The Arnold Chord Conjecture Heather Macbeth “Part III Essay” Cambridge, April 2010 Contents 1 Introduction 2 2 Contact and symplectic geometry 6 2.1 Contact manifolds . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Symplectic manifolds . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Examples and constructions . . . . . . . . . . . . . . . . . . . 10 3 Two problems of Arnold 14 3.1 The chord conjecture . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Lagrangian intersections . . . . . . . . . . . . . . . . . . . . . 18 4 Curves in almost-complex manifolds 22 4.1 Almost-complex structures. . . . . . . . . . . . . . . . . . . . 23 4.2 Banach manifolds and generic behaviour . . . . . . . . . . . . 25 4.3 The Riemann-Roch theorem . . . . . . . . . . . . . . . . . . . 28 4.4 Generic perturbed Cauchy-Riemann equations . . . . . . . . 31 5 Curves in tame almost-complex manifolds 36 5.1 Gromov’s compactness theorem . . . . . . . . . . . . . . . . . 37 5.2 Hamiltonian perturbations . . . . . . . . . . . . . . . . . . . . 39 5.3 A J-holomorphic Fredholm alternative . . . . . . . . . . . . . 45 6 Symplectic corollaries 49 6.1 Lagrangian intersections and displacement energy . . . . . . . 49 6.2 Proof of the chord conjecture . . . . . . . . . . . . . . . . . . 51 1 Chapter 1 Introduction This essay concerns the behaviour, in contact and symplectic manifolds, of certainstructure-preservingflowswithrespecttocertainsubmanifolds. Our two main results are the following: Theorem 1.1 (Arnold chord conjecture). Let (N,α) be a compact simply- connected contact-type hypersurface of R2n. Then for each compact Legen- drian submanifold l of (N,α), some integral curve of α’s Reeb vector field runs from l to l. Theorem 1.2. Let X be a compactly supported time-dependent Hamilto- Ft nian vector field on R2n of Hofer norm less than σ. Let L be a compact Lagrangian submanifold of R2n, such that no disc with boundary on L has positive symplectic area less than σ. Then some length-1 integral curve of X runs from L to L. Ft Versions of both appear in Vladimir Arnold’s classic and influential set [Arn86] of conjectures in symplectic topology, inspired by “nebulous ideas” on analogies between the category of symplectic manifolds and the category of manifolds. Theorem 1.1 was proved in 2001 by Mohnke [Moh01]. His beautiful (and very short) argument deduces the theorem from Theorem 1.2 by con- sidering l’s extensions, within a fixed neighbourhood of N in R2n, to closed Lagrangian submanifolds of R2n. We present this argument in this essay as Proposition 3.1.7 and Theorem 6.2.1. Theorem 1.2 has a more intricate history. The ‘σ = ∞’ version Theorem 1.3 (Arnold conjecture). Let X be a compactly supported time- Ft dependent Hamiltonian vector field on R2n . Let L be a compact Lagrangian submanifold of R2n, such that every disc with boundary on L has zero sym- plectic area. Then some length-1 integral curve of X runs from L to L. Ft was proved by Gromov, in the foundational paper [Gro85] in which he in- troduced techniques of J-holomorphic curves to symplectic geometry. After 2 Hofer’s introduction of the Hofer norm, Polterovich [Pol93] built on Gro- mov’s ideas to prove the weakening of Theorem 1.2 in which the bound σ needed on symplectic areas of discs is increased to 2σ. At roughly the same time, Floer [Flo88] established a new approach to Theorem 1.3 and related problems. His method was to consider strips with edges in L satisfying versions of the PDE ∂ u+J(∂ u−X | ) = 0. s t Ft u Such strips tend to concentrate around integral curves of X , providing a Ft way of discovering such integral curves. Later Chekanov [Che98], building on Floer’s ideas, established the full Theorem 1.2. In fact Floer’s proof of Theorem 1.3, and Chekanov’s proof following Floer of Theorem 1.2, use a much more powerful and general framework, and deduce a refinement of the theorems which we do not need for deduc- ing the chord conjecture Theorem 1.1. The framework is Lagrangian Floer homology, which roughly speaking is a homology theory generated by the length-1 integral curves of X from L to L. The consequent refinement Ft of Theorems 1.3 and 1.2 is, ‘generically,’ the much stronger lower bound dimH (L,Z ) for the numbers of such integral curves – in fact, the bound ∗ 2 originally conjectured by Arnold. On the other hand, the monograph [MS04] proves the basic Theorem 1.3 (their Theorem 9.2.14) using Floer’s equation and analytical content, but withoutexplicitlysettingupLagrangianFloerhomology. Itisthisargument (suitably sharpened), rather than Chekanov’s original presentation, that we use in Section 6.1 to prove Theorem 1.2. The plan of this essay is as follows. Chapter 2 reviews necessary back- ground in contact and symplectic geometry. Chapter 3 is a detailed intro- duction to problems concerning Reeb and Hamiltonian flows’ relation with Legendrian and Lagrangian submanifolds. It includes a number of examples and a further discussion of the literature. Chapters 4 and 5 are the essay’s technical core. The target of these chapters is Proposition 5.3.1. We state it here for convenience: Proposition 1.4. Let (M,ω,J) be a tame compact symplectic manifold, and L a compact Lagrangian submanifold of M, such that no sphere in M or disc in M with boundary on L has positive symplectic area less than σ. Then for each Hamiltonian form H on D2×(M,L) whose curvature satisfies (cid:90) sup R | < σ, H (·,p) p∈M Σ each w ∈ ∂D2 and each p ∈ L, there is a map u : (D2,∂D2,w) → (M,L,p) such that for all z ∈ D2, ∂ u| +X0,1| = 0. J z H (z,u(z)) 3 Observethatthispropositionassertstheexistenceofdiscswithboundaryon L which satisfy perturbations of certain PDE, so long as the perturbations are small in relation to the symplectic areas of spheres and discs in M. There are two main components to the proof of Proposition 5.3.1. The first, discussed in Chapter 4, is a very general pair of theorems, valid in any almost-complex manifold, which describes the moduli space of such discs for atypicalperturbationoftheCauchy-Riemann equations. Theproofofthese theorems uses the Sard-Smale theorem and a version of the Riemann-Roch theorem. Roughly speaking, the theorems say: ‘generically,’ this moduli space is a manifold, and depends smoothly on the perturbation. The second component, developed in Chapter 5, is Gromov’s compact- ness theorem, which severely prescribes the ways in which a sequence of so- lutions in a symplectic manifold to a perturbation of the Cauchy-Riemann equations can fail to have a convergent subsequence. The power of com- bining this result (on compactness of moduli spaces of solutions) with those of Chapter 4 is that we can conclude that the moduli spaces for different perturbation terms are all compact, and cobordant to each other. Results on existence of solutions for the zero perturbation then immediately imply existence of solutions in general. The conclusion of the essay, Chapter 6, uses Proposition 5.3.1 to deduce Theorem 1.2, and thence Theorem 1.1, as previously described. We make one major simplification throughout: we prove Theorems 1.1 and 1.2 for R2n (as we have stated them here), rather than for the class of tame geometrically bounded symplectic manifolds as seen in the versions of these theorems in the literature. This essay was written in 2010 as coursework for the Cambridge “Part III.” I would like to thank Gabriel Paternain, the essay-setter, for his en- couragement and for helpful mathematical discussions. I am also grateful to Chris Elliott for proofreading Chapters 1 and 3. The blame for all errors and misrepresentations is my own. Notation • Let A and B be manifolds. We write pr : A×B → A, pr : A×B → B, 1 2 for the projections from A × B onto its co-ordinates. For a vector bundle E over A, we write pr ∗E for the pullback of E under pr to a 1 1 bundle over A×B. Likewise for a section η of E, we write pr ∗η for 1 η’s pullback to a section of pr ∗E . (By contrast, π ,π ,... denote the 1 1 2 first, second, ... homotopy groups. 4 • Let A and B be manifolds, and E and F vector bundles over A and B respectively. We write E (cid:2)F for the vector bundle pr ∗E ⊗pr ∗F 1 2 overA×B. Wewillsometimesalsowrite,forinstance,E(cid:2)R,todenote the bundle pr ∗E; the idea is that R stands for the trivial bundle over 1 B. • Let X ,X ,Y ,Y be topological spaces, with Y ⊆ X and Y ⊆ X . 1 2 1 2 1 1 2 2 A map f : (X ,Y ) → (X ,Y ) 1 1 2 2 is a continuous function f : X → X such that f(Y ) ⊆ Y . 1 2 1 2 • In a number of related senses defined throughout the essay, a tilde u (cid:101) denotes the graph of a map u. • Let Σ be a Riemann surface. Then Λi,jΣ denotes its bundle of (i,j)- forms. 5 Chapter 2 Contact and symplectic geometry In this chapter we briefly review some basic definitions, facts and examples incontactandsymplecticgeometry. Ourpresentationderivesfromthetexts [CdS01], [MS98] and [Gei08] and the notes [Etn03]. 2.1 Contact manifolds Let N be a manifold of odd dimension 2n−1. Definition. A contact form on N is a nonzero 1-form α, such that the (2n−1)-form α∧(dα)n−1 is nonvanishing. (Equivalently, such that dα has kernel of dimension 1, and ker(dα)∩kerα = (0).) If α is contact, then for each smooth function f : N → R+, the form fα is also contact: for fα∧[d(fα)]n−1 = fα∧(fdα+df ∧α)n−1 = fn(cid:2)α∧(dα)n−1(cid:3). A contact structure on N is an equivalence class of contact forms on N, where α ∼ β if β = fα for some smooth f : N → R+. A contact manifold is a manifold equipped with a contact structure. Remark. Strictly speaking, it is standard to define a contact structure on a manifold N to be a (2n−2)-plane distribution on N which locally is the kernel of a contact form as we have defined them. The two concepts are equivalent if the distribution is co-orientable. For simplicity, we restrict to this case throughout the essay. 6 Example 2.1.1 (Standard contact form on R2n−1). Consider the 1-form 2n−1 (cid:88) α = dz+ xjdyj i=1 on R2n−1. We have 2n−1 (cid:88) dα = dxj ∧dyj; i=1 α∧(dα)n−1 = dz∧(dx1∧dy1)∧···∧(dxn−1∧dyn−1), the standard volume form, so α is contact. Example 2.1.2 (Standard overtwisted form on R3). Consider the 1-form on R3 defined in cylindrical co-ordinates by α = (cosr)dz+(rsinr)dθ. We have dα = −(sinr)dr∧dz+[rcosr+sinr]dr∧dθ; α∧dα = (cosr)[rcosr+sinr]dz∧dr∧dθ−(rsinr)(sinr)dθ∧dr∧dz (cid:20) (cid:21) sinr = 1+ (rdr)∧dθ∧dz. r Since 1+sinr is ≥ 2 for nonnegative r, we conclude α∧dα is nonvanishing. r Thus α is contact. Henceforth in this section fix a contact form α on N. Definition. A Legendrian submanifold of (N,α) is a submanifold l of N, of dimension n−1, such that α vanishes on Tl. Example 2.1.3 (Legendrian submanifolds of standard R3). The following discussion is from [Etn03]. Recall from Example 2.1.1 the standard (not- overtwisted) contact structure α = dz+xdy on R3. We can construct a Legendrian submanifold of (R3,α) as follows: Pick a closed curve γ in R2 which satisfies the following conditions: 1. γ is smooth except for a finite number of points, ‘cusps,’ at which γ’s slope tends to 0 from above from one direction and from below from the other direction, but γ’s ’direction’ reverses. 2. γ is never tangent to vertical. 7 3. At points where γ crosses itself, the slopes of the different branches at the crossing points are all distinct. Now identify R2 with the yz-plane (y horizontal, z vertical). Construct a closed loop l in R3 whose projection to yz is γ, and whose x-co-ordinate is −dz/dy. This gives a continuous map into R3 by conditions (1) and (2), and by condition (3) does not self-intersect. Since x = −dz/dy, the submanifold l is Legendrian. Conversely, it is clear by projection onto the yz-plane that every compact Legendrian submanifold of (R3,α) arises uniquely in this way. It is clear that equivalent contact forms α, fα have the same Legendrian submanifolds. Thus we can talk of Legendrian submanifolds as associated with a contact structure rather than with a particular contact form. Definition. The Reeb vector field of α is the vector field Y on N such that Y ∈ kerdα and α(Y) = 1. (Since dα has kernel of dimension 1, and ker(dα)∩kerα = (0), there indeed exists a unique such vector field.) Lemma 2.1.4. The flow along the Reeb vector field of (N,α) preserves α. 2.2 Symplectic manifolds Let M be a manifold of even dimension 2n. Definition. A symplectic form on M is a closed 2-form ω, such that the 2n-form ωn is nonvanishing. (Equivalently, such that ω has zero kernel.) A symplectic manifold is a manifold equipped with a symplectic form. A symplectomorphism of a symplectic manifold is a diffeomorphism of the manifold under which the symplectic form is preserved. Example 2.2.1 (Standard symplectic form on R2n). Consider the closed 2-form 2n−1 (cid:88) ω = dxj ∧dyj i=1 on R2n. We have ωn = (dx1∧dy1)∧···∧(dxn∧dyn), the standard volume form, so α is contact. 8 Example2.2.2(Cotangentbundles). LetW beasmoothn-manifold. Choose co-ordinates (xi) on a neighbourhood of W, and corresponding co-ordinates ((xi),(ξi)) on the lift of that neighbourhood to T∗W. That is, (ξi) are the co-ordinate functions such that the covector ξ = (cid:80)n ξidxi at a point i=1 (x1,...xn) on W has co-ordinate expression (x1,...xn,ξ1,...ξn). Consider the 1-form n (cid:88) α = − ξidxi, i=1 on T∗W (that is, it is a section of T∗(T∗W).) This 1-form is independent of the defining choice (xi) of co-ordinates. The closed 2-form n (cid:88) dα = dxi∧dξi i=1 on T∗W is therefore also independent of choice of co-ordinates, and has zero kenel. We call dα the canonical symplectic form on T∗W. Example 2.2.3 (Products). Let (M ,ω ) and (M ,ω ) be symplectic man- 1 1 2 2 ifolds. Then (M ×M ,pr ∗ω −pr ∗ω ) 1 2 1 1 2 2 is a symplectic manifold. We now introduce some concepts of symplectic geometry. Definition. A Lagrangian submanifold of (M,ω) is a submanifold L of M, of dimension n, such that ω vanishes on TL. (Since ω has zero kernel, there indeed exists a unique such vector field.) Example 2.2.4. All 1-dimensional submanifolds of R2 are Lagrangian. Example 2.2.5. For each closed 1-form η on a manifold W, the graph of η is a Lagrangian submanifold of the cotangent bundle T∗W with its canonical symplectic structure. Also for each point p ∈ W, the fibre T∗W is a Lagrangian submanifold p of T∗W. Example 2.2.6. Let (M,ω) be a symplectic manifold. Then the graph of a diffeomorphism φ : M → M is a Lagrangian submanifold of (M ×M ,pr ∗ω −pr ∗ω ) 1 2 1 1 2 2 if and only if φ is a symplectomorphism. In particular the diagonal ∆ is a Lagrangian submanifold. 9