → E , g TJ : B → E , T >
1, where g TJ ( w )( z ) = h TJ ( z, w ( z )) ∀ z ∈ S . For λ ∈ [0 ,
1] we define F T,λ : B → E , w ¯ ∂ J w + λg TJ ( w ) . (2.6)Note that τ and the standard complex conjugate c S on ( S , j ) induce an involution τ B : B → B , w τ ◦ w ◦ c − S , (2.7) nd its lifting involution τ E : E → E , (2.8)where for ξ ∈ E w , τ E ( ξ ) ∈ E τ B ( w ) is given by τ E (cid:0) ξ )( z, τ B ( w )( z ) (cid:1) = dτ ( w (¯ z )) ◦ ξ (¯ z, w (¯ z )) ◦ dc S ( z ) ∀ z ∈ S . Let B τ be the set of fixed points of τ B . It is a Banach submanifold in B , and w ∈ B sits in B τ if and only if w (¯ z ) = τ ( w ( z )) for any z ∈ S = C ∪ {∞} . Moreover, theinvolution τ E induces bundles homomorphisms on E| B τ . Denote by E +1 (resp. E − )the eigenspace associated to the eigenvalue +1 (resp. −
1) of this homomorphism.Then both E +1 and E − are Banach subbundles of E| B τ , and E| B τ = E +1 ⊕ E − . Notealso that ¯ ∂ J (cid:0) τ B ( w ) (cid:1) = τ E (cid:0) ¯ ∂ J ( w ) (cid:1) ∀ w ∈ B . (2.9)So the restriction ¯ ∂ J | B τ gives rise to a section of the bundle E + → B τ .Since c S (0) = 0 and c S ( ∞ ) = ∞ , we compute g TJ ( τ B ( w ))( z ) = h TJ ( z, τ B ( w )( z )) = h TJ ( z, τ ( w (¯ z ))) for z ∈ S . (2.10)Note that (1.6) implies that for x ∈ M , ∇ H − t ( x ) = dτ ( τ ( x )) ∇ H t ( τ ( x )) and dτ ( x ) ◦ J ( x ) = − J ( τ ( x )) ◦ dτ ( x ) . From the expression of h TJ ( z, m ) (cid:16) ξ ∂∂x | z + η ∂∂y | z (cid:17) above one easily checks h TJ ( z, τ ( w (¯ z ))) (cid:16) ξ ∂∂x | z + η ∂∂y | z (cid:17) = dτ ( w (¯ z )) h TJ ( z, w (¯ z )) (cid:16) ξ ∂∂x | ¯ z − η ∂∂y | ¯ z (cid:17) , that is, h TJ ( z, τ ( w (¯ z ))) = dτ ( w (¯ z )) ◦ h TJ ( z, w (¯ z )) ◦ dc S ( z ). So (2.8) and (2.10) lead to g TJ ( τ B ( w )) = τ E ( g TJ ( w )) ∀ w ∈ B . (2.11)It follows from (2.9) and (2.11) that F λ in (2.6) satisfies F T,λ ( τ B ( w )) = τ E ( F T,λ ( w )) ∀ w ∈ B , that is, each F T,λ is equivariant with respect to the involutions in (2.7) and (2.8).Hence the restrictions F T,λ | B τ are the sections of the bundle E + → B τ . It is easy toprove that all F T,λ | B τ are Fredholm sections of index n = dim L (by Lemma 2.4 theproof of [Ho2, Prop.6]). Define Z τT,λ := { w ∈ B τ | F T,λ ( w ) = 0 } and Z τT := { ( λ, w ) ∈ [0 , × B τ | F T,λ ( w ) = 0 } . The elliptic regularity arguments show that Z τT,λ is contained in C ∞ c ( S , M ) := { w ∈ C ∞ ( S , M ) | w is contractible } . emma 2.1 For w ∈ Z τT,λ , define u : Z ∞ → M by u = w ◦ φ , where φ : Z ∞ = R × S → S \ { , ∞} , ( s, t ) e π ( s + it ) is the biholomorphism. Then u satisfies ∂ s u ( s, t ) + J ( u ( s, t )) (cid:0) ∂ t u ( s, t ) − λγ T ( s ) X H t ( u ( s, t ) (cid:1) = 0 , (2.12) E ( u ) := Z Z ∞ | ∂ s u | g J dsdt ≤ k H k ≤ k H k C . (2.13) Proof.
The equation dw ( z ) + J ( w ) ◦ dw ( z ) ◦ j + h λJ,T ( z, w ( z )) = 0 yields dw ( z )( ∂∂x ) + J ( w ) ◦ dw ( z ) ◦ j ( ∂∂x ) + h λJ,T ( z, w ( z ))( ∂∂x ) = 0 , that is ∂ x w + J ( w ) ∂ y w + λγ T ( s ) e − πs cos(2 πt )2 π ∇ J H t ( w ) − λγ T ( s ) e − πs sin(2 πt )2 π J ( w ) ∇ J H t ( w ) = 0 . Since ∂ x w + J ( w ) ∂ y w = e − πs cos(2 πt )2 π ( ∂ s u + J ( u ) ∂ t u ) − e − πs sin(2 πt )2 π J ( u )( ∂ s u + J ( u ) ∂ t u ) , it follows that u ( s, t ) = w ( e π ( s + it ) ) satisfies e − πs cos(2 πt )2 π ( ∂ s u + J ( u ) ∂ t u ) − e − πs sin(2 πt )2 π J ( u )( ∂ s u + J ( u ) ∂ t u )+ λγ T ( s ) e − πs cos(2 πt )2 π ∇ J H t ( u ) − λγ T ( s ) e − πs sin(2 πt )2 π J ( u ) ∇ J H t ( u )= e − πs cos(2 πt )2 π (cid:0) ∂ s u + J ( u ) ∂ t u + λγ T ( s ) ∇ J H t ( u ) (cid:1) + e − πs sin(2 πt )2 π J ( u ) (cid:0) ∂ s u + J ( u ) ∂ t u + λγ T ( s ) ∇ J H t ( u ) (cid:1) = 0 . This is equivalent to (2.12) since ∇ H t = − J X H t and g J ( X, J X ) = 0 for any X ∈ T M .As to (2.13), note that the contractility of w : S → M implies0 = Z S w ∗ ω = Z Z ∞ u ∗ ω = Z Z ∞ (cid:0) | ∂ s u | g J + λγ T ( s ) dH t ( ∂ s u ) (cid:1) dsdt = Z Z ∞ | ∂ s u | g J dsdt + λ Z dt Z T +1 − T − γ T ( s ) dds H t ( u )) ds = Z Z ∞ | ∂ s u | g J dsdt − λ Z dt Z T +1 − T − γ ′ T ( s ) H t ( u )) ds. ence E ( u ) = Z Z ∞ | ∂ s u | g J dsdt = λ Z dt Z TT − γ ′ T ( s ) H t ( u ( s )) ds + λ Z dt Z − T +1 − T γ ′ T ( s ) H t ( u ( s )) ds ≤ λ Z sup p H t ( p ) dt Z − T +1 − T γ ′ T ( s ) ds + λ Z inf p H t ( p ) dt Z TT − γ ′ T ( s ) ds = λ Z sup p H t ( p ) dt − λ Z inf p H t ( p ) dt ≤ λ k H k ≤ k H k C , where the first inequality is because γ ′ T ( s ) ≥ − T ≤ s ≤ − T + 1, and γ ′ T ( s ) ≤ T − ≤ s ≤ T . ✷ Lemma 2.2
Suppose that k H k < + ∞ . Then there exists a compact subset W ⊂ M such that w ( S ) ⊂ W for any ( λ, w ) ∈ Z τT , and this W can be assumed to be acompact submanifold of codimension zero and to contain K in its interior. Proof.
Define ∆( w ) := w − ( M \ K ) ⊂ S . As in Lemma 2.1, let u : Z ∞ → M bedefined by u = w ◦ φ . By (2.13) we may derive Z ∆( w ) w ∗ ω ≤ E ( u ) ≤ k H k . Then one can complete the proof as in the proof of [Lu1, Theorem 2.9] or as in theproof of Lemma 2.3(i) below. There exists also another method to prove this. Each( λ, w ) ∈ Z τT satisfies ¯ ∂ J w ( z ) + λh TJ ( z, w ( z )) = 0 for z ∈ S . Thus the “graph map”¯ w : S → S × M given by ¯ w ( z ) = ( z, w ( z )) is holomorphic with respect to the almostcomplex structure J H on S × M by J H,λ ( z, m )( X , X ) = ( iX , − λJ ( m ) ◦ h TJ ( z, m ) X + J ( m ) X ) . Then fixing a metric τ on S and applying [Sik, Prop.4.4.1] to ¯ w : S → ( S × M, J
H,λ , τ ⊕ µ ), the desired conclusion can be obtained. ✷ Let C ∞ c ( S , M ) denote the set of all contractible smooth loops x : S → M , and L ( M, τ ) := { x ∈ C ∞ c ( S , M ) | x ( − t ) = τ ( x ( t )) ∀ t ∈ R } . In the following we always assume that C ∞ ( R × S , M ) is equipped with the compactopen C ∞ -topology. Then it is not necessarily path connected even if M is so. For u ∈ C ∞ ( R × S , M ) and s ∈ R we write u ( s ) : S → M by u ( s )( t ) := u ( s, t ). It isclear that u ( s ) ∈ C ∞ c ( S , M ) ∀ s ∈ R if and only if u ( s ) ∈ C ∞ c ( S , M ) for some s ∈ R .When u ∈ C ∞ ( R × S , M ) satisfies the equation ∂ s u ( s, t ) + J ( u ( s, t ))( ∂ t u ( s, t ) − X H t ( u ( s, t )) = 0 , (2.14) e define its energy by E ( u ) = R Z ∞ | ∂ s u | g J dsdt < + ∞ . Denote by C τ := { u ∈ C ∞ ( R × S , M ) | u ( s ) ∈ L ( M, τ ) ∀ s ∈ R } , (2.15) X τ ∞ := n u ∈ C τ (cid:12)(cid:12)(cid:12) u satisfies (2 . , E ( u ) ≤ k H k o . (2.16)Both are equipped with the topology induced from C ∞ ( R × S , M ). Lemma 2.3 (i)
The compact submanifold W in Lemma 2.2 can be enlarged so that u ( R × S ) ⊂ W for all u ∈ X τ ∞ . (ii) X τ ∞ is a compact metrisable space provided that k H k < m ( M, ω, J ) . (iii) If ♯ P ( H, τ ) is finite, then for every u ∈ C τ satisfying (2.14) and E ( u ) < + ∞ there exist x + , x − ∈ P ( H, τ ) such that lim s →±∞ u ( s, t ) = x ± ( t ) and lim s →±∞ ∂ s u ( s, t ) = 0 , where both limits are uniform in the t -variable. Proof. (i) We may assume that M is noncompact. Let u ∈ C τ satisfy (2.14) and E ( u ) < + ∞ . Then Z + ∞−∞ (cid:18)Z S | ∂ t u ( s, t ) − X H t ( u ( s, t )) | g J dt (cid:19) ds = E ( u ) < + ∞ . Hence there exist sequences s + k ↑ + ∞ and s − k ↓ −∞ such thatlim k → + ∞ (cid:13)(cid:13)(cid:13)(cid:13) ∂u∂t ( s ± k , · ) − X H t ( u ( s ± k , · )) (cid:13)(cid:13)(cid:13)(cid:13) L = 0 . (2.17)Clearly, we may assume 0 < s +1 < s +2 < · · · and 0 > s − > s − > · · · . Since X H t vanishes outside the compact subset K , it follows from (2.17) that there exists aconstant C > (cid:13)(cid:13)(cid:13)(cid:13) ∂u∂t ( s ± k , · ) (cid:13)(cid:13)(cid:13)(cid:13) L ≤ C, ∀ k = 1 , , · · · . (2.18)These imply that for all t ∈ [0 , d g J ( u ( s ± k , t ) , u ( s ± k , ≤ Z t (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂t ( s ± k , τ ) (cid:12)(cid:12)(cid:12)(cid:12) g J dτ ≤ √ t Z t (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂t ( s ± k , τ ) (cid:12)(cid:12)(cid:12)(cid:12) g J dτ ! / ≤ √ C, ∀ k = 1 , , · · · . Since u ( s ± k , ∈ L , it follows that all u k ( { s ± k }× S ) are contained in a compact subsetˆ K of M . Clearly, we can assume that ˆ K is a compact submanifold of codimensionzero and with boundary and that K is contained the interior of ˆ K . ow let us assume that this u belongs to X τ ∞ . Define w = u | [ s − ,s +1 ] × S , w + k = u | [ s + k ,s + k +1 ] × S , w − k = u | [ s − k +1 ,s − k ] × S , k = 1 , , · · · . Then each connected component Σ of w − ( M \ ˆ K ) or ( w + j ) − ( M \ ˆ K ) or ( w − j ) − ( M \ ˆ K ) ( j ∈ N ) , has compact closure and is the increasing union of connected compact Riemanniansurfaces with smooth boundary Σ j , j = 1 , , · · · . For sufficiently large j we havealways ∂ Σ j ⊂ ˆ K = { x ∈ M | d ( x, ˆ K ) ≤ } . Note that the restriction of w or w + j or w − j to each Σ j is J -holomorphic and has the energy ≤ k H k . By [Sik, Prop.4.4.1] wemay deduce that this restriction has the image contained in the τ -neighborhood of( ˆ K ) δ for some τ > M, ω, µ, J ).(ii) By (i) we may assume that M is compact below. As in [HoZe, page 236], itsuffices to prove that there exists a constant C > |∇ u ( s, t ) | g J ≤ C ∀ u ∈ X τ ∞ and ( s, t ) ∈ Z ∞ . (2.19)Arguing indirectly, as on pages 236-238 in [HoZe], we find sequences ε k ↓ { t k } k ⊂ [0 ,
1] and { u k } k ⊂ X τ ∞ such that t k → t ∈ [0 , , ε k R k → + ∞ for R k = |∇ u k (0 , t k ) | g J → + ∞ , |∇ u k ( s, t ) | g J ≤ |∇ u k (0 , t k ) | g J if | s | + | t − t k | ≤ ε k , ≤ t k ≤ ) where we consider the u k as maps defined on R × R by a 1-periodic continuation inthe t -variable. It follows that the new sequence v k ∈ C ∞ ( R , M ) defined by v k ( s, t ) = u k (cid:18) sR k , t k + tR k (cid:19) for s + t ≤ ( ε k R k ) converges, in C ∞ ( R , M ), to v ∈ C ∞ ( R , M ) which satisfies |∇ v (0) | g J = 1 , sup x ∈ R |∇ v ( x ) | g J ≤ , v s + J ( v ) v t = 0 . (2.20)Denote by B ( p, r ) ⊂ R the disk centred at p and of radius r . Then Z B (0 ,ε k R k ) | ∂ s v k | g J dsdt = Z B (0 ,ε k R k ) R k (cid:12)(cid:12)(cid:12)(cid:12) ∂ s u k ( sR k , t k + tR k ) (cid:12)(cid:12)(cid:12)(cid:12) g J dsdt = Z B ((0 ,t k ) ,ε k ) (cid:12)(cid:12) ∂ s u k ( s, t ) (cid:12)(cid:12) g J dsdt ≤ E ( u k ) ≤ k H k for sufficiently large k (so that ε k < / Z C | ∂ s v | g J dsdt ≤ k H k < m ( M, ω, J ) . owever, (2.20) and Gromov’s removable singularity allow us to extend v to a non-constant J -holomorphic sphere v ∞ : S → M with Z S v ∗∞ = Z C | ∂ s v | g J dsdt ≤ k H k < m ( M, ω, J )which contradicts to the definition of m ( M, ω, J ). (2.19) is proved.(iii). Since the condition ♯ P ( H ) < + ∞ is actually sufficient for “(i) ⇒ (ii)” in theproof of [Sa, Prop.1.21], we may complete the proof with the same reason. ✷ Lemma 2.4
Suppose that k H k < m ( M, ω, J ) . Then Z τT,λ and Z τT are compact in C ∞ ( S , M ) and [0 , × C ∞ ( S , M ) , respectively. Proof . By Lemma 2.2 we may assume M to be compact. Using (2.13) we can, asin the proof of Lemma 2.3, prove that there exists a constant C T > λ, w ) ∈ Z τT and u = w ◦ φ : Z ∞ → M as in Lemma 2.1,sup ( s,t ) ∈ Z ∞ |∇ u ( s, t ) | g J ≤ C T . (2.21)It implies that for each multi-index α ∈ N one can find a constant C T,α > u as above, sup ( s,t ) ∈ Z ∞ | ( D α u )( s, t ) | g J ≤ C T,α . (2.22)Now suppose that Z τT is noncompact. Then there exists sequences { ( λ k , w k ) } k ⊂Z τT and { z k } k ⊂ S = C P such that λ k → λ and | dw k ( z k ) | = k dw k k := max z ∈ S | dw k ( z ) | → + ∞ , (2.23)where | dw k ( z ) | is the norm of the tangent map dw k ( z ) : T z S → T w k ( z ) M induced by g J and the standard Riemannian metric on S . We may assume that z k → z ∈ S = C P . By (2.21) this z must be 0 or ∞ in C P . (Otherwise, passing to a subsequencewe may assume inf k d ( z k , > k d ( z k , ∞ ) >
0. Thus there exists a large T > s k , t k ) = φ − ( z k ) ⊂ S × [ − T , T ] for all k . It follows from (2.23)that u k = w k ◦ φ satisfies | du k ( s k , t k ) | → ∞ , which contradicts to (2.22).) By theGromov compactness theorem the sequence { w k } k has a subsequence, still denoted by { w k } k , converges weakly to a connected union of N ≥ J -holomorphicspheres v , · · · , v N : S → M and a smooth map w ∞ : S = C P → M satisfying¯ ∂ J w + λ g TJ ( w ) = 0 . (2.24)In particular, [ v ♯ · · · ♯v N ♯w ∞ ] = 0 ∈ π ( M ). Let u ∞ = w ∞ ◦ φ : Z ∞ → M . Then as n the proof of Lemma 2.1 we have0 = N X k =1 Z S v ∗ k ω + Z S w ∗∞ ω = N X k =1 Z S v ∗ k ω + Z Z ∞ u ∗∞ ω = N X k =1 Z S v ∗ k ω + Z Z ∞ (cid:0) | ∂ s u ∞ | g J + λ γ T ( s ) dH t ( ∂ s u ∞ ) (cid:1) dsdt = N X k =1 Z S v ∗ k ω + Z Z ∞ | ∂ s u ∞ | g J dsdt + λ Z dt Z T +1 − T − γ T ( s ) dds H t ( u ∞ )) ds. It follows that m ( M, ω, J ) ≤ N m ( M, ω, J ) + E ( u ∞ ) ≤ N X k =1 Z S v ∗ k ω + E ( u ∞ )= − λ Z dt Z T +1 − T − γ T ( s ) dds H t ( u ∞ )) ds ≤ λ k H k ≤ k H k < m ( M, ω, J )as in the proof of Lemma 2.1. This contradiction gives the desired conclusion. ✷ For
T > X τT := { u ∈ C ∞ ( Z T , M ) | u (0) ∈ L ( M, τ ) and Z Z T | ∂ s u | g J ≤ k H k} ,X τ,JT := { u ∈ X τT | ∂ s u + J ( u ) ∂ t u + ∇ H t ( u ) = 0 on Z T } . As in the proofs of (i)-(ii) of Lemma 2.3 we may get
Lemma 2.5
The compact submanifold W in Lemma 2.3 can be furthermore enlargedso that u ( Z T ) ⊂ W for all u ∈ X τT . Moreover, there exists a constant e C > suchthat for every T > , sup n |∇ u ( s, t ) | g J (cid:12)(cid:12)(cid:12) ( s, t ) ∈ Z T − o ≤ e C ∀ u ∈ X τT . (2.25)Let γ T ( s ) be as in (2.5). Define σ T : X τT → C τ , u σ T ( u ) (2.26)by σ T ( u )( s, t ) = u ( γ T ( s ) s, t ). Then σ T ( u )( s, t ) = u ( s, t ) ∀ ( s, t ) ∈ Z T +1 . Theorem 2.6
Suppose that k H k < m ( M, ω, J ) . Then for a given open neighborhood U of X τ ∞ in C τ there exists T > such that σ T ( X τ,JT ) ⊂ U for any T ≥ T . Furthermore, this T can be enlarged so that σ T ( u | Z T ) ∈ U ∀ T > T for any u = w ◦ φ with w ∈ Z τT, , where Z τT, is as above Lemma 2.1. roof . Since (2.25) implies that for each multi-index α ∈ N one can find a constant e C α > T > n | ( D α u )( s, t ) | g J (cid:12)(cid:12)(cid:12) ( s, t ) ∈ Z T − o ≤ e C α ∀ u ∈ X τT . (2.27)As in the arguments on pages 244-245 of [HoZe], suppose that there exist an openneighborhood U of X τ ∞ in C τ and sequences T k → + ∞ and u k ∈ X τT k such that u k / ∈ U for all k . From (2.27) we may choose a subsequence { u k j } j of { u k } k such that u k j converges to u in C ∞ loc ( R × S , M ). Clearly, u satisfies ∂ s u + J ( u ) ∂ t u + ∇ H t ( u ) = 0 on Z ∞ ,u (0 , · ) ∈ C ∞ c ( S , M ) and u ( s, − t ) = τ ( u ( s, t )) ∀ ( s, t ) ∈ Z ∞ ,E ( u ) = Z Z ∞ | ∂ s u ( s, t ) | g J dsdt ≤ k H k . That is, u ∈ X τ ∞ . Moreover, all u k j belong to the closed subset C τ \ U of C τ . Hence u / ∈ U , which contradicts u ∈ X τ ∞ ⊂ U . ✷ For C τ in (2.15) we define an evaluation map π : C τ → L, u u (0 , , (2.28)and denote ˇ H ∗ by the Alexander-Spanier cohomology. Then Theorem 1.3 can bederived from the following result. Theorem 2.7
Under the assumptions, for every open neighborhood U of X τ ∞ in C τ the restriction π | U induces an injection ( π | U ) ∗ : ˇ H ∗ ( L, Z ) → ˇ H ∗ ( U, Z ) . So the continuity property of the Alexander-Spanier cohomology implies π | X τ ∞ : ˇ H ∗ ( L, Z ) → ˇ H ∗ ( X τ ∞ , Z ) is injective. If L is orientable, π | X τ ∞ : ˇ H ∗ ( L, Z ) → ˇ H ∗ ( X τ ∞ , Z ) is also injective. Consequently, Cuplength Z ( X τ ∞ ) ≥ Cuplength Z ( L ), and Cuplength Z ( X τ ∞ ) ≥ Cuplength Z ( L ) if L is orientable. Proof of Theorem 1.3 . Clearly, we may assume P ( H, τ ) to be a finite set underthe assumptions of Theorem 1.3. Consider the closed 1-form α on L ( M ) given by α x ( ξ ) = Z ω ( ˙ x ( t ) − X H t ( x ( t ) , ξ ( t )) dt ∀ ( x, ξ ) ∈ T L ( M, τ ) . (2.29)It restricts to a closed 1-form α τ on L ( M, τ ). Let φ ω : π ( M ) → R be the homomor-phism defined by integration of ω . Denote by ˜ L ( M ) the set of all pairs ˜ x = ( x, [ w ]),where x ∈ L ( M ) and [ w ] is an equivalence class of smooth discs w : D → M with w ( e πit ) = x ( t ) ∀ t for the equivalence relation ∼ : w ∼ w ′ if and only if the sphere ♯ ¯ w ′ being vanished by ω . Then Π : ˜ L ( M ) → L ( M ) , ( x, [ w ]) x is a covering whosedesk group is the quotient Γ( ω ) = π ( M ) / ker( φ ω ). The symplectic action functional A H : ˜ L ( M ) → R , ( x, [ w ])
7→ − Z D w ∗ ω + Z H ( t, x ( t )) dt (2.30)is a primitive of Π ∗ α , i.e., d A H ( x, [ w ])[ ξ ] = (Π ∗ α ) ( x, [ w ]) ( ξ ) = α x ( ξ ) for any ξ ∈ C ∞ ( x ∗ T M ). Let A H,τ be the restriction of A H to ˜ L ( M, τ ) := Π − ( L ( M, τ )). Then d A H,τ = α τ .By the assumption the rationality index r ω of ( M, ω ) is positive. If r ω = + ∞ ,i.e., ω | π ( M ) = 0, then ˜ L ( M ) = L ( M ). If r ω ∈ (0 , + ∞ ), A H,τ descends to a map A ∗ H,τ : L ( M, τ ) → R /r ω Z , which is a primitive of α τ . Let p : R → R /r ω Z be thecanonical projection in the latter case. Define a H : L ( M, τ ) → R by a H ( x ) = ( A H,τ ( x ) if r ω = + ∞ , ( p | [0 ,r ω ) ) − ◦ A ∗ H,τ ( x ) if r ω ∈ (0 , + ∞ ) . (2.31)This is continuous and satisfies dds a H ( u ( s )) = Z | ∂ s u ( s, t ) | g J dt ∀ u ∈ X τ ∞ , (2.32)where u ( s )( t ) = u ( s, t ). By Lemma 2 on [HoZe, page 225] (or its proof) dds a H ( u ( s )) | s = s = 0 for some s ∈ R implies that u ( s ) = u ( s ) ∀ s ∈ R and x := u ( s ) = u ( s , · ) belongs to P ( H, τ ). Thisshows that the natural flow on the compact metric space X τ ∞ defined byΦ : R × X τ ∞ → X τ ∞ : ( σ, u ) σ · u, (2.33)where ( σ · u )( s, t ) = u ( σ + s, t ), is gradient-like and has a H as a Ljapunov function.Thus Corollary on [CoZe, page 42] yields ♯ P ( H, τ ) ≥ Cuplength Z ( L ) (or ≥ Cuplength Z ( L ) if L is orientable). (2.34)This and Theorem 2.7 give the desired conclusion immediately. ✷ In order to prove this result let us recall that a
Banach Fredholm bundle of index r and with compact zero sets is a triple ( X, E, S ) consisting of a Banach manifold X , a Banach vector bundle E → X and a Fredholm section S of index r and withcompact zero sets. If the determinant bundle det( S ) → Z ( S ) is oriented, i.e., it istrivializable and is given a continuous section nowhere zero, we said ( X, E, S ) to be oriented . One has the following standard result (cf. [LuT, Theorem 1.5]). heorem 3.1 Let ( X, E, S ) be a Banach Fredholm bundle of index r . Then thereexist finitely many smooth sections σ , σ , · · · , σ m of the bundle E → X such that forthe smooth sections Φ : X × R m → Π ∗ E, ( y, t ) S ( y ) + m X i =1 t i σ i ( y ) , Φ t : X → E, y S ( y ) + m X i =1 t i σ i ( y ) , where t = ( t , · · · , t m ) ∈ R m and Π is the projection to the first factor of X × R m ,the following holds: There exist an open neighborhood W ⊂ O ( Z ( S )) of Z ( S ) and asmall ε > such that: (A) The zero locus of Φ in Cl ( W × B ε ( R m )) is compact. Consequently, for anygiven small open neighborhood U of Z ( S ) there exists a ǫ ∈ (0 , ε ] such that Cl ( W ) ∩ Φ − t (0) ⊂ U for any t ∈ B ǫ ( R m ) . In particular, each set W ∩ Φ − t (0) is compact for t ∈ B ε ( R m ) sufficiently small. (B) The restriction of Φ to W × B ε ( R m ) is (strong) Fredholm and also transversalto the zero section. So U ε := { ( y, t ) ∈ W × B ε ( R m ) | Φ( y, t ) = 0 } is a smooth manifold of dimension m + Ind( S ) , and for t ∈ B ε ( R m ) the section Φ t | W : X → E is transversal to the zero section if and only if t is a regularvalue of the (proper) projection P ε : U ε → B ε ( R m ) , ( y, t ) t , and Φ − t (0) ∩ W = P − ε ( t ) . (Specially, t = 0 is a regular value of P ε if S istransversal to the zero section). Then the Sard theorem yields a residual subset B ε ( R m ) res ⊂ B ε ( R m ) such that: (B.1) For each t ∈ B ε ( R m ) res the set (Φ t | W ) − (0) ≈ (Φ t | W ) − (0) × { t } = P − ε ( t ) is a compact smooth manifold of dimension Ind( S ) and all k -boundaries ∂ k (Φ t | W ) − (0) = ( ∂ k X ) ∩ (Φ t | W ) − (0) for k = 1 , , · · · . Specially, if Z ( S ) ⊂ Int( X ) one can shrink ε > so that (Φ t | W ) − (0) is a closed manifold for each t ∈ B ε ( R m ) res . (B.2) If the Banach Fredholm bundle ( X, E, S ) is oriented , i.e., the deter-minant bundle det( DS ) → Z ( S ) is given a nowhere vanishing continuoussection over Z ( S ) , then it determines an orientation on U ε . In particular,it induces a natural orientation on every (Φ t | W ) − (0) for t ∈ B ε ( R m ) res . (B.3) For any l ∈ N and two different t (1) , t (2) ∈ B ε ( R m ) res the smooth mani-folds (Φ t (1) | W ) − (0) and (Φ t (2) | W ) − (0) are cobordant in the sense that fora generic C l -path γ : [0 , → B ε ( R m ) with γ (0) = t (1) and γ (1) = t (2) theset Φ − ( γ ) := ∪ t ∈ [0 , { t } × (Φ γ ( t ) | W ) − (0) s a compact smooth manifold with boundary { } × (Φ t (1) | W ) − (0) ∪ ( −{ } × (Φ t (2) | W ) − (0)) . In particular, if Z ( S ) ⊂ Int( X ) and ε > is suitably shrunk so that (Φ t | W ) − (0) ⊂ Int( X ) for any t ∈ B ε ( R m ) then Φ − ( γ ) has no corners. (B.4) The cobordant class of the manifold (Φ t | W ) − (0) above is independent ofall related choices. Now we furthermore assume that N is a connected manifold of dimension r and f : X → N is a smooth map. When X has no boundary, by Theorem 3.1(B.1), foreach t ∈ B ε ( R m ) res the section Φ t : X → E is transversal to the zero section andthe set (Φ t | W ) − (0) ⊂ X is a compact smooth manifold of dimension r and withoutboundary. So we may consider the Z -Brouwer degreedeg Z ( f | (Φ t | W ) − (0) )of the restriction f | (Φ t | W ) − (0) : (Φ t | W ) − (0) → N . The elementary properties andTheorem 3.1(B.3) show that deg Z ( f | (Φ t | W ) − (0) ) ∈ Z is independent of the choice of t ∈ B ε ( R m ) res . Moreover, it is claimed in Theorem 3.1(B.4) that the cobordant classof the manifold (Φ t | W ) − (0) above is independent of all related choices. Namely,suppose that σ ′ , σ ′ , · · · , σ ′ m ′ are another group of smooth sections of the bundle E → X such that the sectionΨ : W ′ × B ε ′ ( R m ′ ) → Π ∗ E, ( y, t ′ ) S ( y ) + m ′ X i =1 t ′ i σ ′ i ( y ) , is Fredholm and transversal to the zero and that the set Ψ − t ′ (0) is compact foreach t ′ ∈ B ε ′ ( R m ′ ), where the section Ψ t ′ : W ′ → E is given by Ψ t ′ ( y ) = Ψ( y, t ′ ).Let B ε ′ ( R m ′ ) res ⊂ B ε ′ ( R m ′ ) be the corresponding residual subset such that for each t ′ ∈ B ε ′ ( R m ′ ) res the section Ψ t ′ is transversal to the zero section and that any two t ′ , s ′ ∈ B ε ′ ( R m ′ ) res yield cobordant manifolds (Ψ t ′ ) − (0) and (Ψ s ′ ) − (0). Then it wasshown in the proof of [LuT, Theorem 1.5(B.4)] that there exist a compact submanifoldΘ − t , t ′ ) (0) ⊂ X × [0 ,
1] of dimension r + 1 for any t ∈ B regε ( R m ) and t ′ ∈ B regε ′ ( R m ′ )such that ∂ Θ − t , t ′ ) (0) = (Φ t | W ) − (0) × { } ∪ Ψ − t ′ (0) × { } . This implies thatdeg Z ( f | (Φ t | W ) − (0) ) = deg Z ( f | (Ψ t ′ | W′ ) − (0) ) . Hence we have a well-defined Z -value degreedeg Z ( f, N, X, E, S ) := deg Z ( f | (Φ t | W ) − (0) ) ∈ Z (3.1)for any t ∈ B regε ( R m ), and call it Z -degree of f : X → N relative to ( X, E, S ) . Of course, when both (
X, E, S ) and N are oriented, we may define Z - degree of f : X → N relative to ( X, E, S ) . Let { S λ } λ ∈ [0 , be a smooth family of smooth Fredholm sections of the bundle E → X of index r and with compact zero sets. Then we can still choose finitely any smooth sections σ , σ , · · · , σ m of the bundle E → X , an open neighborhood W λ of each Z ( S λ ) ⊂ X , and a residual subset B ε ( R m ) res for some small ε >
0, suchthat for each t ∈ B ε ( R m ) res the restrictions of the smooth sectionsΦ t : X → E, y S ( y ) + m X i =1 t i σ i ( y ) , Φ t : X → E, y S ( y ) + m X i =1 t i σ i ( y ) , Φ t : X × [0 , → Π ∗ E, ( y, λ ) S λ ( y ) + m X i =1 t i σ i ( y )to W , W and W = ∪ λ ∈ [0 , W λ are transversal to the zero sections respectively. Inparticular, we get ∂ (Φ t | W ) − (0) = (Φ t | W ) − (0) × { } [ (Φ t | W ) − (0) × { } . It follows that deg Z ( f, N, X, E, S ) = deg Z ( f, N, X, E, S ) (3.2)and thus deg Z ( f, N, X, E, S λ ) is independent of λ ∈ [0 , X, E, S λ ) and N are oriented, deg Z ( f, N, X, E, S λ ) is independent of λ ∈ [0 ,
1] as well.
Proof of Theorem 2.7.
Define the evaluation mapΘ : B τ → L, u u (1) , (3.3)where 1 ∈ C ⊂ C ∪{∞} = S . Applying the arguments above to the Banach Fredholmbundle ( B τ , E + , F T,λ | B τ ), λ ∈ [0 , Z (Θ , L, B τ , E + , F T, | B τ ) = deg Z (Θ , L, B τ , E + , F T, | B τ ) (3.4)by (3.2). Since each w ∈ B is contractible, Z τT, = ( F T, | B τ ) − (0 E + ) precisely consistsof the constant maps S → L . It is easily proved that F T, | B τ : B τ → E + is transversalto the zero section, and that (3.1) yieldsdeg Z (Θ , L, B τ , E + , F T, | B τ ) = 1 . (3.5)Let F be a smooth perturbation section of F T, | B τ as Φ t above. Choose l ∈ L to bea regular value for the evaluationsΘ | F − (0 E + ) : F − (0 E + ) → L. Then (3.4) and (3.5) show thatdeg Z (Θ | F − (0 E + ) , l ) = 1 . Hence Θ | F − (0 E + ) : F − (0 E + ) → L induces an injection map(Θ | F − (0 E + ) ) ∗ : ˇ H ∗ ( L, Z ) → ˇ H ∗ ( F − (0 E + ) , Z ) . (3.6) ote that F − (0 E + ) can be chosen so close to Z τT, that it is contained a givensmall neighborhood of Z τT, for which Theorem 2.6 implies for T ≥ T > σ T ( u | Z T ) ∈ U ∀ w ∈ F − (0 E + ) and u = w ◦ φ. (3.7)Here we use F − (0 E + ) ⊂ C ∞ c ( S , M ) due to the arguments above Lemma 2.1. DefineΞ : F − (0 E + ) → X τT,d , w u | Z T for u = w ◦ φ, by (2.26), (2.28), (3.3) and (3.7) it is easy to see that we have for T ≥ T thecommutative diagram X τT ✲ σ T ✻ Ξ U ❄ π | U F − (0 E + ) ✲ Θ | F − (0 E + ) L By (3.6) we get the injectiveness of the map( π | U ) ∗ : ˇ H ∗ ( L, Z ) → ˇ H ∗ ( U, Z ) . If L is orientable, the Banach Fredholm bundles ( B τ , E + , F T, | B τ ), and therefore( B τ , E + , F T,λ | B τ ), λ ∈ [0 , Z -degreedeg Z (Θ , L, B τ , E + , F T,λ | B τ ) and get deg Z (Θ , L, B τ , E + , F T,λ | B τ ) ∈ { , − } . The de-sired conclusion follows immediately. ✷ Example 4.1
Consider the torus T n = R n / Z n with the standard symplectic form ω = dx ∧ dy ). There exist three natural anti-symplectic involutions on it given by • τ : T n → T n , [ x, y ] [ − x, y ], • τ : T n → T n , [ x, y ] [ x, − y ], • τ : T n → T n , [ x, y ] [ y, x ].Clearly, Fix( τ ) = [0] × T n ⊂ T n , Fix( τ ) = T n × [0] ⊂ T n and Fix( τ ) = { [ x, x ] ∈ T n | x ∈ R n } . Let H ∈ C ∞ ( R × R n , R ) be 1-periodic in all its variables so that itmay be viewed as a Hamiltonian function on the standard torus which is 1-periodicin time. Corresponding to the three cases above, suppose that Hamiltonian H also,respectively, satisfies • H ( − t, x, y ) = H ( t, − x, y ) for any t ∈ R and z = ( x, y ) ∈ R n , • H ( − t, x, y ) = H ( t, x, − y )) for any t ∈ R and z = ( x, y ) ∈ R n , • H ( − t, x, y ) = H ( t, y, x ) for any t ∈ R and z = ( x, y ) ∈ R n .Then in each case Theorem 1.3 gives at least n + 1 contractible 1-periodic solutions γ : R → T n of ˙ γ ( t ) = X H ( t, γ ( t )) satisfying γ ( − t ) = τ i ( γ ( t )) for any t ∈ R , i = 1 , , espectively. It is the contractibility of γ that there exists a lift loop z = ( x, y ) : R / Z → R n of it satisfying the associated Hamiltonian system on R n ˙ z = J ∇ H ( t, z ) with J = − I n I n ! and the following conditions • x ( − t ) + x ( t ) ∈ Z n and y ( − t ) − y ( t ) ∈ Z n for any t ∈ R , • x ( − t ) − x ( t ) ∈ Z n and y ( − t ) + y ( t ) ∈ Z n for any t ∈ R , • x ( − t ) − y ( t ) ∈ Z n and y ( − t ) − x ( t ) ∈ Z n for any t ∈ R ,respectively.Clearly, the conclusions in Example 4.1 cannot be derived from [CoZe, Th.1]though the latter yields at 2 n + 1 periodic solutions of ˙ z = J ∇ H ( t, z ) of period 1. Example 4.2 (i)
Consider the standard complex projective space C P n with theFubini-Study form ω F S satisfying R C P ω FS = π . Then the rationality index of( C P n , ω FS ) is equal to π . Let H ∈ C ∞ ( R × C P n , R ) be 1-periodic in the first variable,and also satisfy H ( − t, [ z ]) = H ( t, σ ([ z ])) for any t ∈ R and [ z ] ∈ C P n , where σ isthe standard complex conjugation on C P n with Fix( σ ) = R P n . Then the associatedHamiltonian system ˙ z = X H ( t, z ) on ( C P n , ω FS ) has at least n + 1 contractible pe-riodic solutions z : R → C P n of period 1 satisfying z ( − t ) = σ ( z ( t )) for any t ∈ R provided the Hofer norm k H k < π by Theorem 1.3. (Actually, the final restriction“ k H k < π ” may be removed out with Fortune’s method in [Fo].) (ii) Let (
P, ω ) be a simply connected closed symplectic manifold of dimension 4 andwith c ( T P ) | π ( P ) = 0. By the Hurewicz isomorphism theorem and the Poincar´e dualtheorem there exists a class A ∈ π ( P ) such that β ( A ) >
0. So r ω ∈ [0 , + ∞ ). It easilyfollows from [McSa2, Theorem 3.1.5] that for generic J ∈ J ( P, ω ) there is no noncon-stant J -holomorphic spheres in P , and thus m ( P, ω, J ) = + ∞ . Moreover, if ( P, ω ) isalso real symplectic, by [FuOOO, Proposition 11.10] for generic J ∈ R J ( P, ω ) thereis no nonconstant J -holomorphic sphere in P and so m ( P, ω, J ) = + ∞ . A well-knownexample of such real symplectic manifolds is the K X = (cid:8) [ z : · · · : z ] ∈ C P (cid:12)(cid:12) X j =0 z j = 0 (cid:9) with the canonical symplectic structure ω can induced by the form ω F S on C P n as in(i) above and with the anti-symplectic involution induced by the standard complexconjugation on C P n (cf. [McSa1, Example 4.27]). Hence π ≤ r ω can < + ∞ . Note thatthe real part of X is empty! (iii) A symplectic manifold (
M, ω ) of dimension 2 n is said to be negative monotone if c ( T M ) | π ( M ) = λ · ω | π ( M ) for some negative constant λ , and semipositive if either ω ( M ) | π ( M ) = µ · c | π ( M ) for some constant µ ≥
0, or c | π ( M ) = 0 or the minimalChern number N ≥ n −
2, see [McSa2, Exercise 6.4.3]. Here the minimal Chernnumber N of ( M, ω ) is the positive generator of c ( M ) (cid:0) π ( M ) (cid:1) if c | π ( M ) = 0, and ∞ if c | π ( M ) = 0. Note that a simply connected and closed symplectic manifoldhas always finite rationality index by the Hurewicz isomorphism theorem and theuniversal coefficient theorem. In a negative monotone symplectic manifold ( M, ω )with minimal Chern number N ≥ dim M/
2, for generic J ∈ J ( M, ω ) there is nononconstant J -holomorphic sphere and hence m ( M, ω, J ) = + ∞ by [McSa2], and m ( M, ω, J ) = + ∞ for generic J ∈ R J ( M, ω ) by [FuOOO, Proposition 11.10] if(
M, ω ) is also real.
Here are some concrete examples , which were in detailsdiscussed in [Laz, Appendix A]. For an integer n ≥ d let M n,d = (cid:8) [ z : · · · : z n ] ∈ C P n (cid:12)(cid:12) X j =0 z dj = 0 (cid:9) equipped with a canonical symplectic structure ω n,d induced by the form ω F S on C P n as in (i) above. Then π ≤ r ω n,d < + ∞ . The standard complex conjugationon C P n induces an anti-symplectic τ on M n,d with Fix( τ ) = M n,d ∩ R P n whichis homeomorphic to R P n − . So if H ∈ C ∞ ( R × M n,d , R ) is 1-periodic in the firstvariable, and also satisfy H ( − t, [ z ]) = H ( t, τ ([ z ])) for any t ∈ R and [ z ] ∈ M n,d , wehave ♯ P ( H, τ ) ≥ n provided k H k < π .It was shown in [Laz, Appendix A] that M n,d is simply connected, has a minimalChern number N n,d = | n + 1 − d | , and satisfies c ( M n,d ) | π ( M n,d ) = n + 1 − dr · ω n,d | π ( M n,d ) for some r >
0. Since dim M n,d = 2 n − M n,d is negative monotone if and only if n + 1 < d , and N n,d ≥ dim M n,d if and only if d ≥ n or d = 1. Hence the argumentsabove show that each M n,d with n ≥ d ≥ n or d = 1 satisfies m ( M n,d , ω n,d , J ) = + ∞ for generic J ∈ J ( M n,d , ω n,d ) and for generic J ∈ R J ( M n,d , ω n,d ). Nonsingularalgebraic subvarieties of C P n defined by real equations may provide more examples. Our programme [Lu3] is to construct a real Floer homology
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