fru rv

A Superquadratic Indefinite Elliptic System and its Morse-Conley-Floer Homology Sigurd Angenent

Robertus van der Vorst

March 12, 1998 Abstract

R We study critical points of the indefinite functional fH (u; v ) = u v H (x; u; v ) dx by applying Floer’s homology construction to the ordinary gradient flow of the functional f on a suitable Sobolev space. One of our main observations is that even though this flow is well posed in both time directions and lacks any kind of smoothing property one can still obtain compactness of connecting orbit spaces and thus define the Floer homology for fH .

fr
r ,

g

Contents 1 Introduction 2 The Gradient Flow 2.1 The L2 gradient. . . 2.2 The E gradient flow. 2.3 The flow on X . . . 2.4 The perturbed flows.

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3 Compactness 3.1 Hypotheses on H . . . . . . . . . . . . . . . . . . . 3.2 Boundedness in E for the autonomous equation. . 3.3 Boundedness in E for nonautonomous equations. 3.3.1 Hypotheses for time dependent H , K . . . . 3.4 Boundedness in H¨older spaces. . . . . . . . . . . . 3.5 Action bounds and a Palais Smale condition. . . . . 3.5.1 Corollaries. . . . . . . . . . . . . . . . .

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4 The Morse index and Fredholm properties 4.1 The linearized equation. . . . . . . . . . . . 4.2 Cocycle Property and the renormalized index. 4.3 The spectral flow formula. . . . . . . . . . . 4.4 Upper and lower indices. . . . . . . . . . . .

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5 The space MH;K (z, ; z+ ) and its closure 5.1 Broken orbits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 MH;K (z, ; z+ ) under the assumption of transversality. . . . . . . . .

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6 Gluing

25

7 Generic flows 7.1 Generically a Morse function. . . . . . . . . . . . . . . . . . . . . . 7.2 Generic Kupka-Smale property. . . . . . . . . . . . . . . . . . . . .

26 26 27

8 Floer homology 8.1 Definition in the transverse case. . . . . . . 8.2 Morse relations and Poincar´e polynomials. 8.3 Index pairs. . . . . . . . . . . . . . . . . . 8.4 Continuation. . . . . . . . . . . . . . . . . 8.4.1 Stable isolating neighborhoods. . . 8.4.2 Local continuation. . . . . . . . . 8.4.3 Global continuation. . . . . . . . .

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28 28 30 30 31 32 33 35

9 Existence results for critical points 9.1 The index of constant coefficient linearizations. 9.2 “,” type Hamiltonians. . . . . . . . . . . . . 9.2.1 Existence of a nontrivial solution. . . . 9.2.2 Even “,” type Hamiltonians. . . . . . 9.3 “+” type Hamiltonians. . . . . . . . . . . . . 9.3.1 Existence of a nontrivial solution. . . . 9.3.2 Infinitely many solutions if H is even. 9.4 Estimating the index of critical points. . . . . .

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36 36 37 37 38 39 39 39 40

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1 Introduction In this paper we will use Floer’s homology construction [10] to study the critical points of functionals of the type

fH (u; v) =

Z



fru
rv , H (x; u; v)g dx

(1)

where H :  R2 ! R is a C 2 function, the Hamiltonian, and  Rn is a bounded domain with smooth boundary. The functional fH is strongly indefinite,1 but its Euler-Lagrange equations are elliptic: 


v + Hu (x; u; v) = 0;
u + Hv (x; u; v) = 0:

1 At each critical point z of f there are infinite dimensional subspaces E  H is positive and negative definite respectively

2

(2)

 (C 1 )2 on which d2 fH (z)

We will always assume that the functions u; v vanish on @ , i.e. we will look for solutions of the system (2) with Dirichlet boundary conditions. In section 2 we give precise conditions on H for which our theory works. We consider two different classes of Hamiltonians. These two classes can best be illustrated by the following two special representatives

p+1 q+1 H (x; u; v) = a(x) jpuj+ 1  b(x) jqvj+ 1 ;

(3)

 ) are strictly positive. For such H the Euler Lagrange equations are where a; b 2 C 0 (




v + a(x)up = 0;
u  b(x)vq = 0:

(4)

Here and elsewhere we shall write up for the odd extension of the power function, i.e. up def = ujujp,1 . We will restrict p and q to the subcritical range

1 1 n,2 p+1 + q+1 > n : This turns out to be the range of p, q for which boundedness of Z

,



(5)




jujp+1 + jvjq+1 dx

implies L1 and from there C 2; bounds for the solutions u and v . Inequality (5) is sharp in the following sense. If p and q satisfy the reversed inequality it follows from results by Mitidieri [15] and by Van der Vorst [27] that (2) has no positive solutions. Problem (2) has previously been studied using the variational structure by Felmer and DeFigueiredo [11], and also by Hulshof and Van der Vorst [13]. In these papers a minimax theorem for strongly indefinite functionals due to Benci and Rabinowitz [6] is employed to obtain existence of at least one non-trivial solution to (2). See also Szulkin’s work [24, 14] for a Morse theory for strongly indefinite functionals based on different cohomology theories. The functional fH is very similar to the action functional Z

AH (p; q) = fpq0 , H (p; q; t)gdt whose critical points are periodic orbits of the Hamiltonian system with Hamiltonian H (p; q; t). Floer’s study of the unregularized L2 gradient flow of this functional led to a very succesfull Morse theory for AH . Motivated by this success we tried to develop an analogous Morse theory for the functional fH . Our main conclusion is that this can indeed be done, but that one must not use the unregularized L2 gradient flow, but rather a regular gradient flow in so me fractional power space E = D((,
) )  D((,
)1, ). This is perhaps surprising since the gradient flow in the space E is locally well posed in both time directions and has no smoothing properties, in contrast 3

with the L2 gradient flow which leads to an elliptic initial value problem all of whose solutions are smooth. The best existence result which we have obtained by using Floer’s method is as follows. Theorem 1 Let H (x; u; v ) = a(x) jupj+1 + b(x) jvqj+1 with a and b strictly positive continuous functions. Then fH has an unbounded sequence (in L1 ) of critical points (uk ; vk ) 2 C01 (  )  C01 (  ). If  Rn with n  3 then the generalized Morse indices , (zk ) of the critical points are also unbounded. p+1

q+1

As is to be expected from a Floer-homology approach, we define a generalized Morse index for critical points of fH and obtain Morse inequalities for those H whose associated functional fH is a Morse function. The analogue of theorem 1 can also be obtained for a larger class of non-linearities H , see section 9. To prove that the generalized Morse indices of the critical points in theorem 1 are unbounded we must prove an a priori estimate for solutions in terms of their Morse index analogous to a result of Bahri and Lions [4]. For the current setting we derive the estimate in [2], and we summarize it in section 9.4. In section 2 we discuss the various gradient flows for fH one can choose from. Section 3 establishes compactness of the set of bounded trajectories properties for the “E gradient flow,” which motivates our preference for this flow. In section 4 we discuss the generalized Morse index. The sections 5 upto 8 deal with the definition and continuation properties of the Floer homology groups for isolating neighborhoods for the E gradient flow of fH . Finally in section 9 we use this theory to derive various existence results.

Acknowledgements The first author was supported by a Vilas-fellowship from the UW-Madison and the second author by the Netherlands Organization for Scientific Research, NWO and by grants ARO DAAH-0493G0199 and NIST G-06-605. We would also like to thank the referee for his/her careful reading of the previous version of the manuscript, and in particular for pointing out that we had overlooked the separability hypothesis in the Sard-Smale theorem.

2 The Gradient Flow To define Floer homology groups we must consider a gradient flow for fH , i.e. we must specify an inner product with respect to which we take the gradient of fH .

2.1 The L2 gradient. The L2 gradient of fH is given by

rL fH (z ) = ,@
z , Hz (x; z ); 2

4



where



@
=
0
0 :

(6)

@z = @ z + H (x; z ): z @t


(7)

The L2 gradient flow is therefore equivalent to the following system of PDE’s

The initial value problem ,
for this system is ill-posed in both time directions, as one sees by writing z = uv and noting that w = u  v satisfy forward and backward semilinear heat equations, respectively. Parabolic regularity theory then also implies that bounded (i.e. L1 ) solutions to (7) which are defined for all t 2 R actually have uniformly bounded derivatives. Thus to obtain the necessary compactness one would have to obtain L1 estimates for all connecting orbits between critical points of fH . Originally we found such estimates, but our proof required a stronger hypothesis on the exponents p, q than (5). We then found that our arguments also worked for the ordinary E gradient flow of fH , which we now describe.

2.2 The E gradient flow. We will choose the innerproduct of the Hilbert space

E def = D((,
) )  D((,
)1, )

(8)

where D((,
) ) is the domain of the fractional power of the L2 -realization of the Laplacian with Dirichlet boundary conditions (see [11] and [13] for a more detailed discussion). The number  2 (0; 1) must satisfy

4(1 , ) : q < nn + , 4(1 , )

4 ; p < nn + , 4

(9)

Such an  exists if and only if p and q are subcritical (see (5)). The Sobolev embedding theorems imply that E is continuously embedded in Lp+1 ( )  Lq+1 ( ). Hence if the function H is smooth, and asymptotically grows like a(x)jujp+1  b(x)jvjq+1 , then fH will be a C 2 functional on E , and its E gradient flow will be will a defined local flow. To state this with more precision we observe that the L2 ( )  L2 ( ) and E inner products are related by

(z; z 0)L = (D z; z 0)E ; 2

where2 D

: E  ! E is given by 



,2 0 D = (,
)0 (,
),2(1,) :

2 The inclusion E  L2 L2 induces a triple E L2 L2 E  . The operator is in principle only defined on L2 L2 , but it extends to a continuous operator from E  to E .









5





The L2 and E gradients of f are related by rf is given by

= D rL f , so the E gradient rfH 2





,
),2 f,
v , Hu (x; u; v)g : rfH (u; v) = (,(
) ,2(1,) f,
u , Hv (x; u; v)g

(10)

Assume now, as we shall do throughout this paper, that

H 2 C 2 (   R2 );

(11)

and that H satisfies the following growth condition. 8 > > < > > :

,




jH (x; u; v)j  C 1 + jujp+1 + jvjq+1  pq jHu (x; u; v)j  C 1 + jujp + jvj p  qp jHv (x; u; v)j  C 1 + juj q + jvjq ( +1) +1

(12)

( +1) +1

Then the E gradient rfH defines a continuous vector field on E . By imposing even more growth conditions on the second derivatives of H , we could ensure that the composition operator z 7! D [Hz (
; z (
))] is locally Lipschitz on E , which makes rfH locally Lipschitz on E , and which therefore implies that the gradient flow equation

@z = ,Q z + D H (x; z )   z @t

generates a local flow on E . Here



(13)



1,2 Q = ,D @
= (,
)02,1 (,
)0 (14) Our main observation in this paper is that even though the E gradient flow has no

smoothing properties at all (the initial value problem is well posed in both time directions), connecting orbits between critical points still form precompact sets. Moreover, this compactness can be obtained for a larger range of p, q than one would get for the L2 gradient flow (7). See section 3

2.3 The flow on X . As we just noted, the growth conditions (12) are not sufficient to guarantee that the gradient flow produces a local flow on E . Even if we impose more conditions to make rfH locally Lipschitz, the construction of Floer homology requires genericity arguments which rely on Sard’s theorem, which in turn requires certain differentiability of the functionals involved. Although fH : E ! R is C 1 it will in general not have many more derivatives, even if H is C 1 . Fortunately, as will become clear in a moment, the E gradient flow of fH leaves the space X defined by

,) (  ) X def = h20 (  )  h2(1 0 6

 ) is, by definition invariant. Here h20 (

f' 2 h2 (  ) j 'j@  0g

when 0 <  < 12 , and when 12

f' 2 h1;2,1 (  ) j 'j@  0g

<  < 1. For  = 12 we define

h20 (  ) def = f' 2 C 1 (  ) j 'j@  0g: The space h ( ) is the so-called “little H¨older space”, which is the closure in the usual H¨older space C ( ) of the subspace of smooth functions. The little H¨older spaces have the advantage of being separable. 2(1,) (

) For any  2 (0; 1) the operator (,
)1,2 is an isomorphism from h0 2   to h0 ( ). It follows that Q : X ! X is also an isomorphism (one even has (Q )2 = IX . See for instance [7].) Furthermore, D is a compact operator on X , and, if H is a C m+3 function, then the substitution operator z 7! Hz (
; z (
)) is C m from X to X . Thus we arrive at Lemma 2 If H

2 C m+3 then rfH defines a C m vector field on X .

For H 2 C m+3 the gradient flow equation (13) therefore generates a C m local open flow on X , which we denote by t : Dt ! X (Dt  X is the domain of t ).

2.4 The perturbed flows. To define the Floer homology we need all connecting orbits between critical points to be transverse, at least for a generic set of flows. We achieve this by slightly varying the Riemannian metric on E , thereby perturbing the gradient of fH . Let K : E ! NSym(H ,1 ; C02 ) be a smooth map, where 

NSym(H ,1 ; C02 ) = K 2 L(H ,1; C02 )



K is nuclear and symmetric w.r.t. the inner product of E



 ; R2 ). and where H ,1 = H ,1 ( ; R2 ) = H ,1 ( )  H ,1 ( ), C02 = C02 (

Recall that a bounded operator T : X ! Y between Banach spaces is nuclear if it can be written as an absolutely convergent sum, i.e.

T= with xi

2 X  , yi 2 Y and

1 X i=1

kT knucl def = inf

yi xi

1 X i=1 7

kyi kY kxi kX 

(15)

(16)

Clearly the space of finite rank operators from X to Y is dense in the space of nuclear operators, and since both H ,1 and C02 are separable, the space NSym(H ,1 ; C02 ) is also separable. The set NSym(H ,1 ; C02 ) with the nuclear norm (16) is a Banach space. We define a Riemannian metric on E by ,




gzK (1 ; 2 ) def = 1 ; (I + K (z ))2 E for all z 2 E and i 2 Tz E  = E . Our modified gradient flow will be ,
z 0(t) + I + K (z (t)) rfH (z (t)) = 0: where rfH still denotes the E gradient of fH . We shall assume that kK (z )kE!E  ; 8z 2 E ; for some  < 12 , and also that the map K can be written as K (z ) = e,kzkE K0 (z ); 2

(17)

(18)

(19)

where K0 satisfies the following Gevrey type estimates:

sup kK0(n)(z )kLn(E ;NSym(H , ;C 1

z2E

2 0

))  C (n!)

2

(20)

for some C < 1. ~ to be the set of maps K (
) which satisfy (19, 20). By defining the We define K norm of K (
) to be the infimum of all constants C for which (20) holds, K becomes a ~ is a Banach space which at least contains Banach space. The bounds (20) ensure that K maps of the form (kz , z0 k)K1 ; (21) where K1 2 NSym(H ,1 ; C02 ) is a constant, and (s) = e,1=(1,s ) for s < 1, (s) = 0 for s  1, is a common cutoff function. Since E is a separable Hilbert space, the set of maps of the form (21) is separable. We now define K to be the closed linear ~ spanned by the maps (21). This is the class of metrics which we shall subspace of K allow, and from the construction it follows that this space is a separable Banach space. We will need this in section 7. We write K for the subset of K consisting of those K (
) which also satisfy (18). The assumption (19) directly implies the following uniform bound 2

2 0

(22)

does not depend on z or t. To verify this one considers the two terms in The first term is linear, and bounded in E by C kz kE ; the second involves the Hamiltonian function H , and using (12), (9) and the Sobolev inequalities, one shows that the second term is bounded in H ,1 ( ) by C (1 + kz km E ), with m = max(p; q). The rapid decay we require in (19) now implies (22).

where

C

kK (z )rfH (t; z )kC  C;

rfH (t; z ).

Lemma 3 For H 2 C 1 and K C 1 local flow on X .

: E ! NSym(H ,1 ; C02 ) as above, (17) generates a 8

3 Compactness Our compactness theorems will apply to maximal invariant sets, or to their analogues in the case of nonautonomous equations. We recall that the maximal invariant set S = S (N ) of a closed subset N  X for the flow t consists of all points z 2 N such that t (z ) is defined and lies in N for all t 2 R. If S is contained in the interior of N then S is called an isolated invariant set, and N is an isolating neighborhood for S (compare for instance Conley [8]). For any pair of regular values a and b of fH the set

Na;b = Na;b(H ) def = fz 2 X j a  fH (z )  bg

is an isolating neighborhood for the perturbed gradient flow (17) of rfH .

3.1 Hypotheses on H . The Hamiltonians which we have in mind are the homogeneous examples (3) from the introduction and lower order perturbations of these. For the proofs of the compactness theorems of this section it turns out that only the following properties of the examples (3) are of importance. The jujp+1 + jv jq+1 type Hamiltonians satisfy ,

uHu (x; u; v) + vHv (x; u; v) , 2H (x; u; v)  ,C +  jujp+1 + jvjq+1 and the jujp+1 , jv jq+1 type Hamiltonians satisfy ,
uHu (x; u; v) , vHv (x; u; v)  ,C +  jujp+1 + jvjq+1 for certain constants ; C > 0.




(23)

(24)

3.2 Boundedness in E for the autonomous equation. In this subsection we will prove the following.

 R2 ) satisfies (23) or (24). Then the maximal invariant Lemma 4 Suppose H 2 C 2 (

set of the flow generated by (17) in Na;b is bounded in E . The proof of this lemma proceeds in three steps. Step 1 Let z : R ! E be a solution of (17), with a  f (z )  b. Since (17) is a gradient flow we get the following estimate for free: Z 

R



fI + K (z (t))g,1 z 0 (t); z 0 (t)

Since kK (z )k  1=2, we have

E

dt  b , a:


(I + K ),1 z; z E  21 kz k2E ;

,

9

Z

and hence

R

kz 0(t)k2E dt  2(b , a):

(25)

Step 2 If H satisfies (23) then we take the E inner product of (17) with [I find ,




[I + K (z )],1 z 0(t); z (t) E + 2f (z ) =

Z



+ K (z )],1z and

fuHu + vHv , 2H gdx

By (23) this implies Z



Z

jujp+1 + jvjq+1  C + C kz (t)kE kz 0(t)kE :

If H satisfies (24) then we take the inner product with find Z ,

(26)

  z~ = [I + K (z )],1 ,uv and




[I + K (z )],1z 0 (t); z~(t) E = fuHu , vHv gdx

p +1 From (24) we have uHu , vHv  C fjuj + jvjq+1 g , C so that we again get (26).

By integrating (26) in time and using (25) we get for any T Z

T +1 Z T



jujp+1 +

Z

T +1 Z T



2R

jvjq+1  C + C sup kz (t)kE : R

(27)

Step 3 The gradient flow may be written as

z 0(t) + Q z = D Hz (z ) + b(t) (28)  ) by (22). Since where b(t) = ,K (z (t))rfH (z ) is uniformly bounded in C 2 (

1 , 2  1 ,   (,
) : D((,
) ) ! D((,
) ) is an isometry, Q : E ! E also is an isometry. Moreover, one has Q2 = IE , so that one can split E = E+  E, , with Q jE = IE . If we denote the corresponding projections by P then the operator G(t) = ft>0g e,tP, , ft 0, which proves the boundedness of fzn g. After passing to a subsequence we may assume that the zn converge weakly in E to some z . Conse,1 : E  ! E , quently, since zn ! z in Lp+1  Lq+1 (Sobolev embeddings) and @
the right hand side of

zn = ,@
,1Hz (x; zn ) + Q rfH (zn ): (37) converges in E -norm and kzn , z kE ! 0. The limit z must then be a critical point in E of fH with action fH (z ) = a. In view of our regularization lemma z actually 2 belongs to X . 14

2 X is a sequence for which lim krfH (zn )kE = 0 n!1

Lemma 10 Assume that H satifies (24). If zn

then zn has an E convergent subsequence. Proof. The proof is very similar to that  of the  previous lemma. In the present situation u n we take the inner product with z~n = ,vn . This leads to

C"n kzn kE 

Z



fun Hu , vn Hv g dx

 ,C j j + 

Z







jujp+1 + jvjq+1 dx: 2

From here on the same arguments apply.

3.5.1 Corollaries. In the proof of Lemma 9 we only used fH (zn )  a to extract an E convergent subsequence. Hence it follows that the set of critical points z of fH with fH (z )  a is compact in E . Elliptic regularity then implies that the same set is also compact in X . Lemma 10 does not refer to the function values of the zn at all, so for “, type” Hamiltonians we conclude that the set of critical points is compact.

4 The Morse index and Fredholm properties If z 2 X is a critical point of fH , then the second derivative of fH at z is given by the linear operator

drfH (z )
w = Q w , D [Hzz (x; z )w]:

This operator is a compact perturbation of Q , which is invertible on X , so drfH (z ) is a Fredholm operator of index 0 on X . We will call z a Morse critical point if drfH (z ) : X ! X is invertible (i.e. injective). We will call fH a Morse function if all critical points z 2 X are Morse points.4 In this section we will consider the solutions to (32) for some possibly time dependent Hamiltonian which satisfies the hypotheses of section 3.3.1

fH (1;
) are Morse functions on X : For such H we shall study the space of C 1 solutions z

(38)

: R ! X of (32) which satisfy

lim z (t) = z

t!1

4 We repeat these seemingly standard definitions since X is not a Hilbert space, so that the second 00 z , in its incarnation as a bilinear form on X , can never be coercive. derivative fH

()

15

for certain critical points z+ and z, 2 X of fH+ and fH, , respectively, where H (x; z ) = H (1; x; z ). We denote the space of such solutions by MH;K (z, ; z+ ). By definition, MH;K (z, ; z+ ) is the zeroset of the map H;K : Q1 ! Q0 , where

H;K (z ) = ddzt + (I + K (z ))rfH (t; z ); Q1 (z, ; z+) = z0 (
) + W 1;2 (R; X ); Q0 (z, ; z+) = L2 (R; X ): and where z0 : R ! X is an arbitrary smooth map which satisfies z0 (t)  z, for t  ,1 and z0 (t)  z+ for t  1. Thus Q1 is not a vector space but only an affine space, and z0 is an arbitrarily chosen base point in this space. Lemma 11 If

H (x; t; z ) is a smooth function on   R  R2

K  Q1 ! Q0 defined by

then the map

H :

H (K; z ) = H;K (z )

(39)

is smooth. We leave it to the reader to verify this.

4.1 The linearized equation. The Fr´ech`et derivative of H;K is given by

dH;K (z )
w = ddwt + fI + K (z (t))g drfH (t; z (t))
w(t) + fdK (z (t))
w(t)g rfH (t; z (t)):

(40)

Since we assume that z are critical points, we have limt!1 rfH (t; z (t)) = 0; by  ), and (19) the linear map w 7! fdK (z )
wg rfH (z ) is bounded from X to C 2 (

hence compact from X to X . The following Proposition now implies compactness of the third term in (40). Proposition 12 Let X be a Banach space. If fB (t) j t 2 Rg is a norm-continuous family of compact operators on X , with limt!1 kB (t)kL(X ) = 0, then the multiplication operator w(t) 7! B (t)w(t) is a compact operator from W 1;2 (R; X ) to L2(R; X ). Thus dH;K is a Fredholm operator if and only if the operator

T
w def = ddwt + fI + K (z (t))g drfH (t; z (t))
w(t) is Fredholm. We conjugate this operator with S (t) =

p

I + K (z (t)), with result

S (t),1 TS (t)
w = ddwt + S (t),1 S 0 (t)w(t) + S (t)drfH (t; z (t))S (t)w(t): 16

S (t),1 S 0 (t) is a compact operator from W 1;2 (R; X ) to L2 (R; X ). Proof. By assumption K = K (z (t)) is a compact operator on X whose E operator norm is not more than 12 . Hence the spectrum of K is contained in the disk with Proposition 13

radius 12 and we can define the square root via a Dunford integral of the resolvent R(; t) = ( , K (z (t))),1 of K (z (t)): I p 1 R(; t) 1 + d; I + K (z (t)) = 2i jj=

p

p

with 12 <  < 1, so that 1 +  is analytic inside the contour, and so the contour encloses the spectrum of K (z (t)). The derivative S 0 (t) is given by

1 I @R (; t)p1 + d 2i I @t p = 21i R(; t) fdK
z 0(t)g R(; t) 1 + d:  )) norm of dK (z (t))
z 0(t) also Since z 0 (t) vanishes at t = 1, the L(X ; C02 (

0 2 vanishes at t = 1. By proposition 12 S (t) is therefore compact.

S 0 (t) =

Dropping the term S (t),1 S 0 (t) we see that dH;K is Fredholm if and only if the operator

LA = ddt + A(t) (41) is Fredholm, where A(t) = S (t)drfH (t; z (t))S (t). Furthermore, dH;K (z ) will have the same Fredholm index as LA . Lemma 14 A(t) is a bounded operator on X and symmetric with respect to the E inner product. Moreover, A(t) can be written as A(t) = Q + B (t) (42) where B (t) is compact on X and symmetric with respect to the E inner product. Proof. We already know that A(t) is bounded on X . By definition drfH (t; z ) is selfadjoint on E , since it is an operator representing a symmetric form. For any bounded operator S the operator S  drfH (t; z )S is therefore also selfadjoint on E . Since S (t) is selfadjoint, we find that S (t)drfH (t; z )S (t) is selfadjoint on E , which implies that it is symmetric. To obtain the representation (42) we recall that drfH is a compact perturbation of Q , so, modulo compact operators S (t)drfH S (t) = S (t)Q S (t). We now observe that

p

I + K = I + KM;



p

M = I+ I+K

,1

;

which shows that S (t) = I+ a compact operator on X . Therefore S (t)Q S (t) = Q modulo a compact operator on X . 2 17

We now consider the following subsets of the space of bounded operators on X 

A = Q + B and



B is a compact operator on X and B is E -symmetric.







 = A 2 A j ker(A) 6= f0g : We have just shown that A(t) 2 A for all t 2 R. By (38) drfH (1; z ) and hence A(1) are invertible operators on X . Lemma 15 Let A : R ! A be a continuous family of operators, with A = A(1) 62 . Then the operator LA defined in (41) is a Fredholm operator from W 1;2 (R; X ) to L2(R; X ). Its index (A, ; A+ ) only depends on the limiting operators A . We split the proof in three parts. Step 1 We first consider the case in which A(t)  A is constant. By assumption A must be invertible. Since A is a compact perturbation of Q , its spectrum consists of  (Q ) = f,1; 1g and pure point spectrum. Since A is symmetric with respect to the X inner product, the point spectrum must be real. Hence A is hyperbolic, i.e. its spectrum is disjoint from the imaginary axis. The spectrum of A then divides into a part on the right of the imaginary axis, and a part on the left of this axis. Denoting the projections on the corresponding A-invariant subspaces of X by  , we can write the inverse of LA as Z

(LA ),1 z (t) =

where

R

G(t , s)z (s)ds;



G(t) = e,tA ft>0g + , ft 1 is a large constant. Consider the operators 



L = dtd + A120(t) A 0(t) ; 23   M = dtd + A130(t) A0 : 2 Both are Fredholm operators from W 1;2  W 1;2 to L2  L2 . The first has index(L) = (A1 ; A2 ) + (A2 ; A3 ), while the second has index(M ) = (A1 ; A3 ) + (A2 ; A2 ) = (A1 ; A3 ) (in step 1 of the proof of lemma 15 we saw that (A2 ; A2 ) = 0 for any A2 .) Choose a smooth function  (t) with  (t)  0 for   ,1, and  (t)  =2 for ,  (t=R) , sin  (t=R)
t  1. We now conjugate M with R(t) = cos sin  (t=R) cos  (t=R) and find 

 



0 ) 0 1 + A12 (t + 2R) 0 M^ = R(t),1
M
R(t) = ddt +  (t=R ,1 0 0 A23 (t , 2R) : R

^ with a shift operator S (u; v)(t) = (u(t + 2R); v(t , 2R)), with Next we conjugate M result  

0  0 (t=R , 2)=R : ^ =L+ S ,1MS 0 , (t=R , 2)=R 0 ^ , and that S ,1 MS ^ differs from L by an Thus we see that M is conjugate with S ,1 MS 0 ^ and L operator with norm bounded by k k1 =R. If R is large enough, then S ,1 MS must have the same Fredholm index, and consequently L and M have the same index. 2 This proves the cocycle property for . We can now define the generalized Morse index of critical points of fH . Definition 17 Let z 2 X be a Morse critical point of fH . Then the renormalized Morse index of z is 0 (z ) = 0 (drfH (z )). 4.3 The spectral flow formula. If A(t) 2 A is a continuous family with A(1) invertible, then one can compute the index (A1 ; A2 ) from the way the spectrum of A(t) changes as t increases from ,1 to 1. Since A(t) = Q + K (t) with K (t) compact, the essential spectrum of A(t) is f,1; +1g. Thus there is a neighborhood of  = 0 which only contains point spectrum of A(t). If A(t) is a generic family in A, then there will only be a finite number of times tj 2 R at which A(t) fails to be invertible. At these times kern(A(tj )) will be one dimensional and, near tj , the corresponding eigenvalue j (t) depends smoothly on t, with 0j (t) 6= 0. The Spectral flow formula5 for (A, ; A+ ) states that the index of d 0 0 dt + A(t) is the number of tj with j (tj ) > 0 minus the number of tj with j (tj ) < 0, 5 The

phrase was coined in [3].

20

i.e. (A, ; A+ ) is the number of eigenvalues of A(t) that cross the imaginary axis as t increases from ,1 to +1. In other words,

(A, ; A+ ) =

X

j

sign0j (tj ):

(43)

This was proved in [17] in the context of unbounded operators A(t) with compact resolvent, in particular for elliptic operators on compact manifolds. Our setting is slightly different, since our A(t) are bounded (integral) operators. We now outline an argument that proves (43) still holds in our setting. The idea is simple and standard: first we note that, given A0 2 A n , (A1 ; A0 ) cannot change if A1 varies within one connected component of A n . Then we see how (A1 ; A0 ) changes if A1 crosses  from one component of A n  to another. Lemma 18 Let (A1 ; A2 ) = 0.

A1

and

A2

lie in the same connected component of

A n .

Then

In particular, if A = S (t)drfH (z )S (t), for some nondegenerate critical point z of fH , then 0 (A) is independent of S (t), i.e. of the chosen metric K (
). Indeed, by deforming K (
) linearly to zero one obtains a family A = S (t)drfH (z )S (t) in A n  which connects S (t)drfH (z )S (t) with drfH (z ). Proof of Lemma 18. Choose  2 C 1 (R) with  (t) = 0 for t  ,1, and  (t) = 1 for t  1. Consider for any pair A1 , A2 2 A the operator LA1 ;A2 = d dt + A1 +  (t)(A2 , A1 ). The index of LA1 ;A2 is (A1 ; A2 ). If kA2 , A1 k is sufficiently small then LA1 ;A2 is a small perturbation of LA1 , so that LA1 ;A2 is invertible. Thus we find that (A1 ; A2 ) = 0 for all A2 in a neighborhood of A1 . Since this is true for all A1 we find that the equivalence classes of the relation A1  A2 , (A1 ; A2 ) = 0 are open in A n . Any connected component of A n  must therefore be contained in one such equivalence class, which is what was claimed.

2

One can write  as  = [n1 n with

n = fA 2 A j dim ker(A) = ng

(See [9].) Each n is a smooth submanifold of A of codimension n(n + 1)=2. Indeed, split
H = N  N ? , N = ker A. Any operator A0 2 A near A is of let A 2 n , and ,P 0 ? the form A = Q Q S with P 2 L(N; N ) self-adjoint, Q 2 L(N ; N ) arbitrary, and with S : N ? ! N ? invertible. Such an operator A0 then has the same corank as

A00 =

 

1 ,QS ,1 0 1



P Q Q S 



1

0 ,S ,1 Q 1



,1 Q 0 = P , QS 0 S : Thus A0 has corank n exactly when P = QS ,1 Q , which shows that n is analytic with codimension n(n + 1)=2. We also find that the n(n + 1)=2 dimensional subpace      = A + P 0 j P 2 L(N; N ); P = P  A

00

21

is transverse to n at A. Consider A0 ; A1 2 An , and connect them by a smooth curve fA() j 0    1g in A. After perturbing the curve slightly, if necessary, one may assume it intersects each n transversely; in particular it does not intersect any of the n with n  2 since their codimension is too high. Suppose A(0 ) 2 1 , and let u 2 X span the kernel of A(0 ). We may again modify the family A() for  close to 0 so that either

A(0 + ) = A(0 )  u u (44) holds for small  2 R. Since u u is positive definite both signs can occur. For small  A(0 + ) will have  as isolated simple eigenvalue. The plus sign will occur if an eigenvalue of A() crosses the imaginary axis from the left halfplane to the right halfplane as  increases beyond 0 : if the eigenvalue goes the opposite way, we will get a minus sign in (44). We compute (A(0 ,  ); A(0 +  )) by looking at the Fredholm index of the operator L = ddt + A(0 +  tanh t). If we split X = [u]  [u]? then this operator can be written as the direct sum of the operator L0 = ddt + A0 , where A0 is the restriction of A() to [u]? , and the operator L1 = ddt   tanh t from W 1;2 (R; R ) to L2 (R; R ). The L0 Fredholm index of L is the sum of the indices of L0 and L 1 . Since A0 is invertible + is spanned has index zero; the index of L can be computed directly: the kernel of L 1 1 +  by sech(t), while L, 1 , the adjoint of L1 is injective. Hence L1 has index 1. The same arguments apply each time A() crosses 1 , and we therefore find that (A(0); A(1)) is the number of eigenvalues of A() that went from the left to the right halfplane, minus the number of eigenvalues that crossed the imaginary axis in the other direction. This concludes the proof of (43).

4.4 Upper and lower indices. If A 2 A is degenerate (i.e. A 2 ) then (A) is not defined. Instead we introduce the upper and lower Morse indices of A which are defined by 



A0 2 A n  kA , A0 k < "







A0 2 A n  kA , A0 k < "



+ (A) = lim sup (A0 ) "#0 and

, (A) = lim inf (A0 ) "#0

; ;

respectively.

+ (A) , , (A) = dim kernA. Let m be the dimension of kern A.

Lemma 19

Then any A0 2 A near A will have m Proof. eigenvalues close to zero, while the rest of the spectrum of will be bounded away from zero. Hence, if A0 ; A00 2 A are two operators near A then the spectral flow formu la applied to the linear arc A = (1 , )A0 + A00 implies that (A0 ) and (A00 ) differ at most m. On the other hand if one chooses A0 = A + "P , A00 = A , "P , 22

where P is orthogonal projection on kern A, then for sufficiently small " (A00 ) = (A0 ) , m.

5 The space

MH;K

(z, ; z+ )

> 0 one has 2

and its closure

5.1 Broken orbits.

  R2 ! R be a smooth Hamiltonian satisfying (12), one of (23) or (24), Let H : R 

and the hypotheses in x3.3.1. For given z 2 X we have defined MH;K (z, ; z+ ) to be the set of solutions of z 0 = ,(I + K (z ))rfH (t; z ) with z (t) ! z as t ! 1. Although MH;K (z, ; z+ ) was given as a subset of the affine space Q1 (z, ; z+ ), we will consider a different topology on MH;K (z, ; z+ ), namely, the topology of 0 (R; X ): uniform X norm convergence on bounded intervals. Evaluation of a Cloc z 2 MH;K (z,; z+ ) at t = 0 defines a map z 7! z (0) from MH;K (z, ; z+ ) to X , which is continuous. Since the z 2 MH;K (z, ; z+ ) are solutions of an O.D.E. on a Banach space, the map z 7! z (0) is one-to-one, and its inverse is continuous. We have therefore identified MH;K (z, ; z+ ) with a subset of MH;K  X (all initial values z0 2 X whose corresponding solution z (t) to the O.D.E. exists for all time and has the right limits at t = 1). In general this subset will not be closed, but our compactness theorems do imply that MH;K (z, ; z+ ) is precompact in X . The closure of MH;K (z, ; z+ ) is easiest understood in the case where H is autonomous. Assume that H is time independent and let z0i 2 MH;K (z, ; z+ ) be any given sequence. Consider the corresponding orbits 

i = z i (t) j t 2 R

where z i (t) is the solution to the gradient flow with z i (0) = z0i . The closures  i =

i [ fz,; z+g form a sequence of compact sets in X0, all of which lie in one larger compact set (they are uniformly bounded in C0  C0 ; see Lemma 6.) Along some subsequence they will therefore converge to some other compact set   X which must be invariant under the (K; H )-gradient flow. If the critical points of fH are isolated (e.g. if fH is a Morse function) then  consists of a finite number of critical  and heteroclinic orbits between each pair (zi; zi+1 ). points z1 , : : :, zm If H is time dependent, we make the system autonomous by introducing a new indepedent variable  = tanh(t), and considering a system of O.D.E.s for ; z ). The pair (; z ) then satisfies 8 >
2 : = 1 ,  dt p Although artanh = ln ( + 1)=( , 1) is singular at  = 1, this singularity is nullified by the hypothesis that fH (t; z ) becomes constant for jtj  `. The system (45) therefore defines a smooth O.D.E. on R

 X . 23

By identifying a z 2 MH;K (z, ; z+ ) with its trace = f( (t); z (t)) j t 2 Rg in [,1; 1]  X one can repeat the above arguments to see what happens to an arbitrary sequence in MH;K (z, ; z+ ). After passing to a subsequence one can again show that

the converge to a broken trajectory. Of this broken trajectory one orbit crosses from  = ,1 to  = +1 (i.e. corresponds to a solution of the original nonautonomous equation) while all others have   1 or   ,1, i.e. they correspond to orbits of the autonomous gradient flows at t = 1.

MH;K ( ,

+ ) under the assumption of transversality. Since MH;K (z, ; z+ ) is by definition the zero set of H;K : Q1 ! Q0 , it follows from the implicit function theorem that if H is smooth and if 0 is a regular value of H;K , then MH;K (z, ; z+ ) is a smooth manifold whose dimension is exactly the Fredholm index of dH;K , i.e. 5.2

z

;z

dimMH;K (z, ; z+ ) = 0 (z, ) , 0 (z+ )

(46)

A number of special cases are of interest. Since 0 is a regular value of H;K : Q1 (z, ; z+ ) ! Q0 (z, ; z+ ) for all possible pairs (z, ; z+ ), the derivative dH;K always has nonnegative Fredholm index. Hence MH;K (z, ; z+ ) is empty whenever 0 (z, ) < 0 (z+ ). Index 0 orbits. If z have the same renormalized index then MH;K (z, ; z+ ) is zero dimensional, so that it consists of isolated solutions z 2 Q1 (z, ; z+ ). If, in addition, H and K are time independent, then all z 2 MH;K (z, ; z+ ) must be constant (once MH;K (z, ; z+ ) contains z it also contains all time translates z# (t) = z (t + #); if z is isolated, then one must have z#  z for all #.) Returning to the general non-autonomous case, one observes that any (geometric) limit of zi 2 MH;K (z, ; z+ ) is a broken trajectory z, ! z 0 ! z 00 !


! z+ where each segment z (j ) ! z (j +1) has index 0. Only one of these segments is a solution of the nonautonomous equation. The others are solutions of the autonomous equations at t = 1, which we have just pointed out are constant. Since one does not include constant orbits in broken orbits, it follows that the broken orbit actually only has one segment, and that the zi converge in Q1 (z, ; z+ ) to some other z 2 MH;K (z, ; z+ ). Therefore MH;K (z, ; z+ ) is compact, and consists of a finite number of trajectories of the (H; K ) flow. Index 1 orbits. If H and K are autonomous and 0 (z, ) = 0 (z+ ) + 1 then MH;K (z, ; z+ ) is one dimensional. Since MH;K (z, ; z+ ) contains all time translates z# of any solution z 2 MH;K (z, ; z+ ), one sees that the connected components of MH;K (z, ; z+ ) consist of arcs formed by the time translates of heteroclinic orbits of the (H; K ) flow. Any limit of such trajectories is again a broken orbit only one segment of which can have

24

nonzero index. Since zero index orbits are constant, we again find that MH;K (z, ; z+ ) is compact, after one divides out the time translations.

6 Gluing In general the solution spaces MH;K (z, ; z+ ) are not compact, and a sequence zi 2 MH;K (z, ; z+ ) may converge to a broken orbit. The so-called “gluing construction” provides a more detailed description of the way a sequence zi 2 MH;K (z, ; z+ ) con-

verges to a broken orbit. We illustrate the construction for a broken orbit with only two segments z01 z ,! z12 z z0 ,! (47) 1 2

of an autonomous (H; K ) flow. Thus z01 2 MH;K (z0 ; z1 ) and z12 and we assume 0 (z0 ) > 0 (z1 ) > 0 (z2 ). Given these connecting orbits one defines for any large T > 0 







2 MH;K (z1 ; z2 ),



z02;T (t) = 1 ,  Tt z01 (t + 2T ) +  Tt z12 (t , 2T ); where 0   2 C 1 (R) satisfies  (t)  0 for t  ,1,  (t)  1 for t  1 and  0 (t)  0 for all t. For large T the patched curve z02;T 2 Q1 (z0 ; z2 ) is almost a solution of H;K (z ) = 0. The crux of the gluing construction is that the nondegeneracy of dH;K (z01 ) and dH;K (z12 ) allows one to find an actual solution of H;K (z ) = 0 near z02;T in the

Q1 norm.

The actual solution is constructed as follows. Under the assumption of transversality both dH;K (z01 ) and dH;K (z12 ) are surjective Fredholm operators, so one can choose bounded right inverses M01 ; M12 : L2 (R; X ) ! W 1;2 (R; X ). One “patches” these operators together to form

M T def = T, ,2T M0;1 2T T, + T+ 2T M0;1 ,2T T+ ; in which a '(t) = '(t , a), and T (t) is a pair of smooth functions satifying 1, (t)2 + 1+ (t)2  1, 1+ (t)  0 for t  ,1, 1, (t) = 1+ (,t), T (t) = 1 (t=T ). The operator M T is an approximate right-inverse for dH;K (z02;T ), in the sense that the operator norm of dH;K (z02;T )  M T , IQ tends to zero as T ! 1. One then substitutes the ansatz z = z02;T + M T w, for some w 2 Q0 , in the equation H;K (z ) = 0. Expansion of this equation in a Taylor series results in a fixed point problem for w on Q0 which has a unique small solution when T is large. We omit the 0

details as they are very similar to those in the cited works on Floer homology6 – indeed, our situation is a bit simpler since we only have to estimate the Q0 = L2 (R; X ) norm of w, whereas the usual gluing procedure is done in a Fr´ech`et space of C 1 functions where one has to estimate a sequence of seminorms. If one denotes the solution obtained this way by

z01 #T z12 (t) = z02;T (t) + (M02 w)(t); 6 In

[1] we applied the same procedure in a different setting.

25

then the conclusion of the gluing procedure is that one can approximate the broken orbit (47) by smooth “glued” orbits z01 #T z12 2 MH;K (z0 ; z2 ), and that these glued orbits are the only ones which can approximate the broken orbit (47). The glued orbits thus provide a description of the noncompact ends of the solution spaces MH;K . The solution z01 #T z12 whose construction we have just sketched actually depends on a number of parameters: namely, instead of the ansatz z = z02;T + M T w one could choose any small v 2 ker dH;K (z02;T ) and put z = z02;T + v + M T w. Substitution in H;K (z ) = 0 again leads to a fixed point problem for w with v 2 ker dH;K (z02;T ) as a parameter, and one obtains for small v and large T unique small solutions wT;v . In our setting one can view the gluing procedure from a more dynamical point of view. The solutions z 2 MH;K (z, ; z+ ) are actually orbits of a smooth ODE on a Banach space. The -lemma from dynamical systems then implies that, when all stable and unstable manifolds of critical points intersect transversally, any broken orbit (47) is “shadowed” by actual orbits of the flow. Since both stable and unstable manifolds of all our critical points are infinite dimensional it seems one can not compute the dimension of the intersection of such manifolds without considering the renormalized index 0 (z ) of the critical points.

7 Generic flows In this section we show that nontransversality only occurs for exceptional choices of K and H , and can always be destroyed by arbitrarily small modifications of H and K . We summarize this as follows. Lemma 20

1. fH is a Morse function for generic H

2. Given H 2 C 1 , the generic K 2 K1=2 .

2 C1.

K -gradient flow of fH has the Kupka-Smale property for

3. For generic time dependent H 2 C 1 and K 2 C 1 (R; K1=2 ) solutions of the nonautonomous equation (32) are nondegenerate (0 is a regular value of H;K .)

7.1 Generically a Morse function.   R2 for which Let H be the space of C 1 functions h on

[h]H def =

sup (n!),2 kdn h(x; u; v)k

 R2;n0

is finite. This space is a separable Banach space [21], and the map (h; z ) = rfH +h (z ) is smooth from H  X to X . The derivative of with respect to z is given by dz (h; z )w = Q w + D (Hzz + hzz ) w, which is a compact perturbation of Q , and thus Fredholm. The derivative with respect to h is dh (h; z )k = D [k (x; z (x))], i.e. an evaluation operator followed by the linear operator D . The range of this composition is dense in X . Indeed, the range contains all functions of the form D [k(x)], where we let k be  ), with sup jdn k(x)j  C (n!)2 for some C < 1, independent of z , i.e. k 2 C 1 (

26

 )  C0 (  ), and D maps and k  0 on @ . The set of such k is dense in, say C0 (

this space densely into X . We now claim that 0 is a regular value of . Indeed, let (h; z ) = 0. Then the range of dz (h; k ) has finite codimension, while the range of dh (h; z ) is dense in X . The sum of these ranges must therefore be all of X . Thus , = f(h; z ) 2 H  X j (h; z ) = 0g is a smooth manifold, and the projection 1 (h; z ) = h is a Fredholm map from , to H. The Sard-Smale theorem [22] says that almost every h 2 H is a regular value (in the sense of Baire category). Any h 2 H is a regular value of the project in 1 if and only if fH +h is a Morse function. Thus any H can be approximated by H + h whose corresponding fH +h is a Morse 2 function. The third part of the lemma can be proved using the same strategy, and the same type of pertubations.

7.2 Generic Kupka-Smale property. We use the same strategy as above to prove part 2 of lemma 20; the only difference is   R2 ) we consider that one must choose different perturbations. For given H 2 C 1 (

1 0 the map H : K1=2  Q ! Q given by (39). Lemma 21

0 2 Q0 is a regular value of H .

In the proof of this lemma we will need the following observation which tells us that the class of perturbations we have chosen is large enough. Proposition 22 For any  with k0 ( ) =  . Proof. If (;  )E

2 X ,  6= 0 and  2 C02 there exists a k0 2 NSym(H ,1 ; C02 )

6= 0 then we define )E k0 (x) def = ((x; ; ) : E

Otherwise we choose a 

2 C02 with (;  )E 6= 0 and we put (; x)E : k0 (x) def =  (; x)E(; + ) E

In both cases k0 is an E -symmetric finite rank operator which maps E into C02 . From

(; x)E = (D, ; x)L

2

(48)

one sees that since  2 C02 one also has D,  2 C02 ,! H01 which implies that (48) extends to a continuous functional of x 2 H ,1 . Hence the operators k0 we have 2 defined belong to NSym(H ,1 ; C02 ).

27

Proof of Lemma 21. Let H (K; z ) = 0 and assume dH (K; z ) is not surjective. Then there is a  2 L2 (R; X  ) which annihilates Range(dH (K; z )). The derivative dz H (K; z ) is the same as dH;K (z ), which we know to be a Fredholm operator. Hence the range of dH (H; z ) has finite codimension. By assumption  annihilates the range of dH;K (z ) and thus must be a weak solution of the adjoint equation ,0(t)+(Q + B (t) )(t) = 0, where we have written dH;K (z ) = ddt + Q + B (t). By regularity  belongs to W 1;2 (R; X ); in particular, (t) is continuous. The derivative of H with respect to K is given by

dK H (K; z )
k = k(z (t))rfH (z (t)): By assumption  also annihilates the range of dK H (K; z ). We now choose a fixed k0 2 NSym(H ,1 ; C02 ), some t0 2 R, and put   k" (z ) = 1  kz , z (t0 )kE k

"

with (s) as in (21). Then

"

0

0 = lim h; dK H (K; z )
k" i "#0   Z

 k z ( t ) , z ( t ) k 1 0 E  (t); k rf (z (t))  dt = lim

0 H " R " R 1 ,1=(1,s ) e ds = ,1kz 0(t )k h(t0 ); k0 rfH (z (t0 ))i : 0 E This holds for all k0 2 NSym(H ,1 ; C02 ). Since rfH (z (t0 )) 6= 0 Lemma 22 guarantees us that we can let k0 rfH (z (t0 )) be any vector in C02 we like, by choosing the right k0 . Since C02 is dense in X this implies (t0 ) = 0. This holds for any t0 , so we 2 see that dH (K; z ) is surjective after all. The zero set of H is a smooth manifold. The canonical projection 1 onto K1=2 must be Fredholm, so that Baire-almost every K 2 K1=2 is a regular value of 1 . And such regular values correspond to K for which 0 2 Q0 is a regular value of H;K .

"#0

2

8 Floer homology 8.1 Definition in the transverse case. We now describe the definition of the Floer homology groups FHp (N; H; K ) of a particular isolating neighborhood N  X for some (H; K ) gradient flow. We first treat with the case in which H 2 C 1 , for which fH is a Morse function, and for which K is such that the (H; K ) flow is a Kupka-Smale flow. Let Crp (N; H ) denote the set of critical points of fH which are contained in N and which have renormalized index p 2 Z. Then denote the Z2 module generated by the critical points z 2 Crp (N; H ) by Cp (N; H ). Thus the elements of Cp (N; H ) are finite formal linear combinations a1 [z1 ] +


+ am [zm ] of critical points in N of index p, and with coefficients ai 2 Z2. 28

One defines a boundary operator @p (N; H ) setting X

@ [z ] =

z0 2Cp,1 (N;H )

= @ : Cp (N; H ) ! Cp,1 (N; H ) by

nN;H;K (z; z 0)[z 0 ];

(49)

where nN;H;K (z; z 0 ) is the number (mod 2) of heteroclinic orbits of the gradient flow z 0 = ,(I + K (z ))rfH (z ) from z to z 0 which stay entirely within the isolating neighborhood N . For every pair z; z 0 for which 0 (z ) , 0 (z 0 ) = 1 the number nN;H;K (z; z 0) is well-defined since MH;K (z; z 0) contains only finitely many (modulo translations) orbits (compactness, see section 5). Lemma 23 The sum in (49) is finite. Proof. All critical points z 0 at the end of a heteroclinic orbit starting at z have fH (z 0 ) < fH (z ). The corollaries to lemmas 9 and 10 imply that the set of critical points to which z can connect is compact. Since critical points of a Morse function are isolated the sum 2 in (49) is indeed finite. The proof that the operator @ actually is a boundary operator, i.e. that @p,1  @p : Cp ! Cp,2 vanishes completely standard. The argument begins with the observaP isP , so that one must show that for any tion that @@ [z ] = z00 z0 n(z; z 0 )n(z 0 ; z 00 )[z 00 ]P z 2 Crp (N; H ) and z 00 2 Crp,2 (N; H ) the sum z0 n(z; z 0)n(z 0 ; z 00 ) vanishes (mod 2). One proves this by considering connecting orbits from z to z 00 : these occur in one parameter families which are trapped in N , since N is an isolating neighborhood. The ends of such a one parameter family of connecting orbits from z to z 00 can be used to pair up the possible broken paths z ! z 0 ! z 00 , thereby showing their number to be even. This uses the structure of the sets MH;K (z; z 0 ) for 0 (z ) , 0 (z 0 ) = 2 obtained

from the gluing-construction (section 6). In this argument one needs stable and unstable manifolds of the critical points for the (H; K ) flow to be transverse, and one needs the connecting orbits to stay within compact sets of the phase space X of the flow. We have assumed transversality, and, since the action fH (z ) along a connecting orbit z ! z 0 is always bounded by fH (z ), we have proved compactness in section 3. Thus

@p


! Cp (N; H; K ) ! Cp,1 (N; H; K ) !




is a chain complex. Its homology groups are by definition the Floer homology groups of (N; H; K ): Kern(@p ) FHp (N; H; K ) def = Range (50) (@p+1 ) :

If the set N is not bounded, e.g. if N = X , then there is no a priori reason why the groups FHp (N; H; K ) should be finite. If the Hamiltonian H is of “,” type, i.e. satisfies (24), then the set of critical points is compact (see Lemma 10), and hence finite. So in this case FHp (N; H; K ) will be finite for any choice of N . If the Hamiltonian is of “+” type (satisfies (23)) then we showed in [2] under additional assumptions on H (which are satisfied by our example Hamiltonian (3)) 29

and under the assumption that the space dimension n is at least 3, that Crp (X ; H ) is compact for each p. In this situation each group Cp (N; K ) must therefore be finite, and hence the Floer homology groups are finite.

8.2 Morse relations and Poincar´e polynomials. We recall that the Poincar´e series of any graded Z2 module fCp gp2Z is defined by

PfCpg (t) =

X

rank(Cp )tp :

p2Z

The Poincar´e polynomial in general is a formal power series in t and t,1 , with nonnegative integer coefficients, some or all of which may be +1. One can add such series and multiply them with polynomials with nonnegative integer coefficients without introducing ambiguity. If 0 ! Cp0 ! Cp ! Cp00 ! 0 is an exact sequence, then one has

PfCp g (t) = PfCp0 g (t) + PfCp00 g (t): If fCp ; @p g is a chain complex, then one puts Zp = kern@p  Cp and Bp = range(@p+1 )  Cp , so that the homology of the chain complex is given by Hp = Zp =Bp . From the exact sequence 0 ! Bp ! Zp ! Hp one deduces PfZp g (t) = PfHp g (t) + PfBp g (t): On the other hand Cp

= Zp  Bp,1 , so that we get

PfCpg (t) = PfHp g (t) + PfBp g (t) + PfBp, g (t) = PfHp g (t) + (1 + t)Q(t); 1

(51)

where Q(t) is some formal series in t, t,1 with nonnegative, possibly infinite coefficients (here we have Q(t) = PfBp g (t).) When Cp = Cp (N; H; K ) is the chain complex defined by some (H; K ) flow in an isolating neighborhood N  X , the coefficient of tp in PfCp g (t) is the number of critical points of renormalized index p, and the identity (51) provides the usual lower bounds of the number of critical points (e.g. by substituting t = 1 one obtains that the total number of critical points must be at least the sum of the ranks of the Floer homology groups.) The usual Morse inequalities can be obtained formally from (51) 1 = 1 , t + t2 ,


. by multiplying both sides with 1+ t

8.3 Index pairs. Let N and N0 be isolating neighborhoods for some (H; K ) flow. Assume that N0  N , and let N0 be positively invariant relative to N , i.e. any orbit fz (t) j 0  t < T g which starts in N0 and stays in N must actually stay in N0 . Lemma 24

N1 def = N n N0 is also an isolating neighborhood. 30

Indeed, let z 2 C 1 (R; X ) be an orbit which stays in N1 for all t, but at some t0 touches the boundary @N1 . Then z (t0 ) cannot belong to @N , so z (t0 ) 2 @N1 \ N0 . But then z (t) must remain in N0 for all t  t0 , without entering the interior of N0 . On the other hand, as t ! 1 the orbit z (t) must converge to a critical point of fH , which 2 must lie in the interior of N0 . Consider the following exact sequence

 C (N ) ,! 0 ! Cp (N0 ) ,! Cp (N1 ) ! 0 p

(52)

where ([z ]) = [z ] is just inclusion, and ([z ]) = [z ] when z 2 N1 , ([z ]) = 0 when z 2 N0 . One easily verifies that the sequence (52) is exact, and that  and are chain homomorphisms. We can therefore form the long exact sequence on homology

+1



@p p


! FHp+1 (N1 ) @,! FHp (N0 ) ! FHp (N ) ! FHp (N1 ) ,!


from which we extract a short exact sequence

FHp (N0 ) ! FH (N ) ! kern @  ! 0: 0 ! range( p p @ ) p+1

Writing qp for the rank of range(@p,1 ) we then get

rank FHp (N ) + qp + qp,1 = rank FHp (N0 ) + rank FHp (N1 ); i.e. for some nonnegative formal series Q(t) one has

PN;H;K (t) + (1 + t)Q(t) = PN ;H;K (t) + PN ;H;K (t); 0

1

(53)

where we have written PN;H;K (t) for the Poincar´e polynomial of the Floer homology groups FHp (N; H; K ).

8.4 Continuation. We now consider the way the Floer homology groups change if we modify the Hamiltonian H and metric K . To this end we fix one particular Hamiltonian H0 and only consider other Hamiltonians H for which H , H0 2 H (H was defined in section 7.1.) We would like to show that these groups do not change if one continuously deforms H and K , keeping the isolating neighborhood N fixed. Unfortunately we will need unbounded isolating neighborhoods in the next section, and this makes that we have to strengthen our definition of isolating neighborhood a little. It turns out that we can prove independence of the Floer homology groups for “stable isolating neighborhoods.” After defining these we will show that all isolating neighborhoods we will encounter later are indeed “stable” and discuss the invariance under continuation of the Floer homology.

31

8.4.1 Stable isolating neighborhoods.

  R2 ) satisfy (12), and (23) or (24), and let K 2 K. Let H 2 C 2 (

^ K^ ) with H^ , H 2 C 1 (R; H) and K^ 2 C 1 (R; K) is an We will say that (H; perturbation of (H; K ) if 1. 2.

"

H^ satisfies the conditions (1, 2, 3) of Section 3.3.1 with M < ", and for all t 2 R one has kK^ (t;
) , K kK + kH^ (t;
) , H kH  "

Definition 25 An isolating neighborhood N for the (H; K ) gradient flow is stable if ^ : R ! K and H^ : R ! H any there exists an " > 0 such that for all " perturbations K 1 solution z 2 C (R; X ) of

z 0 = ,(I + K^ (t; z (t)))rfH^ (t; z (t))

(54)

with z (t) 2 N for all t is actually contained in the interior of N . Let N  X be a closed set, and let (N ) be the set of all pairs (H; K ) for which N is a stable isolating neighborhood for the (H; K ) gradient flow on X . By construction (N ) is an open subset of K  (H0 + H). We write reg (N ) for the set of (H; K ) 2 (N ) for which fH is a Morse function and for which the (H; K ) flow is a Kupka Smale flow. In the previous section we showed that reg (N ) is dense in K  (H0 + H). Lemma 26 Let H 2 H0 + H and K 2 K1=2 . 1. Let N be an isolating neighborhood for (H; K ). If fH is bounded on N then N is stable for (H; K ). 2. X is always a stable isolating neighborhood. 3. If a 2 R is a regular value of fH then both Na = fx 2 X j fH (x)  ag and N a = fx 2 X j fH (x)  ag are stable isolating neighborhoods for (H; K ). ^ K^ ) be an " perturbation of (H; K ). Then Proof. 1. (cf. [10, Theorem 3].) Let (H; jfH^ (t; z ) , fH (z )j  j j sup jH^ (t;
) , H (
)j  "j j for any t 2 R. In particular fH^ is also bounded on N . ^ n ; K^ n ) Arguing by contradiction there must exist a sequence of "n perturbations (H with "n ! 0, a sequence of solutions zn 2 C 1 (R; X ) of (54) with z (t) 2 N for all t 2 R, and a sequence tn 2 R for which zn (tn ) 2 @N . Lemmas 5 and 6 apply to ^ n ; K^ n ). Since the fH^ n are uniformly bounded on the solutions of (54), uniformly in (H zn we can therefore extract a convergent subsequence of z^n(t) = zn (tn + t) whose 32

limit z^ will be a solution of the autonomous (H; K ) flow; this solution z^ stays in N but has z (0) 2 @N . This is not possible since N is an isolating neighborhood. 2. The second statement is trivial since @X is empty. 3. We use the Palais-Smale condition proved in Lemmas 9 and 10 to conclude that

8z 2 fH,1 (a)

krfH (z )kE  

^ K^ ) are " perturbations of for some  > 0. Now suppose (H; ,1(a). Then satifies (54) with z (t0 ) 2 @Na = @N a = fH

dfH (z (t)) dt t=t

(H; K ), and that z (t)

D

0

= , rfH (z (t0 )); fI + K^ (t0 ; z (t0 ))grfH^ (t0 ; z (t0 )) D

E

E

D

= , rfH ; (I + K^ )rfH (t0 ; z ) + rfH ; (I + K^ )(rfH , rfH^ )  , 21 krfH k2E + 2krfH kE
krfH , rfH^ kE  , 14 krfH k2E + 4krfH , rfH^ k2E 2

 , 4 + 4

D Hz (x; z (x; t0 )) , H^ z (t0 ; x; z (x; t0 ))

E :    Since D : E ! E is bounded and L1  L1  Epwe can estimate the last ^ z k2L1  4C"2 . So, if we take " = =8 C , then term by 4C kHz , H

n

o 2



dfH (z (t)) < , 2 dt t=t 8

(55)

0

,1 (a). Hence no solution of (54) which remains in Na or in whenever z (t0 ) 2 fH a N can touch the boundary of Na or N a , respectively, since (55) would force such a 2 solution to cross the boundary. 8.4.2 Local continuation. Let (H; K ) 2 (N ), and let " be as in the definition 25. We define consist of all (H 0 ; K 0 ) which satify

U  (N ) to

kH , H 0 kH + kK , K 0 kK < 10" :

For any pair (Hi ; Ki ) 2 U \ reg (N ) (i = 0; 1) we will now define a homomorphism 1;0 : FH (N; H0 ; K0 ) ! FH (N; H1 ; K1 ). The construction, which again is standard [19, 18, 20, 12, 10], proceeds as follows. Let  2 C 1 (R) satisfy  (t)  0 for t  ,1,  (t)  1 for t  1, and  0 (t)  0 for all t. Define 

K^ (t; z ) = (1 , (t))K0 (z ) + (t)K1 (z ); ^ H (t; x; z ) = (1 , (t))H0 (x; z ) + (t)H1 (x; z ) 33

(56)

E

^ K^ ) is an 5" perturbation of (H; K ), which we may assume to be regular in Then (H; ^ K^ ) is not the sense of Lemma 20, i.e. 0 is a regular value of the map H; ^ K^ . If (H; regular then we can replace it by an arbitrary small perturbation which is regular, by Lemma 20. For any pair of critical points z0 2 N of fH0 and z1 2 N of fH1 with the same renormalized index, we now define n1;0 (z0 ; z1 ) 2 Z2 to be the number (mod 2) of solutions z 2 C 1 (R; X ) of (54) which remain in N , and which have boundary values z (,1) = z0 , z (+1) = z1 . For each p 2 Z we define a homomorphism 1;0 : Cp (N; H0 ; K0 ) ! Cp (N; H1 ; K1 ) by 1;0 ([z0 ]) =

X

z1 2Crp (N;H1 )

n1;0 (z0 ; z1)[z1 ]:

(57)

This sum is again finite, since we have fH1 (z1 )  fH0 (z0 ) + C for any z1 occuring in the sum, so that the set of z1 occuring in the sum is compact and hence finite (fH1 is Morse.) Lemma 27

1;0 is a chain homomorphism.

The proof is identical to that given in [19, 18, 20, 12, 10], so we will not reproduce it here. The essential ingredients are compactness of the set of broken orbits of the extended system (45), tranversality of stable and unstable manifolds (which enables one to “glue” broken orbits) and the fact that continua of entire solutions of (54) cannot escape N . Let 1;0 : FH (N; H0 ; K0 ) ! FH (N; H1 ; K1 ) denote the map induced on homology by 1;0 . Lemma 28 The homomorphisms 1;0 are natural in the sense that 1. If (H1 ; K1 ) = (H0 ; K0 ) then 1;0 is the identity,

2. For any three pairs (Hi ; Ki ) 2 reg (N ) one has 2;0

= 2;1  1;0 .

In particular, the 1;0 are isomorphisms. The proof is again largely standard: First one observes that (57) defines a chain ho^ K^ ) of (H; K ) (not necessarily momorphism H; ^ K^ for any regular " perturbation (H; the one we constructed above.) The chain homomorphisms H; ^ K^ obtained in this way ^ ^ may depend on the chosen " perturbation (H; K ), but they all induce the same map on ^ i ; K^ i ) of regular " pertu (Floer-) homology. This is proved by connecting any pair (H rbations of (H; K ) by a homotopy

K^ s = (1 , s)K^ 0 + sK^ 1 ; H^ s = (1 , s)H^ 0 + sH^ 1 and by studying how the solutions of (54) change as one varies the parameter s from 0 to 1. After a making a small perturbation, if necessary, one may again assume that the map n o

(s; z ) = z 0 + I + K^ s (t; z ) rfH^ s (t; z ) 34

from [0; 1]  Q1 ! Q0 has 0 as regular value. ^ K^ ) solutions of (54) with Fredholm For any particular regular " perturbation (H; index -1, i.e. solutions connecting critical points of H0 with index p to critical points of H1 with index p +1 cannot occur. But for one parameter families f(H^ s ; K^ s ) j 0  s  1g of " perturbations solutions with index -1 will occur at isolated s values. Thus there will be a finite number of (sj ; zj ) 2 ,1 (0) where zj connects a critical point of index p to one of index p + 1. Using these connections one can write down a homomorphism Cp (N; H0 ) ! Cp+1 (N; H1 ) which is a chain homotopy from H^ 0 ;K^ 0 to H^ 1 ;K^ 1 . We refer to the quoted works for more details on this construction. The conclusion is that different " perturbations lead to the same homomorphism 1;0 of Floer homology. The second part of the proof of Lemma 28 relies on a gluing argument. Given three ^ 1;0 ; K^ 1;0 ) pairs (Hi ; Ki ) 2 reg (N ) \ U one considers the canonical homotopies (H ^ ^ and (H2;1 ; K2;1 ) defined by (56), and defines another homotopy 

H 2;0 (t; x; z ) = (1 , (t))H^ 1;0 (t + T; x; z ) + (t)H^ 2;1 (t , T; x; z ); K 2;0 (t; z ) = (1 , (t))K^ 1;0 (t + T; z ) + (t)K^ 2;1 (t , T; z ):

 2;0 ; K 2;0 ) is an " perturbation of (H; K ), at least if T is large. One easily shows that (H Given any z 2 Crp (N; H ) one can show by means of a gluing argument [19] that T  (z ) equals the composition T  T (z ). Since for large enough T 2 R H 2;1 1;0 2;0 ;K2;0 we know that the homomorphism induced on Floer homology does not depend on the chosen regular " perturbation, we conclude that 2;0 ([z ]) = 2;1  1;0 ([z ]), for any [z ]. Since the [z ] generate FHp (N; H ) this shows that 2;0 = 2;1  1;0, as claimed. Finally, if (H0 ; K0 ) = (H1 ; K1 ), then the constant homotopy is regular, and the only orbits connecting critical points of the same index are constant orbits. Thus in this case one immediately gets 0;0 = id. 8.4.3 Global continuation. In the previous section we have seen that every (H; K ) 2 (N ) has a neighborhood U such that all (H 0 ; K 0 ) 2 U \ reg (N ) have isomorphic Floer homologies, and that the isomorphisms are natural. Hence, if the given pair (H; K ) is not regular, then we may define the Floer homology of this pair to be the Floer homology of any of the regular pairs (H 0 ; K 0 ) 2 U without introducing any ambiguity. We will say that two Hamiltonian and metric pairs (Hi ; Ki ) are related by continuation if they lie in the same path component of (N ). Since (N ) is an open subset of an affine space (H0 + H)  K it is locally path connected, so that path components and connected components coincide. Given a path f(Hs ; Ks ) 2 (N ) j 0  s  1g, from (H0 ; K0 ) to (H1 ; K1 ) one can find a partition 0 = s0 < s1 <


< sm = 1 such that the Floer homologies of (Hsi ; Ksi ) and (Hsi+1 ; Ksi+1 ) are related by the natural isomorphisms of Lemma 28 (i.e. (Hsi+1 ; Ksi+1 ) 2 U (Hsi ; Ksi )). By composing these isomorphisms one obtains an isomorphism of the Floer homologies of (H0 ; K0 ) and (H1 ; K1 ). This isomorphism depends on the path taken in (N ) from (H0 ; K0 ) to (H1 ; K1 ); but since the Floer homology of a pair (H; K ) is locally constant, the isomorphism from FH (H0 ; K0 ) to FH (H1 ; K1 ) actually only depends on the homotopy class of the path f(Hs ; Ks ) 2 35

(N ) j 0  s  1g. Should (N ) be connected and simply connected all (H; K ) 2 (N ) have naturally isomorphic Floer homologies. If (N ) is multiply connected, then all (H; K ) 2 (N ) still have isomorphic Floer homologies, but the isomorphisms

need not be unique. In particular one obtains a representation of the fundamental group of (N ) in FH (H; K ) by mapping each closed path in (N ) with base point (H; K ) to the corresponding automorphism of FH (H; K ). To conclude this section we remark that one can also vary the isolating neighbourhood N and look for continuation theorems for Floer homology under such variation. Clearly if two isolating neighborhoods N1 and N2 contain the same isolating invaria nt set S = S (N1 ) = S (N2 ), then their Floer homologies must be equal, since the Floer homology groups are defined in terms of the critical points and connecting orbits between them which lie on S . Thus Floer homology will not change under small perturbations of the isolating neighborhood. To go beyond this local continuation observation one could imitate the construction given by Conley in [8, Chapter IV]. We will not do this here.

9 Existence results for critical points 9.1 The index of constant coefficient linearizations. In [2] we studied the index of critical points with the objective of finding a priori estimates for critical points in terms of their index. Along the way we found the following simple properties of the index [2, section 3]. Before describing these properties we note that the index of a critical point z 2 X of fH only depends on the 2  2 matrix function 



uu (x; z (x)) Huv (x; z (x)) x 2 7! Hzz (x; z (x)) = H Hvu (x; z (x)) Hvv (x; z (x)) : Lemma 29 If z0 , z1 2 X are critical points of fH and fH respectively, and if H0;zz (x; z0 (x))  H1;zz (x; z1 (x)) pointwise, then 0 (z0 )  0 (z1 ). Lemma 30 Let z be a critical point with   Hzz (x; z (x)) = ,AC ,BC ; where A, B and C are constants, then p 1. z is a Morse critical point if either AB < 0 or else if C + (AB ) is not an eigen 0

1

value of ,
;

2. if AB

 0 then z has index 0.

,

the number of eigenvalues of ,
in the interval C ,
p(AB>); C0 +thenp(letABk)be ; If A; B > 0 then the index of z is k , if A; B < 0 then z has index ,k .

3. if AB

36

9.2 “,” type Hamiltonians. All existence results follow from the following basic fact. Theorem 31 Let H

2 C 2 (   R2 ) satisfy (12), (24). Then for any K 2 K1=2 FHp (X ; H; K ) =



Z2

0

if p = 0 otherwise

1  2 Proof. We may assume that H is smooth. Let , 0
h 2 C (  R ) have compact ,
support and choose h so that (H + h)z (x; 0; 0) = 0 and (H + h)zz (x; 0; 0)  00 00 . Since h has compact support, H + h satisfies (12), (24) for all  2 [0; 1], and H and H + h are related by continuation. Next put

u2 , v 2 1 + "(u2 + v2 )

~h(u; v) def = def

~ = H + h + h~ are also related by continuation. For large enough  Then H and H and small enough " > 0 one has ,




uH~ u , vH~ v   jujp+1 + jvjq+1 ;

(58) 

,




~ z (x; 0; 0) = 00 and H~ zz (x; 0; 0) = 0 ,0 . It follows for some  > 0, while H from lemma 30 that 0 is a Morse critical point with index 0. Finally, the following computation: 0 = =

Z

fv
u , u
vg dx

Z n

 

Z

o

uH~ u , vH~ v dx





jujp+1 + jvjq+1 dx

shows that (0; 0) is the only critical point of fH~ . Hence fH~ is a Morse function with 2 only one critical point, which happens to have index 0. 9.2.1 Existence of a nontrivial solution. Let H be a “,” type Hamiltonian, and assume that (0; 0) is a Morse critical point with index m 6= 0. It then follows immediately from Theorem 31 that (0; 0) cannot be the only critical point. If it were, then fH would be a Morse function whose only critical point has index m 6= 0; in particular, FHm (X ; H ) would be Z2, contradicting Theorem 31. If fH is a Morse function then we can even assert the existence of two non zero critical points. Indeed, let bp denote the rank (or dimension) of FHp (X ), and let cp be the number of critical points of fH with index p, i.e. the rank of Cp (X ; H ). Then, 37

since FHp (X ) are the homology groups associated to the following relation between the bp and cp : X

cp tp =

X

Cp (X ; H ) we have the

bp tp + (1 + t)Q(t) for some Q(t).

(59)

In the present context we know that all coefficients are finite, since the number of critical points of a “,” type Hamiltonian is finite. One could therefore recover the usual Morse inequalities by dividing both sides by 1 + t. We know that all bp = 0, except for b0 = 1, so that we get X

which, if we write Q(t) =

P

cp tp = 1 + (1 + t)Q(t);

qp tp , leads to cp = p;0 + qp + qp,1 :

(60)

This implies c0  1, so there exists a critical point of index 0. To obtain a second critical point we first consider the case m > 0. Since cm  1 we find that either qm > 0 or qm,1 > 0. If qm > 0 then cm+1  qm > 0, so there must exist a critical point with index m + 1. In this case we therefore have two nontrivial critical points, with indices 0 and m. If qm,1 > 0 then either m > 1 and we again get two nontrivial critical points whose indices are 0 and m , 1; if m = 1 then (60) implies that c0  2, so we still have two nonzero critical points, both of which have index 0. The case m < 0 is similar. 9.2.2 Even “,” type Hamiltonians. If we assume that H is even, i.e.

H (x; ,u; ,v)  H (x; u; v); then the Morse relations (59) imply the existence of more critical points. Suppose fH is Morse, an assume that 0 is a critical point with index m > 0, then we will show that there must exist critical points of indices 0; 1; 2; : : :, m , 1. Indeed, since ,z is critical point for any critical point z , all cp must be even, except cm which must be odd. Thus (60) implies that qp + qp,1 must be even, except when p = 0 or p = m, when qp + qp,1 must be odd. Hence all qp with p < 0 have the same parity. Since H is a “,” type Hamiltonian there are only finitely many critical points so that all cp with p  0 must be even. Then q0 must be odd, to make q,1 + q0 odd. After that all qp with p  m must be odd, and finally all qp with p > m must be even. We conclude for 0  p < m from (60) that cp  2, as claimed. It seems very likely that the nondegeneracy condition is not crucial for this result, but an existence proof of 2m critical points without the nondegeneracy hypotheses would probably require cup-length estimates and thus information about cup products on Floer homology. We will not pursue this here.

38

9.3 “+” type Hamiltonians. We again begin by computing the Floer homology of the whole space X .

  R2 ) satisfy (12), (23). Assume moreover that for any Theorem 32 Let H 2 C 2 (

 2 R there exists R < 1 such that for all x 2 and z 2 R2 with jz j  R one has 



; Hzz (x; z )  j j2 ; 8 2 R2 : (61) Then for any K 2 K1=2 , FHp (X ; H; K ) = 0: Proof. Our hypothesis (61) implies that Hzz is uniformly bounded from below, say by ,C I. Let  be any (large) number; choose R as given in (61). Construct a bounded function h 2 C 1 (R2 ) with hzz (z )  ( + C )I for jz j  R and hzz (z )  , 2 I for all z 2 R2 . Then H and H + h are related by continuation. But H + h satisfies (H + h)zz (x; z )  2 I for all x, z , so Lemmas 29 and 30 imply that any critical point of fH +h has index at least m=2 where m is the number of eigenvalues of ,
less than . This number can be made arbitrarily high, so we see that for any given p 2 Z we can find a Hamiltonian H + h which is related to H by continuation which does not have critical points of index p. Hence Cp (X ; H + h; K ) must be trivial, and thus FHp (X ; H; K ) is also trivial. 2 9.3.1 Existence of a nontrivial solution. In general our computation of the Floer homology groups of X does not force a “+” type Hamiltonian to have critical points at all! However, if fH has one nondegenerate critical point then there must exist another critical point, for otherwise we could conclude that one of the Floer homology groups would be nontrivial. This way we recover the result in [11, 13], in the cases where their assumptions and ours overlap. Indeed, assuming pFF > 2 and qFF > 2, and  = p, = q (notation as in [11]. DeFigeuiredo and Flemer define the exponents p and q differently: our p is pFF , 1) one of their conditions requires the Hamiltonian to vanish super quadratically at z = 0, which implies that 0 is a nondegenerate critical point of index 0. Hulshof and van der Vorst [13] ,
make a similar hypothesis. The presence of one nondegenerate critical point z = 00 then forces the existence of another critical point, for reasons we have just stated. 9.3.2 Infinitely many solutions if H is even.

We again assume that H is even, and for the moment assume that fH is a Morse function. Then z = 0 is a critical point; let m be its index. Theorem 33 Either fH has a pair of critical points fzk ; ,zk g with index k for each k > m, or else there is an m such that fH has infinitely many critical points with index m .

Proof. Assume that fH has only finitely many critical points with any given index p. We have called this number cp . The Morse relations (59) imply cp = qp + qp,1 (62) 39

for all p. Since fH is even, all cp are even, except cm , which is odd. Since Hzz (x; z ) is bounded from below the index of critical points is also bounded from below, so that for p ! ,1 one has cp = 0. For all p < m (62) implies that qp and qp,1 have the same parity; they must all be even since cp = 0 for very negative p. We may conclude that all qp with p  m have the same parity; since qm = cm , qm,1 , and cm is odd, qm and qm,1 have opposite parity, so we find in the end that all qp with p  m are odd. Therefore cp  2 for all p  m, and we find that fH has infinitely 2 many critical points. We now show how to obtain infinitely many critical points without the hypothesis that fH is a Morse function. Theorem 34 Let H be an even “+” type Hamiltonian as above. Then the set of critical points cannot be bounded in L1 . Proof. Suppose that for some “+” type H the critical point set of the functional fH is bounded in L1 . By regularity it would then also be compact in E and there would be an a 2 R such that any a0  a is a regular value of fH . Define

Na = fz 2 X j fH (z )  ag; N a = fz 2 X j fH (z )  ag: Both sets are stable isolating neighborhoods for (H; K ) for any K 2 K1=2 . By hypothesis N a contains no critical points, so its Floer homology groups are all trivial. To compute the Floer homology of (Na ; H; K ) we perturb H and K slightly so H 0 becomes a Morse function and the (H 0 ; K 0) flow is a Kupka Smale flow. We may assume that the perturation H 0 is again even. Thus all critical points of fH 0 occur in pairs, except for 0. The total number of critical points in Na (which is finite, since the action fH 0 of the critical points is bounded by a + o(1)) must therefore be odd. By t = 1 in the Morse relations (51) for (Na ; H 0 ; K 0) we therefore find that substituting P rankFHp (Na ; H 0 ; K 0) is odd. Now apply (53) to the triple (Na ; N = X ; N a ). We get

PX ;H;K (t) + (1 + t)Q(t) = PNa ;H;K (t) + PN a ;H;K (t): The first term on the left and the last term on the right both vanish. If we substitute t = 1 we get an even number on the right, and an odd number on the left. The contradiction 2 shows that the critical point set of our original fH cannot be compact.

9.4 Estimating the index of critical points. Theorem 33 allows the possibility of an H which has infinitely nondegenerate critical points with the same index m , and no other critical points. We would not expect this to happen, but have not been able to prove our expectation in full generality. However, for Hamiltonian functions of the form

p+1 q+1 H (x; u; v) = a(x) jpuj+ 1 + b(x) jqvj+ 1 + h(x; u; v) 40

(63)

with h(x; u; v ) uniformly bounded we have obtained in [2] the following compactness theorem. Proposition 35 Let  Rn with n  3. Assume H is as in (63) and assume also that

Huu ; Hvv  0;

Huv  0:

(64)

Then any critical point z 2 E of fH with lower Morse index m is bounded by kz kL1  C (m; M ) for some constant C (m; M ) < 1 depending only on m and M = supx;u;v jh(x; u; v)j.

In fact we prove this for a slightly larger class of perturbations h (see [2, Theorem 1B] for details.) We have not been able to remove the hypothesis that the domain be at least 3 dimensional; neither have we been able to weaken the assumptions (64). It seems likely to us that these hypotheses are not necessary, and that they could perhaps be removed by following a different method of proof. The proposition can be applied directly to improve the statements of theorems 33 and 34. If fH is a Morse function, and H; are as in the proposition, then fH cannot have infinitely many critical points with any given particular index, so that for each k  m there must be a pair of critical points fzk g with index k . If fH is not a Morse function, then one can still assert that the lower Morse indices of the unbounded sequence of critical points zk obtained in theorem 34 is unbounded, i.e. , (zk ) ! 1.

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Pure and Applied Mathematics,

[27] R.C.A.M. van der Vorst, Variational identities and differential systems, Arch. Rat. Mech. (1991), 116, 375-398. Sigurd Angenent Department of Mathematics University of Wisconsin 480 Lincoln Dr Madison, WI 53706-1313, USA

Robertus van der Vorst Center for Dynamical Systems and Nonlinear Studies Georgia Tech Atlanta, GA 30332-0190, USA and Leiden University P.O. Box 9500 Leiden, The Netherlands

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