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Asymptotically Cylindrical Ricci-Flat Manifolds

Asymptotically cylindrical Ricci-flat manifolds play a key role in constructing Topological Quantum Field Theories. It is particularly important to understand their behavior at the cylindrical ends and the natural restrictions on the geometry. In this paper we show that an orientable, connected, asymptotically cylindrical manifold (M,g) with Ricci-flat metric g can have at most two cylindrical ends. In the case where there are two such cylindrical ends then there is reduction in the holonomy group Hol(g) and (M,g) is a cylinder.


Introduction
Let (M n , g) be a connected asymptotically cylindrical manifold with Ricci-flat metric g. Examples of such manifolds are asymptotically cylindrical Riemannian manifolds whose holonomy groups Hol(g) are subgroups of SU (m), for n = 2m, and subgroups of the exceptional Lie group G 2 , for n = 7. Examples of asymptotically cylindrical Riemannian 6-manifolds with holonomy SU (3) were first constructed by Tian and Yau [12], [13] and Kovalev [5]. In his paper, [5], Kovalev also showed that the Riemannian product of these 6-manifolds with a circle produced asymptotically cylindrical Riemannian 7-manifolds with holonomy SU Other examples of noncompact Ricci-flat manifolds (M n , g) are hyperkähler manifolds, manifolds with holonomy group Hol(g) ⊂ Sp(m), for n = 4m, and Spin(7) manifolds with holonomy group Hol(g) ⊂ Spin (7), for n = 8.
Asymptotically cylindrical Calabi-Yau and G 2 manifolds are the main objects of the Topological Quantum Field Theories, as is explained in [6]. In order to construct consistent TQFT's, it is essential to understand the geometric restrictions at the cylindrical ends of a given orientable, connected, noncompact Ricci-flat manifold with some small decay rate. A very natural and fundamental question is whether it is possible for these manifolds to have multiple cylindrical ends.
In this paper we prove the following theorem.
Theorem 1.1. Let (M, g) be an orientable, connected, asymptotically cylindrical Riemannian manifold with Ricci-flat metric g and have l cylindrical ends. Then l ≤ 2, and in the case when l = 2 there is reduction in the holonomy group Hol(g) and (M, g) is a cylinder.
Remark 1.2. The proof of this theorem also follows from the Cheeger-Gromoll splitting theorem [2]. The key ingredients of the splitting theorem are the maximum principle for continuous functions, Laplacian comparison theorems, constructions of rays, lines and Busemann functions. For more details on the subject see [11]. In this paper we give an alternative proof and show that the reduction in holonomy can be obtained by just using the analytic set-up for Fredholm properties of an elliptic operator on noncompact manifolds. This analytic set-up was developed by Lockhart and McOwen, [8] and by Melrose, [9], [10].

Asymptotically cylindrical manifolds
In this section we first introduce the definitions of cylindrical and asymptotically cylindrical Riemannian manifolds.
Definition 2.1. An n-dimensional Riemannian manifold (M 0 , g 0 ) is called cylindrical if M 0 = X × R and g 0 is compatible with this product structure, that is, where X is a compact, connected (n − 1) manifold with Riemannian metric g X . Definition 2.2. A connected, complete n-manifold (M, g) with l cylindrical ends is called asymptotically cylindrical with decay rate β = (β 1 , . . . , β l ) ∈ R l , β j < 0 for Remark 2.3. For simplicity, we will assume that the decay rates are all equal for each cylindrical end, that is β j = β k , 1 ≤ j, k ≤ l.

2.
1. An Example. If there are additional geometric structures on the cylindrical manifold (M 0 , g 0 ) = (X × R, g 0 ) then they should be also compatible with this product structure. In addition to the metric g, the geometric structures on the asymptotically cylindrical manifold (M, g) should converge to order O(e βt ) to the cylindrical geometric structures on (M 0 , g 0 ), with all of their derivatives.
One example where we have such additional structures on (M, g) is an asymptotically cylindrical G 2 manifold. A 7-dimensional Riemannian manifold (M, g) is called a G 2 manifold if the holonomy group of its Levi-Civita connection of g lies inside of G 2 ⊂ SO(7). A G 2 manifold is equipped with a vector cross product × on its tangent bundle and a harmonic (calibration) 3-form ϕ ∈ Ω 3 (M ) such that for tangent vectors u, v, w ∈ T (M ). For more information on G 2 manifolds see [3].
Next, we define an asymptotically cylindrical G 2 manifold with one end. Let (M 0 , ϕ 0 , g 0 ) be a G 2 manifold with calibration 3-form ϕ 0 and the Riemannian metric g 0 . In this case Definitions 2.1 and 2.2 take the following form, as in [4].
and (ϕ 0 , g 0 ) is compatible with this product structure, that is, where X is a compact, connected Calabi-Yau 3-fold with Kähler form ω, Riemannian metric g X and holomorphic (3,0)-form Ω.
Using similar relations between the geometric structures we can define cylindrical and asymptotically cylindrical Calabi-Yau manifolds. The details of these definitions and the applications will appear in a forthcoming paper, [14].
2.2. The Laplacian operator on asymptotically cylindrical manifolds. In this section we prove Proposition 2.11 and study the properties of the cokernel of asymptotically cylindrical Laplacian operator. This proposition plays an important role in Section 3 in the proof of Theorem 1.1.
First we recall the basic facts about the properties of the Laplacian operator and its (co)kernel on asymptotically cylindrical manifolds with decay rate β < 0. More details can be found in Lockhart and McOwen [7], [8]. Also one can find similar results with a different analytical approach in Melrose, [9], [10].
Definition 2.6. Let (M, g) be an asymptotically cylindrical Riemannian manifold asymptotic to X × R with decay rate β.
be the cylindrical Laplacian operator defined on the Riemannian cylinder which is the asymptotic model for the ends of M and invariant under translations in R. Let also ∆ : C ∞ (M ) → C ∞ (M ) be the Laplacian operator on M . Suppose that Ψ is defined as in Definition 2.2. Then since g is an asymptotically cylindrical metric asymptotic to g 0 , ∆ is asymptotic to ∆ 0 and Ψ * (∆) = ∆ 0 + O(e βt ) as t → ∞ for β < 0 in the sense that their coefficients (in local coordinates on X and R) are exponentially close. We call ∆ the asymptotically cylindrical Laplacian operator.
Choose a smooth function ρ : M → R such that Ψ * (ρ) ≡ t on X × (R, ∞). This prescribes ρ on M \ K, so we only have to extend ρ over the compact set K. For p ≥ 1, k ≥ 0 and α ∈ R we define the weighted Sobolev space L p k,α (M ) to be the set of functions f on M that are locally integrable and k times weakly differentiable and for which the norm is finite. Then L p k,α (M ) 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,α (M ), with equivalent norms.
Let L p k,α (M ) be the weighted Sobolev spaces for some α such that β < α < 0. Then one can show that ∆ extends to bounded linear operators for all p > 1, k 2 and α ∈ R.
In Definition 2.7 we define a set D ∆0 ⊂ R, which determines the value of α so that an asymptotically cylindrical operator is Fredholm.
Definition 2.7. Let ∆ and ∆ 0 be the Laplacian operators on M and X × R. Extend ∆ 0 to the complexifications ∆ 0 : C ∞ (X × R) ⊗ C → C ∞ (X × R) ⊗ C. Then D ∆0 is the set of α ∈ R such that for some δ ∈ R there exists a nonzero section s ∈ C ∞ (X × R) ⊗ C invariant under translations in R such that ∆ 0 (e (α+iδ)t s) = 0.
Then D ∆0 is a discrete subset of R, and for p > 1, k 2 and α ∈ R, the extension Proof. When α / ∈ D ∆0 we see from (3) that Now since ∆ is self adjoint, ∆ = ∆ * , we have Let M be an asymptotically cylindrical Riemannian manifold with one cylindrical end X × (R, ∞). Lockhart and McOwen show [8, Th. 6.2] that for α, δ ∈ R \ D ∆0 with α δ we have where d(ǫ) 1 is the dimension of the vector space of solutions f ∈ C ∞ (X × R) ⊗ C of a prescribed form as in Definition 2.7 with ∆ 0 f = 0.
By (7), ind ∆ q m,−α − ind ∆ p k,α is the dimension of the solution space of ∆ 0 f = 0 on X × R for f polynomial in t ∈ R. By Hodge theory we deduce (8). By Theorems 2.9, 2.10 we can conclude that the dimensions of kernel and cokernel are independent of p, k and we can rewrite (8) as Now let M be an asymptotically cylindrical Riemannian manifold with l cylindrical ends X 1 × (R, ∞), . . . , X l × (R, ∞). In this case we get a contribution of 2 dimensions for each end and the equation (9) becomes Now since we have from equation (6) But by maximum principle the harmonic functions which decay with rate α, α < 0 cannot be nonzero functions and one can easily see that Ker(∆ q m,α ) = 0. This with equation (11)  one can take f ′ ∈ L p k,α (M ) to be smooth which decays like O(e αt ). Therefore f = C i t i + D i + O(e αti ), α < 0, on each end X i × R.
In order to show the existence of the function f such that f = f 0 + f ′ we need to find a function f ′ that satisfies This is equivalent to showing that ∆f 0 ⊥ Coker(∆ p k,α ) and in Proposition 3.1 we prove this fact. Let f 0 = f (Ci,Di) 0 be defined as above.
Proposition 3.1. Let M be an asymptotically cylindrical Riemannian manifold with l cylindrical ends X 1 × (R, ∞), . . . , X l × (R, ∞). Also for h ∈ (Coker(∆ p k,α )) * let Φ be a map defined as where ·, · L 2 is the L 2 -inner product defined on the space of functions on M .
Then for any element (C i , D i ) ∈ Ker Φ the map Ker Φ −→ (Coker(∆ p k,α )) * given by Proof. First note that Φ is linear in (C i , D i ) as we defined f (Ci,Di) 0 to be linear in (C i , D i ). Since there are 2l parameters in (C i , D i ) and also dim Coker(∆ p k,α ) = l from Proposition 2.11, Φ is a linear map between R 2l → R l which guarantees that dim Ker Φ ≥ l. But for each Therefore we have an injective linear map ⊂ (Coker(∆ p k,α )) * and hence we get an injective linear map This implies that dim Ker Φ ≤ dim(Coker(∆ p k,α )) * = l and hence dim Ker Φ = l. Therefore Ker Φ → (Coker(∆ p k,α )) * is an isomorphism as an injective linear map between spaces of dimension l and this completes the proof.

3.2.
Existence of constant harmonic 1-form γ = df on M . After constructing the harmonic function f , one can easily obtain a closed and coclosed 1-form γ = df on M . We now show that γ = df is constant for Ricci-flat manifolds. Let g be the Riemannian metric on M . In the index notation the Weitzenböck formula for a 1-form ξ is (dd * + d * d)ξ a = ∇ * ∇ξ a + R ab g bc ξ c .
(15) where R ab is the Ricci curvature of g. Suppose that R ab = 0 then we have on 1-forms on a Ricci-flat manifold M .
Proposition 3.2. Let M be an asymptotically cylindrical Riemannian manifold with Ricci-flat metric g. Let γ = df be a harmonic 1-form defined as above. Then γ = df is constant, that is ∇γ = 0.
Since γ is harmonic we have (dd * + d * d)γ = 0 and by equation (16) we get and by Stoke's Theorem, for normal vectors η 1 , . . . , η l . The asymptotic behaviour of γ = df guarantees that the right hand side of equation (18) goes to zero as R → ∞ and yields the result.
We now complete the proof of Theorem 1.1. Let (M, g) be an asymptotically cylindrical Ricci-flat manifold with l ends. Then (M, g) is asymptotic to the cylinders X 1 × (R, ∞),..., X l × (R, ∞), for R ≫ 0. Proposition 3.1 guarantees the existence of an harmonic function f on M .
Note that if l = 1, by Proposition 2.11, dim(Coker(∆ p k,α )) = 1 and f should be constant and so df = 0. If l 2 then one can choose f not constant and construct a nonzero, closed and coclosed 1-form γ = df . Since (M, g) is Ricci-flat, by Proposition 3.2, γ is constant. Thus the results [3,Sec.3.2] show that the metric g on M is locally reducible. Rescale f if necessary so that |γ| ≡ 1.
Also note that by Proposition 3.2, df is covariant constant and so |df | is constant. Therefore γ = df never vanishes on M , and f can be viewed as a Morse function with no critical points. Define X to be the level set f −1 (0). This is a nonsingular submanifold of M as df = 0 everywhere, and is compact as f → ±∞ at the ends of M . This implies that M is at least a topological product M = X × R.
We now construct a diffeomorphism X × R → M . For t ∈ R define υ t : M → M to be the gradient flow of f for time t. As M is complete and |df | ≡ 1 this is a well-defined diffeomorphism, which induces a diffeomorphism between X and the level set f −1 (t). Define Υ : X × R → M by Υ(x, t) = υ t (x). This is a bijection as for each t, x → Υ(x, t) = υ t (x) is a bijection between X and f −1 (t), and it is easy to see Υ is a diffeomorphism.
As |df | ≡ 1, the metric Υ * (g) is of the form g(x, t) + dt 2 , where g(x, t) is a 1parameter family of metrics on X depending on t ∈ R. Furthermore, since Υ * (g) is locally reducible and dt is constant, we see that g(x, t) is locally and hence globally independent of t. Thus M is a Riemannian product X × R. Since M is connected X is connected, so M has exactly two ends, so l = 2 and it is a cylinder.  ) is an asymptotically cylindrical G 2 manifold with two cylindrical ends, then Hol(g) preserves γ and there is a reduction in the holonomy group and hence Hol(g) ⊂ SU (3). This implies that M = X × R where X is a Calabi-Yau 3-fold and hence M is a cylinder.