- Email: [email protected]
On a result of Eliashberg and Gromov
C. R. Acad. Sci. Paris, t. 333, Série I, p. 657–661, 2001 Géométrie différentielle/Differential Geometry
On a result of Eliashberg and Gromov Marc CHAPERON Géométrie et Dynamique, Institut de mathématiques de Jussieu, UFR de mathématiques, Université Paris-7, case 7012, 2, place Jussieu, 75251 Paris cedex 05, France E-mail: [email protected] (Reçu le 17 février 2001, accepté le 5 mars 2001)
Abstract.
We give a very simple proof of a theorem of Eliashberg and Gromov implying that intersection between the conormal bundles νM and νN of two proper submanifolds M , N of Rn is persistent under compactly supported Hamiltonian deformations of νM and νN . 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Sur un résultat d’Eliashberg et Gromov Résumé.
Nous donnons une preuve très simple d’un théorème d’Eliashberg et Gromov impliquant que si les fibrés conormaux νM et νN de deux sous-variétés propres M , N de Rn se rencontrent, alors des déformations hamiltoniennes à support compact arbitraires de νM et νN se coupent aussi. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Version française abrégée Dans [2], Eliashberg et Gromov donnent une démonstration assez indirecte du : T HÉORÈME. – Soient M et N deux sous-variétés propres d’une variété V telles que M ∩ N soit compact et non vide. Supposons en outre M et N transverses, c’est-à-dire que l’intersection de leurs fibrés conormaux νM = {(Q, p) ∈ T∗ V | Q ∈ M , p|TQ M = 0} et νN = {(Q, p) ∈ T∗ V | Q ∈ N , p|TQ N = 0} est {(Q, p) ∈ T∗ V | Q ∈ M ∩ N , p = 0}. Alors, pour toute isotopie hamiltonienne (gt )0t1 de T∗ V à support compact, g1 (νN ) ∩ νM contient : – au moins (c (C) + 1) points, où la somme se fait sur les composantes connexes C de M ∩ N et c
désigne la longueur cohomologique (« cuplength ») ; – au moins dim H∗ (M ∩ N ) points si l’intersection g1 (νN ) ∩ νM est transversale (le corps de base étant choisi de manière à maximiser ces bornes). Cette Note le prouve sans détour si V = Rn . Note présentée par Étienne G HYS. S0764-4442(01)01939-5/FLA 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés
657
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Remarques. – Eliashberg et Gromov ne donnent que la seconde estimation bien que la première résulte sans doute de leur théorie. – Pourvu que l’isotopie (gt ) soit à support compact, le théorème est vrai a fortiori si M et N ne sont pas transverses ou ont une intersection non compacte, car g1 (νN ) ∩ νM est alors infini puisque νN ∩ νM va à l’infini. – Le théorème est faux pour les isotopies à support non compact ; en effet, toute isotopie (ht )0t1 de V se relève en l’isotopie hamiltonienne gt (q, p) = (ht (q), p ◦ ∂ht (q)−1 ) de T∗ V , et h1 (N ) ∩ M = ∅ donne g1 (νN ) ∩ νM = ν(h1 (N )) ∩ νM = ∅. – Le cas particulier important où M = V est dû à Lalonde et Sikorav [3], théorème 3 (i). Idée de la démonstration. – Un argument géométrique simple (paragraphe 2 ci-dessous) permet de supposer N compacte. Rappelons (cf. par exemple [1]) que g1 admet une famille génératrice « quadratique », c’est-à-dire une fonction S : T∗ Rn × Rk → R possédant les trois propriétés suivantes : (i) L’application T∗ Rn × Rk (Q, p; v) → ∂v S(Q, p, v) est transverse à 0 ∈ (Rk )∗ . (ii) Il existe une forme quadratique non dégénérée K : Rk → R telle que S(Q, p; v) = K(v) en dehors d’un compact. (iii) Le graphe de g1 est l’ensemble des (q, p), (Q, P ) ∈ (T∗ Rn )2 tels que q = Q + ∂p S(Q, p; v), P = p + ∂Q S(Q, p; v), 0 = ∂v S(Q, p; v) pour un v ∈ Rk , ce qui définit un difféomorphisme de ∂v S −1 (0) sur le graphe de g1 . On prouve sans peine les deux lemmes suivants : L EMME 1.1. – Les points critiques de la fonction F : M × N × (Rn )∗ × Rk → R définie par F (x, y, p; v) = S(x, p; v) + p(x − y) sont en bijection avec νM ∩ g1 (νN ). L EMME 1.2. – La fonction F0 (x, y, p; v) = K(v) + p(x − y) sur M × N × (Rn )∗ × Rk est égale à F en dehors d’un compact et possède les propriétés suivantes : (i) Son ensemble critique Crit(F0 ) = ∆(M ∩ N ) × {(0, 0)} est une sous-variété normalement non dégénérée d’indice de Morse Ind(K) + n, contenue dans F0−1 (0). (ii) Si l’on munit M × N × (Rn )∗ × Rk de la métrique riemannienne (complète) induite par les métriques euclidiennes standard de Rn et Rk , le flot du gradient de F0 est complet et la condition de Palais– Smale est satisfaite. Pour achever la preuve, il ne reste plus qu’à choisir un réel R > 0 tel que F = F0 pour F −R et F R (c’est possible puisque F − F0 est à support compact). Les points critiques de F se trouvent entre les niveaux −R et R, et F satisfait à la condition de Palais–Smale. Or, d’après le lemme 1.2, {F R} = {F0 R} s’obtient à partir de {F −R} = {F0 −R} en lui attachant une « anse », en l’occurrence un fibré en boules fermées de dimension n + Ind(K) sur M ∩ N , le long de son bord. Des arguments standard permettent donc de conclure. Remarques. – On voit apparaître naturellement ici des fonctions qui ne sont pas « quadratiques » à l’infini (l’idée de F est inspirée de la « différence » de deux fonctions génératices introduite par Eliashberg et Gromov [2]). Les fermés {F0 ±R} peuvent être très compliqués mais ce qui compte, c’est-à-dire la différence entre leurs topologies, vient de notre parfaite compréhension des points critiques de F0 . Nous espérons donner d’autres applications de cette remarque à la géométrie symplectique et de contact.
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On a result of Eliashberg and Gromov
In [2], Eliashberg and Gromov use a somewhat indirect argument to prove the following: T HEOREM. – Let M and N be two proper submanifolds of a manifold V such that M ∩N is compact and non-empty. In addition, assume that M and N are transversal, i.e. that the intersection of their conormal bundles νM = {(Q, p) ∈ T∗ V | Q ∈ M , p|TQ M = 0} and νN = {(Q, p) ∈ T∗ V | Q ∈ N , p|TQ N = 0} is {(Q, p) ∈ T∗ V | Q ∈ M ∩ N , p = 0}. Then, for any Hamiltonian isotopy (gt )0t1 of T∗ V with compact support, g1 (νN ) ∩ νM contains – at least c (C) + 1 points, where the sum is over the connected components C of M ∩ N and c
denotes the cuplength; – at least dim H∗ (M ∩ N ) points if the intersection g1 (νN ) ∩ νM is transversal (the base field being so chosen as to get the best estimates). This Note provides a straight-forward proof when V = Rn . Notes. – Eliashberg and Gromov only give the second lower bound, even though their theory probably yields the first as well. – When M and N are not transversal, νM ∩ νN contains lines, hence g1 (νN ) ∩ νM is infinite since the isotopy has compact support. Similarly, if M and N intersect transversally but the intersection is not compact, then g1 (νN ) ∩ νM is infinite. – The theorem is false for non-compactly supported isotopies; indeed, any isotopy (ht )0t1 of V can be lifted to the Hamiltonian isotopy gt (q, p) = (ht (q), p ◦ dht (q)−1 ) of T∗ V , and h1 (N ) ∩ M = ∅ yields g1 (νN ) ∩ νM = ν(h1 (N )) ∩ νM = ∅. – In the important particular case where M = V , the theorem is due to Lalonde and Sikorav [3], théorème 3 (i). 1. Proof of the theorem when V = Rn and N is compact Recall (see for example [1]) that g1 has a “quadratic” generating family, i.e. an S : T∗ Rn × Rk → R with the following three properties: (i) The map T∗ Rn × Rk (Q, p; v) → ∂v S(Q, p, v) ∈ (Rk )∗ is transversal to 0 ∈ (Rk )∗ . (ii) There is a non-degenerate quadratic form K : Rk → R such that S(Q, p; v) = K(v) off a compact subset. (iii) The graph of g1 is the set of those (q, p), (Q, P ) ∈ (T∗ Rn )2 such that q = Q + ∂p S(Q, p; v), P = p + ∂Q S(Q, p; v), 0 = ∂v S(Q, p; v) for some v ∈ Rk , and this defines a diffeomorphism of ∂v S −1 (0) onto the graph of g1 . L EMMA 1.1. – The critical points of the function F : M × N × (Rn )∗ × Rk → R defined by F (x, y, p; v) = S(x, p; v) + p(x − y) are in one to one correspondence with νM ∩ g1 (νN ). Proof of Lemma 1.1. – Setting (as in (iii)) q = x + ∂p S(x, p; v) and P = p + ∂Q S(x, p; v) (where ∂Q denotes derivation with respect to the whole of the first factor Rn , not just M ), we have that ∂p F (x, y, p; v) = q − x + x − y = q − y, ∂x F (x, y, p; v) = (P − p + p)|Tx M = P |Tx M , ∂ y F (x, y, p; v) = −p|Ty N , ∂v F (x, y, p; v) = ∂v S(x, p; v).
(1)
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By (iii) and the last line of (1), ∂v F = 0 if and only if (x, P ) = g1 (q, p); hence, by the first line of (1), ∂p F = 0 and ∂v F = 0 if and only if (x, P ) = g1 (y, p). By the remaining of (1), it follows that (x, y, p; v) is a critical point of F if and only if we have (y, p) ∈ νN and g1 (y, p) = (x, P ) ∈ νM , which proves our lemma. ✷ L EMMA 1.2. – The function F0 (x, y, p; v) = K(v) + p(x − y) on M × N × (Rn )∗ × Rk is equal to F off a compact subset and has the following properties: (i) Its critical set Crit(F0 ) = ∆(M ∩ N ) × {(0, 0)} is a normally non-degenerate submanifold of Morse index Ind(K) + n, lying in F0−1 (0). (ii) If we endow M × N × (Rn )∗ × Rk with the (complete) Riemannian metric induced by the standard Euclidean metrics of Rn and Rk , then the gradient flow of F0 is complete and the Palais–Smale condition is satisfied. Proof of Lemma 1.2. – The partial derivatives ∂x F0 = p|Tx M , ∂y F0 = −p|Ty M and ∂p F0 = x − y vanish simultaneously if and only if y = x ∈ M ∩ N and (as Tx M + Tx N = Rn by transversality) p = 0. Since ∂v F0 = dK and K is non-degenerate, there is no doubt about the critical set of F0 . The Hessian F0 (x, x, 0, 0) of F0 at a critical point (x, x, 0, 0) is the bilinear form (δx, δy, δp, δv), (δ x, δ y, δ p, δ v) → K (0)(δv, δ v) + δp(δ x − δ y) + δ p(δx − δy) 2 on Tx M × Tx N × (Rn )∗ × Rk , hence Ker F0 (x, x, 0, 0) = ∆(Tx M ∩ Tx N ) × {0, 0} = T(x,x) ∆(M ∩ N ) × {0, 0} = T(x,x,0,0) Crit(F0 ), i.e., normal non-degeneracy. If C is any complementary subspace of ∆(Tx M ∩ Tx N ) in Tx M × Tx N , then (δx, δy) → δx − δy is an isomorphism C → Rn sending the quadratic form C × (Rn )∗ (δx, δy, δp) → δp(δx − δy) onto the quadratic form Rn × (Rn )∗ (δq, δp) → δp(δq), which has index n. Hence the index of F0 (x, x, 0, 0) is Ind(K) + n. The gradient flow f0t of F0 is obtained by integrating the differential system x˙ = p|Tx M , y˙ = − p|Ty N , p˙ = (x − y) , v˙ = ∇K(v)
(2)
and of course, applying a translation, we may assume that 0 ∈ / M ∩ N ; it follows that there exists a constant C > 0, such that for every a ∈ M × N × (Rn )∗ × Rk the path γ : t → f0t (a), viewed as a path in d Rn × Rn × (Rn )∗ × Rk , satisfies dt |γ(t)| C |γ(t)| for every t (the left-hand side makes sense because γ cannot vanish for 0 ∈ M ∩ N )). Integrating this differential inequality, we get |f0t (a)| eC|t| |a|, which shows that a solution of (2) cannot go to infinity in a finite time. Finally, if |dF0 (xk , yk , pk , vk )| → 0, then |xk − yk | → 0 and therefore the distance from (xk , yk ) to ∆(M ∩ N ) tends to 0. From this, we shall deduce that pk → 0, which will prove that the Palais– Smale condition is satisfied since vk → 0 (K is non-degenerate). If this were not the case, then, taking subsequences, we might assume that |pk | is bounded away from 0, that uk = pk /|pk | converges to some u and that xk , yk both converge to z ∈ M ∩ N . Since ∂x F0 = p|Tx M and ∂y F0 = −p|Ty M , we would have |pk |Txk M | → 0 and |pk |Tyk N | → 0, therefore |uk |Txk M | → 0 and |uk |Tyk N | → 0, hence u|Tz M = 0 and u|Tz N = 0. As Tz M + Tz N = Rn by transversality, this would yield u = 0, contradicting |u| = 1. ✷
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End of the proof. – Choose a large positive number R such that F = F0 for F −R and for F R (this is possible since F − F0 has compact support). The critical points of F lie between the levels −R and R and F satisfies the Palais–Smale condition. Now, by Lemma 1.2, {F R} = {F0 R} is obtained from {F −R} = {F0 −R} by attaching to it a “handle”, namely a ball bundle of rank n + Ind(K) over M ∩ N along its boundary. Hence the theorem follows from standard arguments in critical point theory. 2. Proof of the theorem for arbitrary M , N when V = Rn As the problem is symmetric with respect to M and N (replacing gt by gt−1 ), we may assume N = Rn and reduce the problem to the previous one as follows: – choose a large ball B centered at 0 in Rn , containing both M ∩ N and the projection of the support of (gt ) in its interior, and whose boundary ∂B is transversal to both M and N ; – use an isotopy to make M and N “radial" near ∂B; – identify B to the south hemisphere of Sn by stereographic projection from the north pole and take the unions of M ∩ B and N ∩ B with their symmetrics in the north hemisphere to obtain compact and N
of Sn . Similarly, consider the Hamiltonian isotopy (
gt ) of T∗ Sn obtained by submanifolds M gluing to (gt |T∗ B ) its symmetric on the northern side;
, nor in the projection of the support of (
gt ), and – take away a point a ∈ Sn which lies neither in N n n identify S \ {a} to R .
) ∩ νM contains at least (c (C)
+ 1) points, where the By the previous version of the theorem,
g1 (ν N
sum is over the connected components C of M ∩ N , and at least dim H∗ (M ∩ N ) points if the intersection
) ∩ νM is transversal. As M ∩N
(resp.
) ∩ νM ) is just the disjoint union of two copies of g 1 (ν N g1 (ν N M ∩ N (resp. g1 (νN ) ∩ νM ), this completes the proof when V = Rn . Final remark. – The lesson of this proof is twofold: – Functions which are not naturally “quadratic” at infinity do appear naturally in symplectic geometry (the idea of F was inspired by the Eliashberg–Gromov “difference” of two generating functions [2]). – The sublevels {F0 ±R} can be hard to understand, but the critical behaviour of F0 tells us the relevant fact, which is the topological difference between them. We plan to give other applications of this remark to symplectic and contact geometry. Acknowledgements. I wish to thank Mikhael Gromov for sending me a copy of [2], Alain Chenciner and David Hermann for useful discussions and Santiago López de Medrano, whose invitation to lecture on symplectic geometry at the Instituto de Matemáticas of the Universidad Nacional Autónoma de México stimulated my vacillating interest in the subject.
References [1] Chaperon M., On generating families, in: Hofer H., Taubes C.H., Weinstein A., Zehnder E. (Eds.), The Floer Memorial Volume, Progress in Mathematics, Vol. 133, Birkhäuser, 1995, pp. 283–296. [2] Eliashberg Y., Gromov M., Lagrangian Intersection Theory: Finite-Dimensional Approach, Amer. Math. Soc. Transl., Series 2, Vol. 186, 1998, pp. 27–188. [3] Lalonde F., Sikorav J.C., Sous-variétés lagrangiennes et lagrangiennes exactes des fibrés cotangents, Commun. Math. Helv. 66 (1991) 18–33.
661
On a result of Eliashberg and Gromov Marc CHAPERON Géométrie et Dynamique, Institut de mathématiques de Jussieu, UFR de mathématiques, Université Paris-7, case 7012, 2, place Jussieu, 75251 Paris cedex 05, France E-mail: [email protected] (Reçu le 17 février 2001, accepté le 5 mars 2001)
Abstract.
We give a very simple proof of a theorem of Eliashberg and Gromov implying that intersection between the conormal bundles νM and νN of two proper submanifolds M , N of Rn is persistent under compactly supported Hamiltonian deformations of νM and νN . 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Sur un résultat d’Eliashberg et Gromov Résumé.
Nous donnons une preuve très simple d’un théorème d’Eliashberg et Gromov impliquant que si les fibrés conormaux νM et νN de deux sous-variétés propres M , N de Rn se rencontrent, alors des déformations hamiltoniennes à support compact arbitraires de νM et νN se coupent aussi. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Version française abrégée Dans [2], Eliashberg et Gromov donnent une démonstration assez indirecte du : T HÉORÈME. – Soient M et N deux sous-variétés propres d’une variété V telles que M ∩ N soit compact et non vide. Supposons en outre M et N transverses, c’est-à-dire que l’intersection de leurs fibrés conormaux νM = {(Q, p) ∈ T∗ V | Q ∈ M , p|TQ M = 0} et νN = {(Q, p) ∈ T∗ V | Q ∈ N , p|TQ N = 0} est {(Q, p) ∈ T∗ V | Q ∈ M ∩ N , p = 0}. Alors, pour toute isotopie hamiltonienne (gt )0t1 de T∗ V à support compact, g1 (νN ) ∩ νM contient : – au moins (c (C) + 1) points, où la somme se fait sur les composantes connexes C de M ∩ N et c
désigne la longueur cohomologique (« cuplength ») ; – au moins dim H∗ (M ∩ N ) points si l’intersection g1 (νN ) ∩ νM est transversale (le corps de base étant choisi de manière à maximiser ces bornes). Cette Note le prouve sans détour si V = Rn . Note présentée par Étienne G HYS. S0764-4442(01)01939-5/FLA 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés
657
M. Chaperon
Remarques. – Eliashberg et Gromov ne donnent que la seconde estimation bien que la première résulte sans doute de leur théorie. – Pourvu que l’isotopie (gt ) soit à support compact, le théorème est vrai a fortiori si M et N ne sont pas transverses ou ont une intersection non compacte, car g1 (νN ) ∩ νM est alors infini puisque νN ∩ νM va à l’infini. – Le théorème est faux pour les isotopies à support non compact ; en effet, toute isotopie (ht )0t1 de V se relève en l’isotopie hamiltonienne gt (q, p) = (ht (q), p ◦ ∂ht (q)−1 ) de T∗ V , et h1 (N ) ∩ M = ∅ donne g1 (νN ) ∩ νM = ν(h1 (N )) ∩ νM = ∅. – Le cas particulier important où M = V est dû à Lalonde et Sikorav [3], théorème 3 (i). Idée de la démonstration. – Un argument géométrique simple (paragraphe 2 ci-dessous) permet de supposer N compacte. Rappelons (cf. par exemple [1]) que g1 admet une famille génératrice « quadratique », c’est-à-dire une fonction S : T∗ Rn × Rk → R possédant les trois propriétés suivantes : (i) L’application T∗ Rn × Rk (Q, p; v) → ∂v S(Q, p, v) est transverse à 0 ∈ (Rk )∗ . (ii) Il existe une forme quadratique non dégénérée K : Rk → R telle que S(Q, p; v) = K(v) en dehors d’un compact. (iii) Le graphe de g1 est l’ensemble des (q, p), (Q, P ) ∈ (T∗ Rn )2 tels que q = Q + ∂p S(Q, p; v), P = p + ∂Q S(Q, p; v), 0 = ∂v S(Q, p; v) pour un v ∈ Rk , ce qui définit un difféomorphisme de ∂v S −1 (0) sur le graphe de g1 . On prouve sans peine les deux lemmes suivants : L EMME 1.1. – Les points critiques de la fonction F : M × N × (Rn )∗ × Rk → R définie par F (x, y, p; v) = S(x, p; v) + p(x − y) sont en bijection avec νM ∩ g1 (νN ). L EMME 1.2. – La fonction F0 (x, y, p; v) = K(v) + p(x − y) sur M × N × (Rn )∗ × Rk est égale à F en dehors d’un compact et possède les propriétés suivantes : (i) Son ensemble critique Crit(F0 ) = ∆(M ∩ N ) × {(0, 0)} est une sous-variété normalement non dégénérée d’indice de Morse Ind(K) + n, contenue dans F0−1 (0). (ii) Si l’on munit M × N × (Rn )∗ × Rk de la métrique riemannienne (complète) induite par les métriques euclidiennes standard de Rn et Rk , le flot du gradient de F0 est complet et la condition de Palais– Smale est satisfaite. Pour achever la preuve, il ne reste plus qu’à choisir un réel R > 0 tel que F = F0 pour F −R et F R (c’est possible puisque F − F0 est à support compact). Les points critiques de F se trouvent entre les niveaux −R et R, et F satisfait à la condition de Palais–Smale. Or, d’après le lemme 1.2, {F R} = {F0 R} s’obtient à partir de {F −R} = {F0 −R} en lui attachant une « anse », en l’occurrence un fibré en boules fermées de dimension n + Ind(K) sur M ∩ N , le long de son bord. Des arguments standard permettent donc de conclure. Remarques. – On voit apparaître naturellement ici des fonctions qui ne sont pas « quadratiques » à l’infini (l’idée de F est inspirée de la « différence » de deux fonctions génératices introduite par Eliashberg et Gromov [2]). Les fermés {F0 ±R} peuvent être très compliqués mais ce qui compte, c’est-à-dire la différence entre leurs topologies, vient de notre parfaite compréhension des points critiques de F0 . Nous espérons donner d’autres applications de cette remarque à la géométrie symplectique et de contact.
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On a result of Eliashberg and Gromov
In [2], Eliashberg and Gromov use a somewhat indirect argument to prove the following: T HEOREM. – Let M and N be two proper submanifolds of a manifold V such that M ∩N is compact and non-empty. In addition, assume that M and N are transversal, i.e. that the intersection of their conormal bundles νM = {(Q, p) ∈ T∗ V | Q ∈ M , p|TQ M = 0} and νN = {(Q, p) ∈ T∗ V | Q ∈ N , p|TQ N = 0} is {(Q, p) ∈ T∗ V | Q ∈ M ∩ N , p = 0}. Then, for any Hamiltonian isotopy (gt )0t1 of T∗ V with compact support, g1 (νN ) ∩ νM contains – at least c (C) + 1 points, where the sum is over the connected components C of M ∩ N and c
denotes the cuplength; – at least dim H∗ (M ∩ N ) points if the intersection g1 (νN ) ∩ νM is transversal (the base field being so chosen as to get the best estimates). This Note provides a straight-forward proof when V = Rn . Notes. – Eliashberg and Gromov only give the second lower bound, even though their theory probably yields the first as well. – When M and N are not transversal, νM ∩ νN contains lines, hence g1 (νN ) ∩ νM is infinite since the isotopy has compact support. Similarly, if M and N intersect transversally but the intersection is not compact, then g1 (νN ) ∩ νM is infinite. – The theorem is false for non-compactly supported isotopies; indeed, any isotopy (ht )0t1 of V can be lifted to the Hamiltonian isotopy gt (q, p) = (ht (q), p ◦ dht (q)−1 ) of T∗ V , and h1 (N ) ∩ M = ∅ yields g1 (νN ) ∩ νM = ν(h1 (N )) ∩ νM = ∅. – In the important particular case where M = V , the theorem is due to Lalonde and Sikorav [3], théorème 3 (i). 1. Proof of the theorem when V = Rn and N is compact Recall (see for example [1]) that g1 has a “quadratic” generating family, i.e. an S : T∗ Rn × Rk → R with the following three properties: (i) The map T∗ Rn × Rk (Q, p; v) → ∂v S(Q, p, v) ∈ (Rk )∗ is transversal to 0 ∈ (Rk )∗ . (ii) There is a non-degenerate quadratic form K : Rk → R such that S(Q, p; v) = K(v) off a compact subset. (iii) The graph of g1 is the set of those (q, p), (Q, P ) ∈ (T∗ Rn )2 such that q = Q + ∂p S(Q, p; v), P = p + ∂Q S(Q, p; v), 0 = ∂v S(Q, p; v) for some v ∈ Rk , and this defines a diffeomorphism of ∂v S −1 (0) onto the graph of g1 . L EMMA 1.1. – The critical points of the function F : M × N × (Rn )∗ × Rk → R defined by F (x, y, p; v) = S(x, p; v) + p(x − y) are in one to one correspondence with νM ∩ g1 (νN ). Proof of Lemma 1.1. – Setting (as in (iii)) q = x + ∂p S(x, p; v) and P = p + ∂Q S(x, p; v) (where ∂Q denotes derivation with respect to the whole of the first factor Rn , not just M ), we have that ∂p F (x, y, p; v) = q − x + x − y = q − y, ∂x F (x, y, p; v) = (P − p + p)|Tx M = P |Tx M , ∂ y F (x, y, p; v) = −p|Ty N , ∂v F (x, y, p; v) = ∂v S(x, p; v).
(1)
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By (iii) and the last line of (1), ∂v F = 0 if and only if (x, P ) = g1 (q, p); hence, by the first line of (1), ∂p F = 0 and ∂v F = 0 if and only if (x, P ) = g1 (y, p). By the remaining of (1), it follows that (x, y, p; v) is a critical point of F if and only if we have (y, p) ∈ νN and g1 (y, p) = (x, P ) ∈ νM , which proves our lemma. ✷ L EMMA 1.2. – The function F0 (x, y, p; v) = K(v) + p(x − y) on M × N × (Rn )∗ × Rk is equal to F off a compact subset and has the following properties: (i) Its critical set Crit(F0 ) = ∆(M ∩ N ) × {(0, 0)} is a normally non-degenerate submanifold of Morse index Ind(K) + n, lying in F0−1 (0). (ii) If we endow M × N × (Rn )∗ × Rk with the (complete) Riemannian metric induced by the standard Euclidean metrics of Rn and Rk , then the gradient flow of F0 is complete and the Palais–Smale condition is satisfied. Proof of Lemma 1.2. – The partial derivatives ∂x F0 = p|Tx M , ∂y F0 = −p|Ty M and ∂p F0 = x − y vanish simultaneously if and only if y = x ∈ M ∩ N and (as Tx M + Tx N = Rn by transversality) p = 0. Since ∂v F0 = dK and K is non-degenerate, there is no doubt about the critical set of F0 . The Hessian F0 (x, x, 0, 0) of F0 at a critical point (x, x, 0, 0) is the bilinear form (δx, δy, δp, δv), (δ x, δ y, δ p, δ v) → K (0)(δv, δ v) + δp(δ x − δ y) + δ p(δx − δy) 2 on Tx M × Tx N × (Rn )∗ × Rk , hence Ker F0 (x, x, 0, 0) = ∆(Tx M ∩ Tx N ) × {0, 0} = T(x,x) ∆(M ∩ N ) × {0, 0} = T(x,x,0,0) Crit(F0 ), i.e., normal non-degeneracy. If C is any complementary subspace of ∆(Tx M ∩ Tx N ) in Tx M × Tx N , then (δx, δy) → δx − δy is an isomorphism C → Rn sending the quadratic form C × (Rn )∗ (δx, δy, δp) → δp(δx − δy) onto the quadratic form Rn × (Rn )∗ (δq, δp) → δp(δq), which has index n. Hence the index of F0 (x, x, 0, 0) is Ind(K) + n. The gradient flow f0t of F0 is obtained by integrating the differential system x˙ = p|Tx M , y˙ = − p|Ty N , p˙ = (x − y) , v˙ = ∇K(v)
(2)
and of course, applying a translation, we may assume that 0 ∈ / M ∩ N ; it follows that there exists a constant C > 0, such that for every a ∈ M × N × (Rn )∗ × Rk the path γ : t → f0t (a), viewed as a path in d Rn × Rn × (Rn )∗ × Rk , satisfies dt |γ(t)| C |γ(t)| for every t (the left-hand side makes sense because γ cannot vanish for 0 ∈ M ∩ N )). Integrating this differential inequality, we get |f0t (a)| eC|t| |a|, which shows that a solution of (2) cannot go to infinity in a finite time. Finally, if |dF0 (xk , yk , pk , vk )| → 0, then |xk − yk | → 0 and therefore the distance from (xk , yk ) to ∆(M ∩ N ) tends to 0. From this, we shall deduce that pk → 0, which will prove that the Palais– Smale condition is satisfied since vk → 0 (K is non-degenerate). If this were not the case, then, taking subsequences, we might assume that |pk | is bounded away from 0, that uk = pk /|pk | converges to some u and that xk , yk both converge to z ∈ M ∩ N . Since ∂x F0 = p|Tx M and ∂y F0 = −p|Ty M , we would have |pk |Txk M | → 0 and |pk |Tyk N | → 0, therefore |uk |Txk M | → 0 and |uk |Tyk N | → 0, hence u|Tz M = 0 and u|Tz N = 0. As Tz M + Tz N = Rn by transversality, this would yield u = 0, contradicting |u| = 1. ✷
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End of the proof. – Choose a large positive number R such that F = F0 for F −R and for F R (this is possible since F − F0 has compact support). The critical points of F lie between the levels −R and R and F satisfies the Palais–Smale condition. Now, by Lemma 1.2, {F R} = {F0 R} is obtained from {F −R} = {F0 −R} by attaching to it a “handle”, namely a ball bundle of rank n + Ind(K) over M ∩ N along its boundary. Hence the theorem follows from standard arguments in critical point theory. 2. Proof of the theorem for arbitrary M , N when V = Rn As the problem is symmetric with respect to M and N (replacing gt by gt−1 ), we may assume N = Rn and reduce the problem to the previous one as follows: – choose a large ball B centered at 0 in Rn , containing both M ∩ N and the projection of the support of (gt ) in its interior, and whose boundary ∂B is transversal to both M and N ; – use an isotopy to make M and N “radial" near ∂B; – identify B to the south hemisphere of Sn by stereographic projection from the north pole and take the unions of M ∩ B and N ∩ B with their symmetrics in the north hemisphere to obtain compact and N
of Sn . Similarly, consider the Hamiltonian isotopy (
gt ) of T∗ Sn obtained by submanifolds M gluing to (gt |T∗ B ) its symmetric on the northern side;
, nor in the projection of the support of (
gt ), and – take away a point a ∈ Sn which lies neither in N n n identify S \ {a} to R .
) ∩ νM contains at least (c (C)
+ 1) points, where the By the previous version of the theorem,
g1 (ν N
sum is over the connected components C of M ∩ N , and at least dim H∗ (M ∩ N ) points if the intersection
) ∩ νM is transversal. As M ∩N
(resp.
) ∩ νM ) is just the disjoint union of two copies of g 1 (ν N g1 (ν N M ∩ N (resp. g1 (νN ) ∩ νM ), this completes the proof when V = Rn . Final remark. – The lesson of this proof is twofold: – Functions which are not naturally “quadratic” at infinity do appear naturally in symplectic geometry (the idea of F was inspired by the Eliashberg–Gromov “difference” of two generating functions [2]). – The sublevels {F0 ±R} can be hard to understand, but the critical behaviour of F0 tells us the relevant fact, which is the topological difference between them. We plan to give other applications of this remark to symplectic and contact geometry. Acknowledgements. I wish to thank Mikhael Gromov for sending me a copy of [2], Alain Chenciner and David Hermann for useful discussions and Santiago López de Medrano, whose invitation to lecture on symplectic geometry at the Instituto de Matemáticas of the Universidad Nacional Autónoma de México stimulated my vacillating interest in the subject.
References [1] Chaperon M., On generating families, in: Hofer H., Taubes C.H., Weinstein A., Zehnder E. (Eds.), The Floer Memorial Volume, Progress in Mathematics, Vol. 133, Birkhäuser, 1995, pp. 283–296. [2] Eliashberg Y., Gromov M., Lagrangian Intersection Theory: Finite-Dimensional Approach, Amer. Math. Soc. Transl., Series 2, Vol. 186, 1998, pp. 27–188. [3] Lalonde F., Sikorav J.C., Sous-variétés lagrangiennes et lagrangiennes exactes des fibrés cotangents, Commun. Math. Helv. 66 (1991) 18–33.
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