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Deformations of asymptotically cylindrical coassociative submanifolds with fixed boundary

McLean proved that the moduli space of coassociative deformations of a compact coassociative 4-submanifold C in a G_2-manifold (M,phi,g) is a smooth manifold of dimension equal to b^2_+(C). In this paper, we show that the moduli space of coassociative deformations of a noncompact, asymptotically cylindrical coassociative 4-fold C in an asymptotically cylindrical G_2-manifold (M,phi,g) is also a smooth manifold. Its dimension is the dimension of the positive subspace of the image of H^2_cs(C,R) in H^2(C,R).


Introduction
Let (M, g) be a Riemannian 7-manifold whose holonomy group Hol(g) is the exceptional holonomy group G 2 (or, more generally, a subgroup of G 2 ). Then M is naturally equipped with a constant 3-form ϕ and 4-form * ϕ. We call (M, ϕ, g) a G 2 -manifold. Complete examples of Riemannian 7-manifolds with holonomy G 2 were constructed by Bryant and Salamon [3], and compact examples by Joyce [7] and Kovalev [11]. Now ϕ and * ϕ are calibrations on M , in the sense of Harvey and Lawson [5]. The corresponding calibrated submanifolds in M are called associative 3folds and coassociative 4-folds, respectively. They are distinguished classes of minimal 3-and 4-submanifolds in (M, g) with a rich structure, that can be thought of as analogous to complex curves and surfaces in a Calabi-Yau 3-fold.
Harvey and Lawson [5] introduced four types of calibrated geometries. Special Lagrangian submanifolds of Calabi-Yau manifolds, associative and coassociative submanifolds of G 2 manifolds and Cayley submanifolds of Spin(7) manifolds. Calibrated geometries have been of growing interest over the past few years and represent one of the most mysterious classes of minimal submanifolds [12], [13]. A great deal of progress has been made recently in the field of special Lagrangian submanifolds that arise in mirror symmetry for Calabi-Yau manifolds and plays a significant role in string theory, for references see [8]. As one might expect, another promising direction for future investigation is calibrated submanifolds in G 2 and Spin(7) manifolds. Recently, some progress has been made in constructing such submanifolds [6,17,18] and in understanding their deformations [1,14].
The deformation theory of compact calibrated submanifolds was studied by McLean [22]. He showed that if C is a compact coassociative 4-fold in a G 2manifold (M, ϕ, g), then the moduli space M C of coassociative deformations of C is smooth, with dimension b 2 + (C). This paper proves an analogue of McLean's theorem for a special class of noncompact coassociative 4-folds. The situation we are interested in is when (M, ϕ, g) is an asymptotically cylindrical G 2 -manifold, that is, it is a noncompact 7-manifold with one end asymptotic to the cylinder X × R on a Calabi-Yau 3-fold X . The natural class of noncompact coassociative 4-folds in M are asymptotically cylindrical coassociative 4-folds C , asymptotic at infinity in M to a cylinder L × R, where L is a special Lagrangian 3-fold in X , with phase i. Understanding the deformations of such submanifolds when the ambient G 2 -manifold decomposes into connected sum of two pieces will provide the necessary technical framework towards completing the Floer homology program for coassociative submanifolds, [13].
In particular, we prove the following theorem. Theorem 1.1 Let (M, ϕ, g) be a G 2 -manifold asymptotic to X ×(R, ∞) with decay rate α < 0, where X is a Calabi-Yau 3-fold. Let C be a coassociative 4-fold in M asymptotic to L × (R ′ , ∞) for R ′ > R with decay rate β for α β < 0, where L is a special Lagrangian 3-fold in X with phase i.
If γ < 0 is small enough then the moduli space M γ C of asymptotically cylindrical coassociative submanifolds in M close to C , and asymptotic to L × (R ′ , ∞) with decay rate γ , is a smooth manifold of dimension dim V + , where V + is the positive subspace of the image of H 2 cs (C, R) in H 2 (C, R).
The principal analytic tool we shall use to prove this is the theory of weighted Sobolev spaces on manifolds with ends, developed by Lockhart and McOwen [15,16]. The important fact is that elliptic partial differential operators on exterior forms such as d + d * or d * d + dd * on the noncompact 4-manifold C are Fredholm operators between appropriate Banach spaces of forms, and we can describe their kernels and cokernels.
Results similar to Theorem 1.1 on the deformations of classes of noncompact special Lagrangian m-folds were proved by Marshall [19] and Pacini [24] for asymptotically conical special Lagrangian m-folds, and by Joyce [7,9] for special Lagrangian m-folds with isolated conical singularities. Marshall and Joyce also use the Lockhart-McOwen framework, but Pacini uses a different analytical approach due to Melrose [20,21]. Note also that Kovalev [11] constructs compact G 2 -manifolds by gluing together two noncompact, asymptotically cylindrical G 2 -manifolds.
We begin in Section 2 with an introduction to G 2 -manifolds and coassociative submanifolds, including a sketch of the proof of McLean's theorem on deformations of compact coassociative 4-folds, and the definitions of asymptotically cylindrical G 2 -manifolds and coassociative 4-folds. Section 3 introduces the weighted Sobolev spaces of Lockhart and McOwen, and determines the kernel and cokernel of the elliptic operator d + + d * on C used in the proof. Finally, Section 4 proves Theorem 1.1, using Banach space techniques and elliptic regularity.
Remark 1.2 In [11], Kovalev constructs asymptotically cylindrical manifolds X with holonomy SU (3). Then X × S 1 is an asymptotically cylindrical G 2manifold, though with holonomy SU (3) rather than G 2 . One can find examples of asymptotically cylindrical coassociative 4-folds C in X × S 1 of two types: (a) C = C ′ × pt, for C ′ an asymptotically cylindrical complex surface in X ; or (b) C = L×S 1 , for L an asymptotically cylindrical special Lagrangian 3-fold in X , with phase i.
Examples of type (a) can be constructed using algebraic geometry: if X = X \D for X a Fano 3-fold and D a smooth divisor in X , then we can take C = C \ D for C a smooth divisor in X intersecting D transversely. Examples of type (b) can be found by choosing the Calabi-Yau 3-fold (X, J, ω, Ω) to have an antiholomorphic involution σ : X → X with σ * (J) = −J , σ * (ω) = −ω and σ * (Ω) = −Ω. Then the fixed points L of σ are a special Lagrangian 3-fold with phase i, and each infinite end of L is asymptotically cylindrical.
We can then apply Theorem 1.1 to these examples. One can show that if C is a small deformation of a coassociative 4-fold C of type (a) or (b) then C is also of type (a) or (b) and thus, Theorem 1.1 implies analogous results on the deformation theory of asymptotically cylindrical complex surfaces and special Lagrangian 3-folds in asymptotically cylindrical Calabi-Yau 3-folds.
2 Introduction to G 2 geometry We now give background material on G 2 -manifolds and their coassociative submanifolds that will be needed later. A good reference on G 2 geometry is Joyce [7,, and a good reference on calibrated geometry is Harvey and Lawson [5].
A G 2 -structure on a 7-manifold M is a principal subbundle of the frame bundle of M , with structure group G 2 . Each G 2 -structure gives rise to a 3-form ϕ and a metric g on M , such that every tangent space of M admits an isomorphism with R 7 identifying ϕ and g with ϕ 0 and g 0 respectively. By an abuse of notation, we will refer to (ϕ, g) as a G 2 -structure.
Proposition 2.1 Let M be a 7-manifold and (ϕ, g) a G 2 -structure on M . Then the following are equivalent: We call ∇ϕ the torsion of the G 2 -structure (ϕ, g), and when ∇ϕ = 0 the G 2 -structure is torsion-free. A triple (M, ϕ, g) is called a G 2 -manifold if M is a 7-manifold and (ϕ, g) a torsion-free G 2 -structure on M . If g has holonomy Hol(g) ⊆ G 2 , then g is Ricci-flat. For explicit, complete examples of G 2manifolds see Bryant and Salamon [3], and for compact examples see Joyce [7] and Kovalev [11]. Here are the basic definitions in calibrated geometry, due to Harvey and Lawson [5].
Definition 2.2 Let (M, g) be a Riemannian manifold. An oriented tangent k -plane V on M is a vector subspace V of some tangent space T x M to M with dim V = k , equipped with an orientation. If V is an oriented tangent k -plane on M then g| V is a Euclidean metric on V , so combining g| V with the orientation on V gives a natural volume form vol V on V , which is a k -form on V . Now let ϕ be a closed k -form on M . We say that ϕ is a calibration on M if for every oriented k -plane V on M we have ϕ| V vol V . Here ϕ| V = α · vol V for some α ∈ R, and ϕ| V vol V if α 1. Let N be an oriented submanifold of M with dimension k . Then each tangent space T x N for x ∈ N is an oriented tangent k -plane. We call N a calibrated submanifold if ϕ| TxN = vol TxN for all x ∈ N .
Calibrated submanifolds are automatically minimal submanifolds (see [5,Theorem II.4.2]). There are two natural classes of calibrated submanifolds in G 2manifolds. Definition 2.3 Let (M, ϕ, g) be a G 2 -manifold, as above. Then the 3-form ϕ is a calibration on (M, g). We define an associative 3-fold in M to be a 3-submanifold of M calibrated with respect to ϕ. Similarly, the Hodge star * ϕ of ϕ is a calibration 4-form on (M, g). We define a coassociative 4-fold in M to be a 4-submanifold of M calibrated with respect to * ϕ.
McLean [22,Prop. 4.4] gives an alternative description of coassociative 4-folds: Proposition 2.4 Let (M, ϕ, g) be a G 2 -manifold, and C a 4-dimensional submanifold of M . Then C admits an orientation making it into a coassociative 4-fold if and only if ϕ| C ≡ 0.

Deformations of compact coassociative 4-folds
Here is the main result in the deformation theory of coassociative 4-folds, proved by McLean [22,Theorem 4.5]. As our sign conventions for ϕ 0 , * ϕ 0 in (1) are different to McLean's, we use self-dual 2-forms in place of McLean's anti-selfdual 2-forms.
Theorem 2.5 Let (M, ϕ, g) be a G 2 -manifold, and C a compact coassociative 4-fold in M . Then the moduli space M C of coassociative 4-folds isotopic to C in M is a smooth manifold of dimension b 2 + (C).
Sketch proof Suppose for simplicity that C is an embedded submanifold. There is a natural orthogonal decomposition T M | C = T C ⊕ ν , where ν → C is the normal bundle of C in M . There is a natural isomorphism ν ∼ = Λ 2 + T * C , constructed as follows. Let x ∈ C and V ∈ ν x . Then V lies in T x M , so V ·ϕ| x ∈ Λ 2 T * x M , and (V · ϕ| x )| TxC ∈ Λ 2 T * x C . Moreover (V · ϕ| x )| TxC actually lies in Λ 2 + T * x C , the bundle of self-dual 2-forms on C , and the map V → (V · ϕ| x )| TxC defines an isomorphism ν ∼ = −→ Λ 2 + T * C . For small ǫ > 0, write B ǫ (ν) for the subbundle of ν with fibre at x the open ball about 0 in ν| x with radius ǫ. Then the exponential map exp : ν → M induces a diffeomorphism between B ǫ (ν) and a small tubular neighbourhood Under this identification, submanifolds C in T C ⊂ M which are C 1 close to C are identified with the graphs Γ ζ 2 + of small smooth sections ζ 2 , and so C = exp(Γ ζ 2 + ) is a 4-submanifold of T C . We need to know: which 2-forms ζ 2 + correspond to coassociative 4-folds C in T C ?
C is coassociative if ϕ| C ≡ 0. Now π| C : C → C is a diffeomorphism, so we can push ϕ| C down to C , and regard it as a function of ζ 2 + . That is, we define Q : Then the moduli space M C is locally isomorphic near C to the set of small self-dual 2-forms ζ 2 + on C with ϕ| exp(Γ ζ 2 + ) ≡ 0, that is, to a neighborhood of 0 in Q −1 (0).
To understand the equation Q(ζ 2 + ) = 0, note that at x ∈ C , Q(ζ 2 + )| x depends on the tangent space to Γ ζ 2 + at ζ 2 + | x , and so on ζ 2 + | x and ∇ζ 2 + | x . Thus the functional form of Q is where F is a smooth function of its arguments. Hence Q(ζ 2 + ) = 0 is a nonlinear first order PDE in ζ 2 + . As ϕ is closed, ϕ| C ≡ 0, and Γ ζ 2 + is isotopic to C , we see Therefore Ker(dQ(0)) is the vector space H 2 + of closed self-dual 2-forms β on C , which by Hodge theory is a finite-dimensional vector space isomorphic to H 2 + (C, R), with dimension b 2 + (C). This is the Zariski tangent space of M C at C , the infinitesimal deformation space of C as a coassociative 4-fold.
To complete the proof we must show that M C is locally isomorphic to its Zariski tangent space H 2 + , and so is a smooth manifold of dimension b 2 + (C). To do this rigorously requires some technical analytic machinery, which is passed over in a few lines in [22, p. 731]. Here is one way to do it.
As Q maps from Λ 2 + T * C with fibre R 3 to Λ 3 T * C with fibre R 4 , it is overdetermined, and not elliptic. To turn it into an elliptic operator, define Then the linearization of P at (0, 0) is are not Banach spaces, we extend P in (2) to act on Sobolev spaces L p l+2 (Λ 2 + T * C), L p l+2 (Λ 3 T * C) for p > 4 and l 0, giving Then P is a smooth map of Banach manifolds.
Let H 3 be the vector space of closed and coclosed 3-forms on C , so that H 3 ∼ = H 3 (C, R) by Hodge theory, and V p l+1 be the Banach subspace of L p l+1 (Λ 3 T * C) L 2 -orthogonal to H 3 . Then one can show that P maps into V p l+1 , and the linearization is a smooth map of Banach manifolds, with d P (0, 0) surjective. The Implicit Mapping Theorem for Banach spaces (Theorem 4.4) now implies that P −1 (0) is near 0 a smooth submanifold, locally isomorphic to Ker(d P (0)). But P (ζ 2 + , ζ 4 ) = 0 is an elliptic equation for small ζ 2 + , ζ 4 , and so elliptic regularity implies that solutions (ζ 2 + , ζ 4 ) are smooth.

Asymptotically cylindrical G 2 -manifolds and coassociative 4-folds
We first define cylindrical and asymptotically cylindrical G 2 -manifolds.
Definition 2.7 A connected, complete G 2 -manifold (M, ϕ, g) is called asymptotically cylindrical with decay rate α if there exists a cylindrical G 2 -manifold (M 0 , ϕ 0 , g 0 ) with M 0 = X × R as above, a compact subset K ⊂ M , a real number R, and a diffeomorphism Ψ : where ∇ is the Levi-Civita connection of the cylindrical metric g 0 .
The point of this is that M has one end modelled on X × (R, ∞), and as t → ∞ in (R, ∞) the G 2 -structure (ϕ, g) on M converges to order O(e αt ) to the cylindrical G 2 -structure on X × (R, ∞), with all of its derivatives. We suppose M and X are connected, that is, we allow M to have only one end.
This is because one can use Cheeger-Gromoll splitting theorem [4] to show that an orientable, connected, asymptotically cylindrical Riemannian manifold with Ricci-flat metric g can have at most 2 cylindrical ends. In the case when there are 2 cylindrical ends then there is reduction in the holonomy group Hol(g) and (M, g) is a cylinder. One can also show that reduction in holonomy can be obtained by just using the analytic set-up for Fredholm properties of an elliptic operator on noncompact manifolds, [25].
Here are the analogous definitions for coassociative submanifolds.
Definition 2.8 Let (M 0 , ϕ 0 , g 0 ) and X be as in Definition 2.6. A submanifold C 0 of M 0 is called cylindrical if C 0 = L × R for some compact submanifold L in X , not necessarily connected. C 0 is coassociative if and only if L is a special Lagrangian 3-fold with phase i in the Calabi-Yau 3-fold X . Definition 2.9 Let (M 0 , ϕ 0 , g 0 ), X , (M, ϕ, g), K, Ψ and α be as in Definitions 2.6 and 2.7, and let C 0 = L × R be a cylindrical coassociative 4-fold in M 0 , as in Definition 2.8.
Here we require C but not L to be connected, that is, we allow C to have multiple ends. The point of Definition 2.9 is to find a good way to say that a submanifold C in M is asymptotic to the cylinder C 0 in M 0 = X × R as t → ∞ in R, to order O(e βt ). We do this by writing C near infinity as the graph of a normal vector field v to C 0 = L × R in M 0 = X × R, and requiring v and its derivatives to be O(e βt ).

Infinitesimal deformations of C
Let (M, ϕ, g) be an asymptotically cylindrical G 2 -manifold asymptotic to X × (R, ∞), and C an asymptotically cylindrical coassociative 4-fold in M asymptotic to L × (R ′ , ∞). We wish to study the moduli space M γ C of asymptotically cylindrical deformations C of C in M with rate γ . To do this we modify the proof of Theorem 2.5 in Section 2, for the case when C is compact. There we modelled M C on P −1 (0) for a nonlinear map P between Banach spaces, whose linearization d P (0, 0) at 0 was the Fredholm map between Sobolev spaces Now when C is not compact, as in the asymptotically cylindrical case, (4) is not in general Fredholm, and the proof of Theorem 2.5 fails. To repair it we use the analytical framework for asymptotically cylindrical manifolds developed by Lockhart and McOwen in [15,16], involving weighted Sobolev spaces L p k,γ (Λ r T * C). Roughly speaking, elements of L p k,γ (Λ r T * C) are L p k r-forms on C which decay like O(e γt ) on the end L × (R ′ , ∞). This has the advantage of building the decay rate γ into the proof from the outset. This section will study the weighted analogue of (4), for small γ < 0. It will be shown in Section 4 to be the linearization at 0 of a nonlinear operator P for which M γ C is locally modelled on P −1 (0). Section 3.1 introduces weighted Sobolev spaces, and the Lockhart-McOwen theory of elliptic operators between them. Then Sections 3.2 and 3.3 compute the kernel and cokernel of (5) for small γ < 0, and Section 3.4 determines the set of rates γ for which (5) is Fredholm.

Elliptic operators on asymptotically cylindrical manifolds
We now sketch parts of the theory of analysis on manifolds with cylindrical ends due to Lockhart and McOwen [15,16]. We begin with some elementary definitions.
Definition 3.1 Let (C, g) be an asymptotically cylindrical Riemannian manifold. That is, there is a Riemannian cylinder (L × R, g 0 ) with L compact, a compact subset K ′ ⊂ C and a diffeomorphism Φ : Let E be a vector bundle on C equipped with metrics h on the fibres, and a connection ∇ E on E preserving h. We say that E, h, ∇ E are asymptotic to Choose a smooth function ρ : This prescribes ρ on C \ K ′ , so we only have to extend ρ over the compact set K ′ . For p ≥ 1, k ≥ 0 and γ ∈ R we define the weighted Sobolev space L p k,γ (E) to be the set of sections s of E that are locally integrable and k times weakly differentiable and for which the norm is finite. Then L p k,γ (E) is a Banach space. Since ρ is uniquely determined except on the compact set K ′ , different choices of ρ give the same space L p k,γ (E), with equivalent norms.
For instance, the r-forms E = Λ r T * C on C with metric g and the Levi-Civita connection are automatically asymptotically cylindrical, and if C is an oriented 4-manifold then the self-dual 2-forms Λ 2 + T * C are also asymptotically cylindrical. We consider partial differential operators on asymptotically cylindrical manifolds.
Definition 3.2 In the situation of Definition 3.1, suppose E, F are two asymptotically cylindrical vector bundles on C , asymptotic to cylindrical vector bun- Then we call A an asymptotically cylindrical operator. It is easy to show that A extends to bounded linear operators for all p > 1, l 0 and γ ∈ R.
Now suppose A is an elliptic operator. (7) is Fredholm if and only if γ does not lie in a discrete set D A 0 ⊂ R, which we now define.  Theorem 3.4 Let (C, g) be a Riemannian manifold asymptotic to (L×R, g 0 ), and A : C ∞ (E) → C ∞ (F ) an elliptic partial differential operator on C of order k between vector bundles E, F on C , asymptotic to the cylindrical elliptic Then D A 0 is a discrete subset of R, and for p > 1, l 0 and γ ∈ R, the extension A p k+l,γ :  10] then implies that e ∈ L r k+m,δ (E) for all r > 1, m 0 and δ > γ , and e is smooth. But Ker(A p k+1,γ ) is invariant under small changes of γ in R \ D A 0 , so e ∈ L r k+m,γ (E) for all r > 1 and m 0. This proves: is independent of p, l, and is a finite-dimensional vector space of smooth sections of E .
This is also an asymptotically cylindrical linear elliptic partial differential operator of order k on C , with the property that for compactly-supported e ∈ C ∞ (E) and f ∈ C ∞ (F ).
Then for p > 1, l 0 and γ / ∈ D A 0 , the dual operator of (7) is where q > 1 is defined by 1 , and these isomorphisms identify (A * ) q −l,−γ with the dual linear map to (7). Now there is a problem with (8), as it involves Sobolev spaces with negative numbers of derivatives −l, −k − l. Such Sobolev spaces exist as spaces of distributions. But we can avoid defining or using these spaces, by the following trick. We are interested in Ker (A * ) q −l,−γ , as it is dual to Coker(A p k+l,γ ). The elliptic regularity argument above showing Ker(A p k+l,γ ) is independent of l also holds for negative differentiability, so we have Ker (A * ) q −l,−γ = Ker (A * ) q k+m,−γ for m ∈ Z, and in particular for m 0. So we deduce: Lockhart and McOwen show [16, Theorem 6.2] that for γ, δ ∈ R \ D A 0 with γ δ we have 3.2 d+d * and d * d+dd * on an asymptotically cylindrical manifold Let (C, g) be an oriented asymptotically cylindrical Riemannian n-manifold asymptotic to a Riemannian cylinder (L × R, g 0 ), where g 0 = g L + dt 2 and (L, g L ) is a compact oriented Riemannian (n − 1)-manifold. Consider the asymptotically cylindrical linear elliptic operators We shall apply the theory of Section 3.1 to study the extensions for p > 1, l 0 and γ ∈ R, and their kernels and cokernels.
For |γ| close to zero we can say more about the kernels of (13) and (14).
Proposition 3.8 Suppose p, q > 1, l, m 0 and γ < 0 with 1 Moreover all four kernels consist of smooth closed and coclosed forms.
This is because from (11), the l.h.s. of (18) is the dimension of the solution space of (d + d * ) 0 χ = 0 on L × R for χ independent of t ∈ R. The space of such χ is the direct sum over k = 0, . . . , n − 1 of the spaces of k -forms η and (k+1)-forms η ∧ dt for η ∈ C ∞ (Λ k T * L) with dη = d * η = 0. By Hodge theory we deduce (18). Now d+d * is formally self adjoint, that is, A * = A in the notation of Section 3.1. Thus by (10), and equation (17) follows from (18).
It remains to show the four kernels consist of smooth closed and coclosed forms. Let χ lie in one of the kernels, and write χ = n k=0 χ k for χ k a k -form. Since (d * d + dd * )χ = 0 we have (d * d + dd * )χ k = 0, as d * d + dd * takes k -forms to k -forms. Thus χ k lies in the same kernel, so (d + d * )χ k = 0 by (15) or (16). But dχ k and d * χ k lie in different vector spaces, so dχ k = d * χ k = 0 for all k . Hence dχ = d * χ = 0, and χ is closed and coclosed. Smoothness follows by elliptic regularity.
As the forms χ in Ker (d + d * ) p l+2,γ are closed we can map them to de Rham cohomology H * (C, R) by χ → [χ]. We identify the kernel and image of this map. Proof Lockhart [15,Ex. 0.14] shows that the vector space H 2 (Λ k T * C, g) of closed, coclosed k -forms in L 2 (Λ k T * C) on an asymptotically cylindrical R). Taking the direct sum over k = 0, . . . , n, this implies that for l 0 the map is injective, with image that of the natural map H * cs (C, R) → H * (C, R).

d + + d * on a 4-manifold
Now we restrict to dim C = 4, so that (C, g) is an oriented asymptotically cylindrical Riemannian 4-manifold asymptotic to a Riemannian cylinder (L×R, g 0 ). In Section 4 we will take C to be an asymptotically cylindrical coassociative 4fold. Consider the asymptotically cylindrical linear elliptic operator Here d + is the restriction of d to the self-dual 2-forms. We use this notation to distinguish d + + d * from d + d * in (12). Roughly speaking, d + + d * is a quarter of d + d * in (12), as it acts on half of the even forms, rather than on all forms. Its formal adjoint is is the projection of d * to the self-dual 2-forms. We shall apply the results of Section 3.2 to study the extension (20) for p > 1, l 0 and γ ∈ R. We begin with some algebraic topology.
Suppose for simplicity that C has no compact connected components, so that H 4 (C, R) = H 0 cs (C, R) = 0. Then L is a compact, oriented 3-manifold, and C is the interior (C) • of a compact, oriented 4-manifold C with boundary ∂C = L. Thus we have a long exact sequence in cohomology: where  (21) is written so that each vertically aligned pair of spaces are dual vector spaces, and each vertically aligned pair of maps are dual linear maps.
Let V ⊆ H 2 (C, R) be the image of the natural map H 2 cs (C, R) → H 2 (C, R). Taking alternating sums of dimensions in (21) shows that Now the cup product ∪ : H 2 cs (C, R)×H 2 (C, R) → R restricted to H 2 cs (C, R)×V is zero on the product of the kernel of H 2 cs (C, R) → H 2 (C, R) with V . Hence it pushes forward to a quadratic form ∪ : V × V → R, which is symmetric and nondegenerate.
Suppose V = V + ⊕ V − is a decomposition of V into subspaces with ∪ positive definite on V + and negative definite on V − . Then dim V + and dim V − are topological invariants of C, L. That is, they depend only on C as an oriented 4-manifold, and not on the choice of subspaces V ± .
Under this isomorphism, the cup product on V is given by for χ, ξ ∈ H 2 . The Hodge star * maps H 2 ∼ = V to itself with * 2 = 1. Let V ± be the ±1 eigenspaces of * on V . Then V = V + ⊕ V − , and (24) implies that ∪ is positive definite on V + and negative definite on V − . Hence dim V + is a topological invariant, from above. But Ker (d + + d * ) p l+2,γ is the self-dual 2-forms in H 2 . Thus χ → [χ] induces an isomorphism of Ker (d + + d * ) p l+2,γ with V + , as we have to prove.
We can say more about the cokernel (22). Its dimension is . So we can think of Ker (d * + + d) q m+2,−γ as a space of closed and coclosed 3-forms filling out all of H 3 cs (C, R) and H 3 (C, R). But we will not need these facts, so we shall not prove them.

Conditions on the rate γ for d + + d * to be Fredholm
Finally we determine the set D (d + +d * ) 0 for the cylindrical operator (d + +d * ) 0 on L × R, which by Theorem 3.4 gives the set of γ for which (20) is not Fredholm. Proposition 3.11 Let (C, g) be an oriented, asymptotically cylindrical Riemannian 4-manifold asymptotic to (L × R, g 0 ), let p > 1, l 0 and γ ∈ R, and define (d + + d * ) p l+2,γ as in (20).

Proof of Theorem 1.1
We now prove Theorem 1.1. Let (M, ϕ, g) be an asymptotically cylindrical G 2 -manifold asymptotic to X × (R, ∞), R > 0, with decay rate α < 0. Let C be an asymptotically cylindrical coassociative 4-fold in X asymptotic to L × (R ′ , ∞) for R ′ > R with decay rate β for α β < 0. Write g C = g| C for the metric on C , g X for the Calabi-Yau metric on X , and g L = g X | L for the metric on L. Then (C, g C ) is an asymptotically cylindrical Riemannian 4-manifold, with rate β .

29) as in Section 3.3. Then Proposition 3.11 and the conditions on γ imply that
l+2,γ is Fredholm by Theorem 3.4. Also, Theorem 3.10 applies to (d + + d * ) p l+2,γ . Let ν L be the normal bundle of L in X , regarded as the orthogonal subbundle to T L in T X| L , and exp L : ν L → X the exponential map. For r > 0, write B r (ν L ) for the subbundle of ν L with fibre at x the open ball about 0 in ν L | x with radius r. Then for small ǫ > 0, there is a tubular neighbourhood T L of L in X such that exp L : B 2ǫ (ν L ) → T L is a diffeomorphism. Also, ν L ×R → L×R is the normal bundle to L × R in X × R with exponential map exp L ×id : ν L × R → X × R. Then T L × R is a tubular neighborhood of L × R in X × R, and exp L ×id : Let K, R, Ψ : X ×(R, ∞) → M \K , and K ′ , R ′ > R, Φ : L×(R ′ , ∞) → C \K ′ , and the normal vector field v on L × (R ′ , ∞) be as in Section 2.3, so that (3) commutes. Then v is a section of ν L × (R ′ , ∞) → L × (R ′ , ∞), decaying at rate O(e βt ). Therefore making K ′ and R ′ larger if necessary, we can suppose the graph of v lies in B ǫ (ν L ) × (R ′ , ∞).
Here w is a point in B ǫ (ν L ) × (R ′ , ∞), in the fibre over π(w) ∈ L × (R ′ , ∞). Thus v| π(w) is a point in the same fibre, which is a ball of radius ǫ in a vector space.
Identify L × (R ′ , ∞) with the zero section in B ǫ (ν L ) × (R ′ , ∞). Then as (3) commutes we see that Ξ| L×(R ′ ,∞) ≡ Φ. Let ν C be the normal bundle of C in M , which we regard not as the orthogonal subbundle to T C in T M | C , but rather as the quotient bundle T M | C /T C . Define an isomorphism ξ between the vector bundles ν L × (R ′ , ∞) and Φ * (ν C ) over L × (R ′ , ∞) as follows.
As Ξ is a diffeomorphism with its image, dΞ : is an isomorphism. Restricting this to the zero section L × (R ′ , ∞) and noting that Ξ| L×(R ′ ,∞) ≡ Φ, we see that is an isomorphism. As Φ : is an isomorphism. But (32) is the restriction of (31) to a vector subbundle. Quotienting (31) by (32) gives an isomorphism Now choose a small ǫ ′ > 0, a tubular neighborhood T C of C in M , and a diffeomorphism Θ : B ǫ ′ (ν C ) → T C satisfying the conditions: is the zero section.
(ii) By (i), dΘ| C : T (B ǫ ′ (ν C ))| C → T M | C is an isomorphism, which restricts to the identity on the subbundles T C of each side. Hence it induces an isomorphism dΘ| C : This isomorphism is the identity map.
Notice that (iii) determines Θ and T C uniquely on B ǫ ′ (ν C )| C\K ′ , and by construction here it satisfies (i) and (ii). Thus, it remains only to choose T C and Θ satisfying (i), (ii) over the compact set K ′ ⊂ C , which is possible by standard differential topology.
The point of all this is that we have chosen a local identification Θ between ν C and M near C that is compatible in a nice way with the asymptotic identifications Φ, Ψ of C , M with L × R and X × R. Using Θ, submanifolds C of M close to C are identified with small sections s of ν C , and importantly, the asymptotic convergence of C to C , and so to L × R, is reflected in the asymptotic convergence of s to 0.
As in the proof of Theorem 2.5, the map V → (V · ϕ| x )| TxC defines an isomorphism ν C → Λ 2 + T * C . (Note that since ϕ| C ≡ 0, this map is well-defined for V ∈ T x M/T x C , rather than just for V ∈ T x M orthogonal to T x C .) We now identify ν C with Λ 2 + T * C , and regard Θ as a map Θ : is an open subset of the Banach space L p l+2,γ (Λ 2 + T * C).
The point of this definition is that if Γ ζ 2 + is the graph of ζ 2 + in B ǫ ′ (Λ 2 + T * C) and C = Θ(Γ ζ 2 + ) its image in M , then C is coassociative if and only if ϕ| C ≡ 0, which holds if and only if Q(ζ 2 + ) = 0. So Q −1 (0) parametrizes coassociative 4-folds C close to C . We now consider which class of 3-forms Q maps to.
Proof As in the proof of Theorem 2.5, the functional form of Q is where F is a smooth function of its arguments. Since p > 4 and l 1 we have L p l+2,γ (Λ 2 + T * C) ֒→ C 1 γ (Λ 2 + T * C) by Sobolev embedding. General arguments then show that locally Q(ζ 2 + ) is L p l+1 . To show Q(ζ 2 + ) lies in L p l+1,γ (Λ 3 T * C), we must know something of the asymptotic behavior of F at infinity. Essentially Q(ζ 2 + ) is the restriction to Γ(ζ 2 + ) of the 3-form Θ * (ϕ) on B ǫ ′ (Λ 2 + T * C). Using the identifications Φ : But the asymptotic conditions on Φ, Ψ and v imply that Ξ * (ϕ) is the sum of a translation-invariant 3-form on B ǫ (ν L ) × R, the pullback of the cylindrical G 2 form ϕ 0 on X × R, and an error term which decays at rate O(e βt ), with all its derivatives. As β < γ < 0, it is not difficult to see from this that Q maps to L p l+1,γ (Λ 3 T * C). Smoothness of Q holds by general principles. Finally, the linearization of Q is d, by the calculation of McLean alluded to in Theorem 2.5. As the calculation is local, it does not matter that we are on a noncompact manifold C .
Next we show that the image of Q consists of exact 3-forms. Notice that we lose one degree of differentiability: although the image of Q lies in L p l+1,γ (Λ 3 T * C), we claim only that it lies in the exact 3-forms in L p l,γ (Λ 3 T * C).
Proof Consider the restriction of the 3-form ϕ to the tubular neighborhood T C of C . As ϕ is closed, and T C retracts onto C , and ϕ| C ≡ 0, we see that ϕ| T C is exact. Thus we may write ϕ| T C = dθ for θ ∈ C ∞ (Λ 2 T * T C ). Since ϕ| C ≡ 0 we may choose θ| C ≡ 0. Also, as ϕ is asymptotic to O(e βt ) with all its derivatives to a translation-invariant 3-form ϕ 0 on X × R, we may take θ to be asymptotic to O(e βt ) with all its derivatives to a translation-invariant 2-form on T L × R.
The proof of Proposition 4.1 now shows that the map ζ 2 As in the proof of Theorem 2.5, we augment Q by a space of 4-forms on C to make it elliptic. Define . Proposition 4.1 implies that the linearization dP (0, 0) of P at 0 is the Fredholm operator (d + +d * ) p l+2,γ of (29). Define C to be the image of (d + +d * ) p l+2,γ . Then C is a Banach subspace of L p l+1,γ (Λ 3 T * C), since (d + + d * ) p l+2,γ is Fredholm. We show P maps into C .
We must show it lies in C . Since γ / ∈ D (d + +d * ) 0 , from Section 3 this holds if and only if where 1 p + 1 q = 1 and m 0. By Theorem 3.10, Ker (d * + + d) q m+2,−γ consists of closed and coclosed 3-forms χ. By Proposition 4.2 we have Q(ζ 2 ,−γ , as χ is closed and coclosed, and the inner products and integration by parts are valid because of the matching of rates γ, −γ and L p , L q with 1 p + 1 q = 1. So (34) holds, and P maps into C .
We now apply the Implicit Mapping Theorem for Banach spaces, [7, Theorem 1.2.5]. Define A to be Ker (d + +d * ) p l+2,γ , and B to be the subspace of L p l+2,γ (Λ 2 + T * C) ⊕ L p l+2,γ (Λ 4 T * C) which is L 2 -orthogonal to A. As A is finite-dimensional and the L 2 inner product is continuous on L p l+2,γ (Λ 2 + T * C) ⊕ L p l+2,γ (Λ 4 T * C), both A, B are Banach spaces, and A⊕B = L p l+2,γ (Λ 2 Let P, C be as above. Then by Propositions 4.1 and 4.3, P : U × V → C is a smooth map of Banach manifolds with P (0, 0) = 0, and linearization dP (0, 0) = (d + + d * ) p l+2,γ : A ⊕ B → C . By definition A is the kernel and C is the image of (d + + d * ) p l+2,γ . Hence dP (0,0) | B : B → C is an isomorphism of vector spaces, and as it is a continuous linear map of Banach spaces, it is an isomorphism of topological spaces by the Open Mapping Theorem.
Thus Theorem 4.4 applies, and gives a connected open neighborhood U ′ of 0 in U , and a smooth map G : U ′ → V such that G(0) = 0 and P (x, G(x)) ≡ 0.
Write ∂ y F and ∂ z F for the partial derivatives of F in the y, z directions. That is, with x held constant the domains Λ 2 + T * x C and T * x C ⊗ Λ 2 + T * x C of y, z and the range Λ 3 T * x C of F are vector spaces, so the map (y, z) → F (x, y, z) is a smooth map between vector spaces, and has well-defined partial derivatives.
The operator L ζ 2 + is essentially ∆ C (d + + d * ), and so is a linear third-order elliptic operator. Note that we allow 0 n m + 1 in (37). The reason for this is that the coefficients of L ζ 2 + depend on ζ 2 + , ∇ζ 2 + , and so lie in L p m+1 locally. So the maximum regularity we can expect for L ζ 2 + (ξ 2 + , ξ 4 ) is L p m+1 , forcing n m + 1.
The most obvious value for n in (37) is n = m − 1, as then the domain of L ζ 2 + is the space L p m+2,γ containing (ζ 2 + , ζ 4 ). However, our next lemma is an elliptic regularity result for L ζ 2 + when n = m. This is because we will use it to increase the regularity of (ζ 2 + , ζ 4 ) from L p m+2,γ to L p m+3,γ by 'bootstrapping'.
Proof If L ζ 2 + were a smooth, asymptotically cylindrical elliptic operator the lemma would follow from Lockhart (39), where A > 0 depends on m, p, the domains involved, C m bounds on the coefficients of L ζ 2 + , and a modulus of continuity for their m th derivatives.
Such a modulus of continuity is provided by a Hölder C 0,α bound for the m th derivatives, for α ∈ (0, 1). Thus, we can prove local estimates of the form (39) provided we have local C m,α bounds for the coefficients of L ζ 2 + , which follow from local C m+1,α bounds for ζ 2 + . Now ζ 2 + ∈ L p m+2,γ (Λ 2 + T * C) and p > 4, so the Sobolev Embedding Theorem shows that L p m+2 embeds in C m+1,α for α = 1 − 4/p. Therefore we do have C m,α control on the coefficients of L ζ 2 + , and can show that L ζ 2 + is asymptotic in a weighted Hölder C m,α γ sense to a cylindrical operator. So (39) holds as in [16], and the lemma is proved.
Proposition 4.8 S is a homeomorphism from Q −1 (0) to a neighborhood of C in M γ C .
To show S is a homeomorphism requires us to specify the topology on M γ C , which we have not done. The natural way to define a topology on a space of submanifolds is to identify submanifolds C near C with sections of the normal bundle ν C to C , and induce the topology from some Banach norm on a space of sections of ν C . In our case, this just means that the topology on M γ C is induced from some choice of Banach norm on sections ζ 2 + of Λ 2 + T * C , say the C 1 γ topology.
But since Q −1 (0) with its L p l+2,γ topology is locally homeomorphic to the finitedimensional vector space Ker (d + +d * ) p l+2,γ , any choice of topology on sections ζ 2 + gives the same topology on M γ C , as all Banach norms give the same topology on a finite-dimensional space. So S is a local homeomorphism.

Remark 4.9
In a forthcoming paper, we study the deformation space of asymptotically cylindrical coassociative submanifolds with moving boundary. Similarly to the setting here, we work with an asymptotically cylindrical G 2manifold M with a Calabi-Yau boundary X at infinity. We also assume that the boundary of the coassociative submanifold at infinity is a special Lagrangian submanifold of X and is allowed to move. The details of this construction will appear in [26].