Generating Functions, Symplectic Geometry, and ... - Semantic Scholar

Generating Functions, Symplectic Geometry, and Applications CLAUDE V I T E R B O

Département de Mathématiques, Bâtiment 425 Université de Paris-Sud and URA 1169 du C.N.R.S. F-91405 Orsay Cedex, France

Contents 1. 2. 3. 4. 5.

Symplectic manifolds, their Lagrange submanifolds and generating functions Existence and uniqueness theorems for generating functions Symplectic invariants, solutions of Hamilton-Jacobi equations and applications. Generalized generating functions and applications to Floer homology computations. Applications to Hamiltonian dynamics and obstructions to Lagrange embeddings

1 Symplectic manifolds, their Lagrange submanifolds and generating functions A symplectic form on a manifold is a closed two form UJ, nondegenerate as a skewsymmetric bilinear form on the tangent space at each point. Integration of the form on a two-dimensional submanifold S with boundary 8S in M associates to S a real number (positive or negative) the "area of 5", which due to Stokes' formula only depends on the curves dS, and the homology class of S rei dS. If moreover the form is exact, that is UJ = dX, the area of S is obtained by integrating A over dS. In this case it is also possible to integrate A on loops nonhomologous to zero and we get the notion of "area enclosed by a loop ". However this area depends on the choice of A. If this choice is fixed once for all, we shall then talk about an exact manifold. One should be careful about the fact that this notion is slightly different from that of a symplectic manifold with exact symplectic form (because in the latter case we have not chosen the primitive of CJ). It is a theorem of Darboux that the simplest example, R2n with the constant symplectic form o = Y^j=i ^xj A dy3, is also the universal local model (i.e. any symplectic manifold is locally symplectomorphic to (M2n,cr). The main example for us will be the exact manifold T*N, the phase space of Af, where the exact form is the "contraction tensor" given in local coordinates by £]j =1 7Jjdç J , where qj are local coordinates on N and pj are the dual coordinates. Submanifolds of a sjunplectic manifold inherit naturally the 2 form induced by UJ. This is naturally a closed form, but only exceptionally nondegenerate. In this Proceedings of the International Congress of Mathematicians, Zürich, Switzerland 1994 © Birkhäuser Verlag, Basel, Switzerland 1995

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case the submanifold is called symplectic. Other remarkable cases occur if the form vanishes on the submanifold, which is then called isotropic (if 2-dim(y) < dim(M)) or Lagrangian (if 2 • dim(Vr) = dim(M)). It is then maximal among isotropic submanifolds. In terms of area, any contractible curve on a Lagrange submanifold has zero area. In an exact manifold, any two homologous curves on L have the same area. If moreover any two curves have the same area (which is then necessarily zero), the Lagrange submanifold is called exact. This is equivalent to the exactness of the pullback of A on L. In T*N there is a particularly simple family of Lagrange submanifolds. To any closed one form a we may associate La = {(q,a(q)) \ q E N}. The form induced by A on La is just a; thus, UJ = dX induces da = 0. In particular, if a = df is exact we get an exact Lagrange submanifold. A remarkable property of a Lagrange submanifold in T* N is that it intersects the zero section more often than a differential topologist would expect. For Lf, we see that points in Lf Ci ON are in one-to-one correspondence with critical points of / . In any case, as AT is compact, there are at least two such points. One of the Arnold conjectures, partially solved by Hofer in 1983 (see [H]), claims that for L exact and obtained from the zero section by a Hamiltonian isotopy, the number of points in L D OJV is bounded from below by the Lusternik-Shnirelman category of N (i.e. the minimal number of critical points for a function on N). For Lf this conjecture is obvious. However, the Lagrange submanifolds that may be written as Lf are exactly those for which the projection p : T*N —> N restricts to a diffeomorphism. Our main interest will be on Lagrange manifolds, and we shall represent them through their generating functions, an idea first introduced by Hörmander for different purposes (see [Hö]). A generating function for the Lagrange submanifold L is a function S : E —• R, defined on a vector bundle E over N, and such that L = < ( x, —— ) 9 —- = 0 > where x is in N, and £ in the fibre dx

IV

J

£

J

(assuming that 0 is a regular value for | f ) . Generating functions have the advantage of preserving the following interesting property: the points in L$ H ON ^G in one-to-one correspondence with the critical points of S. The apparent drawback is that there are of course functions on E with no critical points (because E is noncompact), but this may be restored if we restrict ourselves to Generating Functions Quadratic at Infinity (abbreviated as G.F.Q.I.s): DEFINITION (G.F.Q.I.). A generating function Sis aG.F.Q.I. if and only if there exists a fibrewise quadratic nondegenerate form Q(x, £) such that S(x, £) — Q(x, £) has compact support.

The main example is associated to a symplectic diffeomorphism / that we assume to be the time one flow of a compact supported Hamiltonian. Then T^ = {(z,(j)(z)) | z G R2n} is a Lagrange submanifold of R2n_x R2n (R2n is simply R2n with the symplectic form -UJ). We shall identify R2n x R2TI with T*(A) = T*(TId)

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(where A is the diagonal). Note that if £ is denoted by TT, and GL will be ir~1(L). For S,T two G.F.Q.I. on E and F, we denote by S © T the G.F.Q.I. on E © F defined by (S © T)(x,£, rj) = S(x, £) + T(x, rj). Let us introduce the following two equivalence relations on Q\ (a) S\ ~ S2 if a n d only if there are nondegenerate quadratic forms Q\,Q2 on Fi, F% and a fibre-preserving diffeomorphism $ ; E\ © F\ —> E"2 © Fi such that (Si©Qi) = ( S 2 © Ç 2 ) o $ . (b) S\ ~ S*2 if and only if there are G.F.Q.I. for the zero section Xq,£2 defined on F\, F R is a smooth function satisfying the Palais-Smale condition. The smoothness and transversality assumptions clearly extend to the Banach space setting. And finally, our proof of uniqueness can be also extended without adding any new ingredient to this case. Note however that the G.F.Q.I. should be such that the quadratic function at infinity should have finite index. A typical example is as follows. Let L(t, q, q) be a Lagrangian on TN such that §^L is invertible. S(q)=

/ Jo

L(t,q,q) L(t,q,q)dt

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defined on the Banach bundle V = {q : [0,1] -> AT | q(0) = 0} and n : V -> AT is given by 7r(ç) = ç(l). We claim that S is a generating function for ^i(Ojv) where (j)t is the flow associated to the Hamiltonian H obtained from L by Legendre duality. The computation is omitted, since a very similar one follows. S will never be quadratic at infinity, but if L(t, q, q) = \\q\\2 outside a compact set, it is easy to show that H*(EC,E~C) = H*(N) for c large enough (it is easy to prove this by comparison with L— ||#|| 2 ). Now we turn to the more subtle version of infinite-dimensional generating functions, which we shall call Floer generating functions. This relies on Floer's idea to deal with the variational theory of the action functional, as there was not, prior to Floer's work, any reasonable approach to the variational study of this functional on a general manifold (see [R] for a finitedimensional approach in the case of R2n). One way to understand the introduction of generating functions is to consider the action functional as such a function. Let S be the set of paths, S = {7 = (q,p) : [0,1] -> T*N \ p(0) = 0} (we do not specify the regularity of the path, as it is of no interest for the moment) and TT : S —> N be the map 7 —> q(l). Then consider the function AH defined as

AH(l)=

I

\pJi-H(t,q,p)}dt.

Jo

We have that DAH(1)81

/•l f)TI F)TI = / [(Jq - — )6p - (Jp - —)6q}(t,q,p)

dt + p(l)6q(l).

Thus, the set of 7 such that the derivative of AH in the direction of the fibres of 7T vanishes corresponds to solutions of •

9H

.

n

8H

n

that is 7(t) = FH*(T*N) = H*(AN), satisfying the following algebraic property:

*Ü)(a:UAOTG/)) = $Ü)(aOUy. It is easy to construct, for many pairs of manifolds (L,N), obstructions to the existence of such maps. We refer to [V6] for many examples. Let us just quote the solution to a previously open question (cf. [Lal-S]). THEOREM.

There is no exact embedding from T2 to T*S2.

REMARK. The method of proof in [V7] (see also [V5]) is more complicated, because the relation between Floer cohomology and generating functions had not been established yet. Thus, the whole proof is based on generating functions methods. We also mention another application of the above computation to the Weinstein conjecture in a cotangent bundle. A compact hypersurface in a symplectic manifold is said to be of contact type if there is a conformai vector field (i.e. Lçu) — UJ) defined in a neighborhood of the hypersurface, and transverse to it. The characteristic flow on a hypersurface is the flow of a time-independent Hamiltonian

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having t h e hypersurface as a regular energy level. T h e special feature of a contact t y p e hypersurface is t h a t a neighborhood of it is foliated by hypersurfaces having diffeomorphic characteristic flow. T h e conjecture of Weinstein claims t h a t such a hypersurface always has a periodic characteristic. This was first proved in M 2 n by the author ([VI]), later extended in joint work with Hofer and Floer ([HV1], [FHV], [HV2]). In particular in [HV1], it is proved t h a t the conjecture holds for a contact hypersurface in T*N surrounding the zero section. But, strangely enough, it was left open for a general contact hypersurface inT*N, even though, as pointed out by Chaperon, if N has a Lagrange embedding in R2n, then the Weinstein conjecture in T*N holds as a consequence of it holding in M 2 n . T h e above computation, together with the information on the structure of t h e ring H' * (AN) in the simply connected case due t o Goodwillie (see [Go]), may be exploited to prove (see [V7]): THEOREM

instein

( W E I N S T E I N C O N J E C T U R E IN S.C. C O T A N G E N T B U N D L E S ) .

conjecture

holds in T*N for N a simply connected compact

The

We-

manifold.

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[Hö] L., Hörmander, Fourier integral operators I, Acta Math. 127 (1971), 79-183. [J] T. Joukovkskaia, Thèse d'université, Université de Paris 7, Denis Diderot, 1993. [Lal-S] F. Lalonde and J.-C. Sikorav, Sous-variétés lagrangiennes et lagrangiennes exactes des fibres cotangents, Comm. Math. Helv. (1991), 18-33. [LS] F. Laudenbach and J.-C. Sikorav, Persistance d'intersection avec la section nulle au cours d'une isotopie hamiltonienne dans un fibre cotangent, Invent. Math. 82 (1985), 349-357. [R] P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appi. Math. 31 (1978), 157-184. [S] J.-C. Sikorav, Problèmes d'intersection et de points fixes en géométrie hamiltonienne, Comm. Math. Helv. 62 (1987), 61-72. [Tli] D. Théret, Equivalence globale des fonctions génératrices, preprint, Université de Paris 7, 2 Place Jussieu, 75230 Paris Cedex 05. [TV] L. Traynor, Symplectic homology via generating functions, preprint, Math. Department, Bryn Mawr College, Bryn Mawr, PA. [VI] C. Viterbo, A proof of Weinstein conjecture in R n, Ann. Inst. H. Poincaré Analyse Non Linéaire 4 (1987), 337-356. [V2] C. Viterbo, A new obstruction to embedding Lagrangian tori, Invent. Math. 100 (1990), 301-320. [V3] C. Viterbo, The cup product on the Thorn-Smale-Witten complex and Floer cohomology, to appear in Floer Memorial volume, Birkhäuser, Basel and Boston. [V4] C. Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann. 692 (1992), 685-710. [V5] C. Viterbo, Properties of embedded Lagrange submanifolds, Proc. First Europ. Congress Math., Paris 1992; Birkhäuser, Basel and Boston 1994. [V6] C. Viterbo, Some remarks on Massey products, tied cohomology classes and Lusternik-Shnirelman category, preprint. [V7] C. Viterbo, Exact Lagrange submanifolds, periodic orbits, and the cohomology of loop spaces, in preparation. [V8] C. Viterbo, Floer generating functions and applications to symplectic topology, in preparation.