diff --git "a/2tE1T4oBgHgl3EQf5gWl/content/tmp_files/2301.03513v1.pdf.txt" "b/2tE1T4oBgHgl3EQf5gWl/content/tmp_files/2301.03513v1.pdf.txt" new file mode 100644--- /dev/null +++ "b/2tE1T4oBgHgl3EQf5gWl/content/tmp_files/2301.03513v1.pdf.txt" @@ -0,0 +1,3869 @@ +arXiv:2301.03513v1 [math.DG] 9 Jan 2023 +Analysis and spectral theory of neck-stretching problems +Thibault Langlais +Mathematical Institute, +University of Oxford +Abstract +We study the mapping properties of a large class of elliptic operators PT in gluing +problems where two non-compact manifolds with asymptotically cylindrical geometry +are glued along a neck of length 2T . +In the limit where T → ∞, we reduce the +question of constructing approximate solutions of PT u = f to a finite-dimensional +linear system, and provide a geometric interpretation of the obstructions to solving +this system. Under some assumptions on the real roots of the model operator P0 on +the cylinder, we are able to construct a Fredholm inverse for PT with good control on +the growth of its norm. As applications of our method, we study the decay rate and +density of the low eigenvalues of the Laplacian acting on differential forms, and give +improved estimates for compact G2-manifolds constructed by twisted connected sum. +We relate our results to the swampland distance conjectures in physics. +Contents +1 +Introduction and motivation +2 +2 +Setup and discussion of results +3 +2.1 +Model gluing problem +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2.2 +Substitute kernel and cokernel . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +2.3 +Results and strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +3 +Translation-invariant elliptic PDEs on cylinders +12 +3.1 +Analysis on cylinders by separation of variables . . . . . . . . . . . . . . . . +12 +3.2 +Polyhomogeneous sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . +17 +3.3 +Existence of solutions +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +20 +4 +Construction of solutions +24 +4.1 +Analysis on EAC manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . +25 +4.2 +Characteristic system +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +27 +4.3 +Main construction +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +31 +5 +Spectral aspects +36 +5.1 +Approximate harmonic forms . . . . . . . . . . . . . . . . . . . . . . . . . . +36 +5.2 +Density of low eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . +40 +6 +Improved estimates for twisted connected sums +45 +6.1 +G2-manifolds +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +45 +6.2 +The twisted connected sum construction . . . . . . . . . . . . . . . . . . . . +47 +6.3 +Proof of the quadratic estimates +. . . . . . . . . . . . . . . . . . . . . . . . +49 +1 + +References +52 +1 +Introduction and motivation +This paper is concerned with analytical aspects of the study of special geometric structures +near degenerate limits. Typically, such structures are defined as solutions of a non-linear +PDE and explicit solutions can be quite challenging to find, especially in the absence of +symmetries or in the compact setting. A standard way to produce such solutions is to +glue simpler building blocks together in order to construct approximate solutions, and +then perturb them so as to obtain a genuine solution using a fixed-point argument. This +idea has been applied in various contexts for differential-geometric constructions, such +as self-dual metrics [13, 15], hyperkähler metrics on K3 surfaces [27], minimal surfaces +[21, 22], or compact manifolds with special holonomy [18, 19], to only cite a few of them. +The question of whether such construction can be successfully performed can often be +reduced to understanding the linearised problem. Therefore we shall be concerned with +linear operators during most of the paper. The broad question that we are interested in is +how to invert such operators and with which optimal bound in Sobolev and Hölder norms. +For formally self-adjoint operators this corresponds to understanding the low eigenvalues. +In order to give a precise formulation of these questions we will restrict ourselves to +constructions where the gluing region is modelled on a cylinder and study the limit where +this neck is infinitely stretched. Our main focus throughout this paper will be the case +where the model operator on the cylinder has real roots, which substantially complicates +the analysis. Our aim is to develop a systematic method to deal with such cases. +As a prototypical example of construction that the techniques developed in this paper +are aimed at, we shall study compact manifolds with holonomy G2 constructed by twisted +connected sum. The exceptional Lie group G2 ⊂ SO(7) is one of the two exceptional +groups (along with Spin(7) ⊂ SO(8)) appearing in the Berger’s list of possible Riemannian +holonomy groups of non-symmetric spaces [5]. At this time, it was not known whether +metrics with full holonomy G2 actually existed, and the first non-complete examples on a +ball B7 ⊂ R7 were only constructed three decades later by Bryant [7], shortly before the +first examples of complete metrics with exceptional holonomy by Bryant–Salamon [8]. The +first compact G2-manifolds were exhibited by Joyce [18, 19] using a gluing construction to +resolve the singularities of a flat orbifold. This construction was more recently extended +by Joyce–Karigiannis in [17]. The only other known examples of compact manifolds with +full holonomy G2 rely on the twisted connected sum construction, first due to Kovalev [25] +and later improved by Corti–Haskins–Nordström–Pacini [10, 11] and further extended by +Nordström [33]. The twisted connected sum fits into our general setup, and we are able +to give improved estimates for this construction. Moreover we can derive the decay rate +and the density of the low eigenvalues of the Laplacian on twisted connected sums. +The work presented here was partly motivated by our interest in the swampland conjec- +tures in physics [37] (see also [6] and [36] for detailed reviews). Indeed, compact manifolds +with special geometry play an important role for compactifications in quantum gravity +theories. For instance, it is relevant to consider Calabi–Yau threefold compactifications +in string theory or compactifications on G2-manifolds in M-theory. The geometry of the +internal manifold then governs the low-energy physics in the remaining dimensions (four in +the previous cases). Typically the number of massless fields can be deduced from the Betty +numbers of the internal manifold and the masses of the so-called Kaluza–Klein modes are +2 + +determined by the eigenvalues of certain self-adjoint operators, such as the Laplacian for +instance. Thus deforming the geometry of the internal manifold modifies the parameters +of the low-energy effective field theory. +Regardless of a particular choice of internal geometry, the swampland distance con- +jectures [34] concern universal features that are expected to hold for any moduli space +arising from the low-energy limit of a consistent quantum gravity theory. It is in particu- +lar conjectured that as one moves towards an infinite distance limit in such a moduli space, +the effective field theory breaks down due to the appearance of an infinite tower of light +states. Therefore it is an interesting problem to study the decay of the low eigenvalues +of Laplacian on manifolds with special holonomy near singular limits. Such a problem +has been studied in [3] for quintic Calabi–Yau threefolds using numerical methods. In +M-theory compactified on G2-manifolds, the eigenvalues of the Laplacian acting on co- +closed q-forms for q = 0, 1, 2, 3 correspond the squared masses of Kaluza–Klein states. As +mentioned above, our analysis notably yields precise estimates for the decay rate and the +distribution of the low eigenvalues of the Laplacian on twisted connected sums. We hope +that this could be of interest to the study of the distance conjectures and shed some light +on the geometric origin of the towers of light states, even if in a very specific example. +Organisation of the paper. +In Section 2, we describe the general gluing problem that +we are interested in and the notations and conventions that will be used throughout the +paper. We also define the notions of adapted operators and their substitute kernel and +cokernel that are central to our analysis and state our main results, Theorems 2.6 and 2.8 +and Corollary 2.9. Section 3 is concerned with the analysis of translation-invariant PDEs +on cylinders and contains most of the technical ingredients underlying our results. Our +exposition is self-contained, and for this purpose we include a brief review of the standard +results that we need. Section 4 is dedicated to the analysis of the mapping properties of +adapted operators in the limit where the length of the neck joining the building blocks +tends to infinity. Under some assumptions we prove a general theorem on the invertibility +of such operators, but our method is more general and we also comment on how to adapt it +in different contexts. In the subsequent sections we apply our techniques to the study of the +low eigenvalues of the Laplacian (Section 5) and the twisted connected sum construction +of compact G2-manifolds (Section 6). +Acknowledgements. +I would like to thank Tristan Ozuch for helpful discussions which +clarified some points in the analysis presented in this paper. I am also grateful to my su- +pervisor Jason Lotay for his support and advice. My research is supported by a Clarendon +Scholarship. +2 +Setup and discussion of results +In this section we explain our setup and formulate the main results of this paper. The +gluing problem under consideration is described in §2.1, in which we introduce the building +blocks and the class of adapted operators that we are interested in. In §2.2 we motivate and +introduce the notions of substitute kernel and cokernel for adapted operators, following +ideas present for instance in [21, 23] or [26]. Our main results are discussed in §2.3 and +related to the swampland distance conjectures in (2.3.4). Last, we give an overview of our +strategy in (2.3.5). +3 + +2.1 +Model gluing problem +(2.1.1) +Before describing the type of gluing problems we are interested in, we need to +recall a few standard definitions, starting with the notion of Exponentially Asymptotically +Cylindrical (EAC) manifold. Let Z be an oriented non-compact manifold of dimension n +and X an oriented compact manifold of dimension n − 1. We say that Z is asymptotic to +the cylinder Y = R × X at infinity if there exist a compact K ⊂ Z and an orientation- +preserving diffeomorphism φ : (0, ∞) × X → Z\K. The compact manifold X is called the +cross-section of Z. It will often be useful to pick a positive function ρ : Z → R such that +ρ(φ(t, x)) = t for (t, x) ∈ [1, ∞) × X and ρ < 1 outside of φ([1, ∞) × X). Following the +terminology of [16], we will call such function a cylindrical coordinate function. +We say that a Riemannian metric g on Z is EAC of rate µ > 0 if we have, for all +integers l ≥ 0: +|∇l +Y (φ∗g − gY )|gY = O +� +e−µt� +(2.1) +as t → ∞, where gY = dt2 + gX is a cylindrical metric on Y = R × X, ∇Y the associated +Levi-Civita connection and | · |gY the associated norm on tensor bundles. +Given the above data, we may define a notion of adapted bundle as follows. Any vector +bundle E0 → X equipped with a metric h0 and a connection ∇0 can be extended to +a vector bundle E0 → Y with translation-invariant metric and connection (h0, ∇0) (see +Section 3 for more details). We call such bundles on Y = R × X translation-invariant +bundles. Let E → Z be a vector bundle on Z, endowed with a metric h and a connection +∇. We say that E is an adapted bundle on (Z, g) if there exist a translation-invariant +vector bundle (E0, h0, ∇0) and a bundle isomorphism Φ : E0|(0,×X) → E|Z\K covering φ, +such that for all integers l ≥ 0: +|∇l +0(Φ∗h − h0)|0 = O(e−µt), +and |∇l +0(Φ∗∇ − ∇0)|0 = O +� +e−µt� +(2.2) +as t → ∞. +Remark 2.1. This definition is valid for both real and complex vector bundles. As we +will make heavy use of the Fourier transform in Section 3, we usually assume that we are +dealing with complex vector bundles and that the bundle metric is hermitian. Our result +will also apply to real vector bundles by considering their complexification. +Last, we may also define the notion of adapted differential operator between adapted +bundles. Let E, F be adapted bundles on Z and P : C∞(E) → C∞(F) be a differential +operator of order k ≥ 1. If u is a smooth section of E0 defined over the half-cylinder +(0, ∞), let: +�Pu = Φ−1 +F PΦEu. +(2.3) +This defines a differential operator �P : C∞(E0) → C∞(F 0) over the cylinder (0, ∞) × X, +modelling the action of P on sections supported in Z\K. The operator �P can be written +in the form: +�P = +k +� +j=0 +Ak−j(t)∂j +t +(2.4) +where ∂t is the covariant derivative in the direction +∂ +∂t for the connection ∇0 on E, and +Ak−j(t) : C∞(E0) → C∞(F0) are differential operators depending smoothly on t. We say +that P is adapted (with exponential rate µ > 0) if there exists a translation-invariant +differential operator P0 : C∞(E0) → C∞(F0) of the form: +P0 = +k +� +j=0 +Ak−j∂j +t +(2.5) +4 + +such that for any smooth section u of E0 defined on (0, ∞) × X and for any l ≥ 0 and +0 ≤ j ≤ k we have: +|∇l +0(Ak−j(t)u − Ak−ju)|0 = O + +e−µt � +i≤l +|∇i +0u|0 + + +(2.6) +as t → ∞. That is, we essentially want the coefficients of �P − P0 and all their derivatives +to have exponential decay when t → ∞. The operator P0 is called the indicial operator of +P. Note that the formal adjoint of an adapted P is also adapted, and its indicial operator +is naturally P ∗ +0 . +Example 2.2. On an EAC manifold (Z, g) with cross section X, the bundles TZ, TZ⊗l +and ΛqT ∗Z are adapted, endowed with the Levi-Civita connection and the metric induced +by g. The operators d + d∗ and ∆ = dd∗ + d∗d are adapted. +(2.1.2) +We now describe the general gluing problem we are interested in. Let Z1 and +Z2 be two EAC manifolds and assume that the cross-section of Z2 is the same as the +cross-section X of Z1, but with opposite orientation. By definition there exists compacts +Ki ⊂ Zi and diffeomorphisms φi : (0, ∞) × Xi → Zi\Ki where X1 = X = X2, and we can +pick cylindrical coordinate functions ρi : Zi → R>0. For T ≥ 0 we can construct a compact +oriented manifold MT by gluing the domains {ρ1 ≤ T + 2} ⊂ Z1 and {ρ2 ≤ T + 2} ⊂ Z2 +along the annulus {T ≤ ρi ≤ T + 2} ≃ [−1, 1] × X with the identification: +φ1(T + 1 + t, x) ≃ φ2(T + 1 − t, x), +∀(t, x) ∈ [−1, 1] × X. +(2.7) +Define a smooth function ρT on MT by: +ρT ≡ +� +ρ1 − T − 1 +in {ρ1 ≤ T + 2} +T + 1 − ρ2 +in {ρ2 ≤ T + 2} . +(2.8) +This is well defined as ρ1 − T − 1 coincides with T + 1 − ρ2 under the identification +(2.7). Intuitively the function ρT parametrises the neck of MT . In particular the domain +{|ρ| ≤ T} is diffeomorphic to the finite cylinder [−T, T] × X. Our goal is to study the +mapping properties of elliptic operators defined on MT as T becomes very large, and relate +it to the corresponding properties of operators on Zi. +To that end, suppose that the manifolds Zi are endowed we EAC metrics gi both +asymptotic to a translation-invariant metric gY = dt2 + g0 on Y = R × X. It will also be +useful to fix a cutoff function χ : R → [0, 1] such that χ ≡ 0 on (−∞, − 1 +2] and χ ≡ 1 on +[1 +2, +∞). If T ∈ R we denote χT (t) = χ(t − T). Then for T large enough +gi,T = (1 − χT (ρi))gi + χT (ρi)gY +(2.9) +is a Riemannian metric on Zi which coincides with gi on {ρi ≤ T − 1 +2} and with gY on +{ρi ≥ T + 1 +2}. Moreover, the difference gi − gi,T and all their derivatives are uniformly +bounded by O(e−µT ). Note that here we implicitly identify Zi\Ki with the half cylinder +(0, ∞) × Xi to make notations lighter. We can patch g1,T and g2,T to form a Riemannian +metric gT on MT , defining: +gT ≡ +� +g1,T +if ρT ≤ 0 +g2,T +if ρT ≥ 0 +. +(2.10) +Similarly, if we have adapted bundles (Ei, hi, ∇i) on Zi such that their asymptotic models +are both isomorphic to the same translation-invariant vector bundle (E0, h0, ∇0) on R×X, +we can use the same cutoff procedure to patch them up on MT and form a vector bundle +ET equipped with a metric hT and a connection ∇T. +5 + +(2.1.3) +Consider matching adapted bundles Ei, Fi on Zi (i = 1, 2) asymptotic to the +same translation-invariant bundles E0, F 0, and adapted elliptic operators Pi : C∞(Ei) → +C∞(Fi) of order k. Denote Pi,0(x, ∂x, ∂t) : C∞(E0) → C∞(F 0) the indicial operator of Pi, +where we use ∂x as a loose notation for the derivatives along the cross-section X. In order +to patch up these operators we need to assume the following compatibility condition [26]: +P2,0(x, ∂x, ∂t) = P1,0(x, ∂x, −∂t). +(2.11) +Assuming that it is satisfied define: +Pi,T = (1 − χT (ρi))Pi + χT (ρi)Pi,0 +(2.12) +which coincides with Pi for ρi ≤ T − 1 +2 and with Pi,0 for ρi ≥ T + 1 +2. For large enough +T, the operators Pi,T are elliptic, and moreover the coefficients of Pi − Pi,T and all their +derivatives are uniformly bounded by O(e−µT ). +Patching P1,T and P2,T as above, we +obtain a family of operators PT : C∞(ET ) → C∞(FT ) which are elliptic for large enough +T. Elliptic regularity on compact manifolds implies that the action of PT on Sobolev +or Hölder spaces of sections induce Fredholm maps. Our goal is to construct Fredholm +inverses for these maps, with a good control on their norm as T → ∞. +Remark 2.3. In our applications, we will also be interested in a variation of the above +gluing construction where the EAC manifolds Z1 and Z2 are glued along a non-trivial +isometry γ : X → X. This corresponds to replacing the identification (2.7) by: +φ1(T + 1 + t, x) ≃ φ2(T + 1 − t, γ(x)), +∀(t, x) ∈ [−1, 1] × X. +(2.13) +All the matching conditions have to be changed accordingly, but otherwise everything +we will do applies without modification up to a mere change of notations. This will be +important in particular for the construction of compact G2-manifolds by twisted connected +sums that we study in §6.2. +(2.1.4) +Before explaining our results in more details in the next part, let us make our +conventions for Sobolev and Hölder norms explicit. For p ≥ 1 and l ≥ 0 an integer, the +W l,p-norm of a section u ∈ C∞(ET ) can be defined as: +∥u∥W l,p = +� +j≤l +∥∇j +T u∥Lp +(2.14) +where the fibrewise norm of ∇j +Tu is computed with the metrics hT , gT and we integrate +over the volume form of gT . The Sobolev space W l,p(ET ) is defined as the completion of +C∞(ET ) for the W l,p-norm. +For the definition of Hölder norms, note that the injectivity radius of (MT , gT ) is +uniformly bounded below by some r > 0. The Cl,α norm of a smooth section u ∈ C∞(ET ) +is defined as +∥u∥Cl,α = +� +j≤l +∥∇j +T u∥C0 + sup +x∈MT +sup +dT (x,x′) 1, α ∈ (0, 1) and l ≥ 0. Then the maps +PT : W k+l,p(ET ) → W l,p(FT ), +PT : Ck+l,α(ET ) → Cl,α(FT ) +are uniformly bounded as T → ∞. Moreover there exist constants C, C′ > 0 such that for +T large enough and for any u ∈ W k+l,p(ET ) and v ∈ Ck+l,α(ET ) we have: +∥u∥W k+l,p ≤ C (∥PT u∥W l,p + ∥u∥Lp) , +∥v∥Ck+l,α ≤ C′(∥Pv∥Cl,α + ∥v∥C0). +Remark 2.5. In the same way, it follows from the embedding theorems of [28] for weighted +spaces on EAC manifolds that the constants in the Sobolev embeddings W r,p ֒→ W s,q and +W r,p ֒→ Cl,α, in the range where such embeddings exist and are continuous, are in fact +uniformly bounded as T → ∞. This will be useful in Sections 5 and 6. +By elliptic regularity, the above maps can be inverted in the L2-orthogonal complements +of the kernels of PT and of its adjoint. However, since the dimensions of these spaces is not +continuous (only the index is), they do depend on the precise way we take cutoffs to define +our gluing, and so will the norm of the inverse. In order to make general statements, we +would like to define notions of substitute kernel and cokernel in the fashion of [21] (see +also [23, §18]), determined by the gluing data and in the complement of which we have +a good control on the norm of the inverse of PT . Under the restricting assumption that +the indicial operators P0 = P1,0 is invertible, these have been defined and studied in [26]. +However in many cases of interest this is not satisfied, as the indicial operator may have +real roots. In the next section we will define notions of substitute kernel and cokernel +adapted to the case where there are real roots. +2.2 +Substitute kernel and cokernel +(2.2.1) +In order to define the substitute kernel and cokernel, a good understanding of +the mapping properties of translation-invariant operators on cylinders and of adapted +operators on EAC manifolds is crucial. For completeness, the results that we need are +gathered in §3.1 and §4.1. Original references are [1] and [29], as well as [30] for a detailed +discussion of analysis in the broader context of b-calculus. +In the situation described in the previous part, let P0 = P1,0 : C∞(E0) → C∞(F 0) +denote the indicial operator of P1, acting on the cylinder Y = R × X. Points in Y will be +denoted y = (t, x). A particularly important role in our analysis is played by solutions of +the homogeneous equation P0u = 0 of the form: +u(t, x) = +m +� +j=1 +eiλjtuj(t, x) +(2.16) +where λ1, ..., λm are real numbers and the sections uj are polynomial in the variable t. +Such solutions are called polyhomogeneous solutions of rate 0, and we denote by E the +vector space they span. As a matter of general theory, this is a finite-dimensional space, +and in particular there are only finitely many values λ ∈ R such that the homogeneous +equation P0u = 0 admits a non-trivial solution of the form u(t, x) = eiλtuλ(t, x), where +uλ is polynomial in t. These values are called the real roots of P0 (see Section 3 for a +detailed discussion). Let us point out here that the real roots of the formal adjoint P ∗ +0 are +the same as the real roots of P0, and denote E ∗ the space of polyhomogeneous solutions +7 + +of rate 0 of P ∗ +0 u = 0. It follows from the compatibility condition (2.11) that the space of +polyhomogeneous solutions of P2,0u = 0 of rate 0 is {u(−t, x), u ∈ E }, and similarly for +the adjoint operators. +Let us denote by Ki the space of solutions of Piu = 0 with sub-exponential growth and +Ki,0 the subspace of decaying solutions, for i = 1, 2. By Lockhart–McOwen theory ([29], +see also §4.1 for more details), Ki has finite dimension and each of its elements is asymp- +totic to a polyhomogeneous solution of Pi,0u = 0 with rate 0, up to terms exponentially +decaying terms. More precisely, for any u ∈ K1, there exists a polyhomogeneous solutions +u ∈ E such that for any l ≥ 0: +|∇l +0(u(t, x) − u0(t, x))|0 = O +� +e−δt� +(2.17) +when t → ∞, for any sufficiently small δ > 0. Here we implicitly identify u over Z1\K1 +with a section of E0 over (0, ∞) × X. Therefore we can define a linear map κ1 : K1 → E , +such that for any u ∈ K1, the difference u − κ1[u] and all its derivatives have exponential +decay at infinity. Taking care of the fact that we need to change the sign of the variable +t, we can similarly define a map κ2 : K2 → E such that |u(x, t) − κ2[u](x, −t)|0 = O(e−δt) +as t → ∞ for all u ∈ K2, with the usual identifications. For i = 1, 2, the kernel of the map +κi in Ki is Ki,0. Considering adjoint operators, we may also define K ∗ +i , K ∗ +i,0 and linear +maps κ∗ +i : K ∗ +i → E ∗. +(2.2.2) +With these notations in hand, let u1 ∈ K1 and u2 ∈ K2 and fix T > 0. We say +that u1 and u2 are matching at T if the following condition is satisfied: +κ1[u1](t + T + 1, x) = κ2[u2](t − T − 1, x), +∀(t, x) ∈ R × X. +(2.18) +Given a matching pair of solutions (u1, u2), we can define a section of the bundle ET → MT +as follows: +uT = (1 − χT+1(ρ1))u1 + (1 − χT+1(ρ2))u2 +(2.19) +where we consider χ(ρi)ui as a section of ET supported in the domain {ρi ≤ T +2} ⊂ MT . +In particular, uT ≡ u1 in the domain {ρ1 ≤ T + 1 +2}, uT ≡ u2 in {ρ2 ≤ T + 1 +2} and it +smoothly interpolates between the two in {|ρT | ≤ 1 +2}. It is easy to see that PT uT ≡ 0 +outside of the annulus {|ρT | ≤ 3 +2}. Moreover the matching condition (2.18) insures that +for any l ≥ 0 there exists a constant C > 0 independent from T such that: +∥PT uT ∥Cl ≤ Ce−δT (∥u1∥ + ∥u2∥) +(2.20) +where we fix arbitrary norms on K1 and K2 and small enough δ > 0. In that sense, uT +is an approximate solution of PT u = 0. The substitute kernel KT of PT is defined as the +finite-dimensional subspace of C∞(ET ) of approximate solutions constructed in this way +from a matching pair (u1, u2) of solutions of Piu = 0. Similarly we define the substitute +cokernel K ∗ +T as the substitute kernel of P ∗ +T . +(2.2.3) +For these notions of substitute kernel and cokernel to be convenient to handle +in practice, it is simpler to assume that for T large enough the dimensions of KT and K ∗ +T +do not depend on T. This is automatically satisfied if the indicial operator P0 has only +one root. Indeed, in this case we can express the matching condition at T as a finite- +dimensional linear system depending polynomially on T by choosing convenient basis for +im κ1, im κ2 and E . The minors of this system are polynomial in T, and therefore are +either identically 0 or do not vanish for T large enough. Hence the rank of the system do +8 + +not depend on T for T large enough, and neither does the dimension of its kernel. We can +argue similarly for the substitute cokernel K ∗ +T . +For more general operators, the matching condition will be expressed as a finite- +dimensional linear system with coefficients depending analytically on T, and although +the non-trivial minors of the system only have isolated zeroes we cannot insure that there +are only finitely many of them. This is the situation that we want to avoid. Therefore we +will assume that P has only one real root to state our main result about the existence of +a Fredholm inverse for P in the complement of the substitute kernel and cokernel. This +is sufficient for our applications in Sections 5 and 6. However the method we develop is +more general, and for most of the paper we need not to take any restricting assumptions +on the roots of the indicial operator. +Assuming that the spaces KT and K ∗ +T have constant dimension for T large enough, +it follows from (2.20) that for any Hölder norm Ck,α and any small enough δ > 0, there +exists a constant C > 0 such that for T large enough: +∥PT u∥Ck,α ≤ Ce−δT ∥u∥C0,α. +(2.21) +Similar bounds hold for Sobolev norms and for P ∗ +T . Hence there is no hope to have a +control on the norm of the inverse of PT better than O(eδT ) if we do not work in the +complement of KT and K ∗ +T . +2.3 +Results and strategy +(2.3.1) +We keep the setup and notations of the previous part. Our main result is the +following theorem, which says that under the limiting assumption described above we can +find a Fredholm inverse for PT in the complement of the substitute kernel and cokernel, +with norm bounded by a power of T: +Theorem 2.6. Let p > 1 and l ≥ 0 be an integer, and assume that P0 has only one real +root. Then there exists constants C, C′ > 0 and an exponent β ≥ 0 such that for T large +enough the following holds. +For any f ∈ W l,p(FT ), there exist a unique u ∈ W k+l,p(ET ) orthogonal to KT and a +unique w ∈ K ∗ +T such that f = PT u + w. Moreover, u satisfies the bound: +∥u∥W k+l,p ≤ C∥f∥W l,p + C′T β∥f∥Lp. +Remark 2.7. The statement also holds with Hölder norms. If l ≥ 0 is an integer, α ∈ (0, 1) +and P0 has only one real root, then there exist constants C, C′ > 0 and an exponent β ≥ 0 +such that the following is true. +For any f ∈ Cl,α(FT ), there exist a unique u ∈ Ck+l,α(ET ) orthogonal to KT and a +unique w ∈ K ∗ +T such that f = PT u + w. Moreover, u satisfies the bound: +∥u∥Ck+l,α ≤ C∥f∥Cl,α + C′T β∥f∥C0,α. +(2.3.2) +In some cases, we are also able to determine the optimal exponent β. This is +the case for instance of the Laplacian operator ∆T of the metric gT acting on differential +forms, where β = 2 is optimal and the substitute kernel gives a good approximation of +harmonic forms, measured in the Cl,α or W l,p sense for any choice of l ≥ 0, p > 1 and +α ∈ (0, 1) (Corollary 5.3). +If we consider the L2-range, our estimates imply that whenever 0 is a root of the +Laplacian acting on q-forms, which is equivalent to saying that the Betty numbers of the +9 + +cross-section satisfy bq−1(X) + bq(X) > 0, the lowest eigenvalue of ∆T satisfies a bound of +the type λ1(T) ≥ +C +T 2 for some constant C > 0. It is an interesting problem to determine +the distribution the eigenvalues that have the fastest decay rate. Let us define the densities +of low eigenvalues as: +Λq,inf(s) = lim inf +T→∞ # +� +eigenvalues of ∆T acting on q-forms in +� +0, π2sT −2�� +Λq,sup(s) = lim sup +T→∞ +# +� +eigenvalues of ∆T acting on q-forms in +� +0, π2sT −2�� +where we count eigenvalues with multiplicity. The normalisation by T −2 comes from the +fact that we expect the lowest eigenvalues to be decaying at precisely this rate, while the +factor π2 is just a matter of convenience. We can similarly define the densities Λ∗ +q,inf(s) and +Λ∗ +q,sup(s) of low eigenvalues of the Laplacian acting on co-exact q-forms. We are interested +in understanding the asymptotic behavior of these densities as s → ∞. In §5.2 we prove +the following: +Theorem 2.8. As s → ∞, the densities of low eigenvalues satisfy: +Λq,inf(s) ∼ 2(bq−1(X) + bq(X))√s ∼ Λq,sup(s). +If moreover bq(X) ̸= 0 then: +Λ∗ +q,inf(s) ∼ 2bq(X)√s ∼ Λ∗ +q,sup(s). +This proposition essentially says that the lowest eigenvalues of ∆T are distributed as +the low eigenvalues of the Laplacian acting on the product S1 +2T × X, where the first factor +is a circle of length 2T. We shall moreover see in (5.2.4) that stronger statements relating +the lower spectrum of ∆T to the spectrum of the Laplacian on S1 +2T × X would not hold. +The reason is that the interaction between the building blocks of the construction seem +to create a shift in the spectrum. We shall also briefly discuss generalisations of the above +statement to other self-adjoint operators. +(2.3.3) +As a second application of Theorem 2.6 and Corollary 5.3, we shall give improved +estimates for compact G2-manifolds constructed by twisted connected sum in Section 6, +thereby addressing some issues in the literature about the analysis involved. The building +blocks of the twisted connected sum construction, are a pair of EAC G2 manifolds (Z1, ϕ1) +and (Z2, ϕ2) with rate µ > 0, satisfying a certain matching condition (see [25, 11]). These +building blocks can be glued along a non-trivial isometry γ of the cross-section X (see +Remark 2.3) to form a 1-parameter family ϕT of compact manifolds MT equipped with +closed G2-structures with exponentially decaying torsion, measured in any Ck-norm (see +§6.2 for more details). For large enough T, ϕT can be deformed to a nearby torsion-free +G2-structure �ϕT with ∥ �ϕT − ϕT ∥C0 = O(e−δT ) [25, Theorem 5.34]. Using Theorem 2.6 we +are able to prove that stronger estimates hold: +Corollary 2.9. Let k ≥ 0 be an integer, and δ > 0 smaller than µ and the square roots of +the smallest non-zero eigenvalues of the Laplacian acting on 2-and 3-forms on X. Then +there exist a constant C > 0 such that for T large enough the following estimates hold: +∥ �ϕT − ϕT ∥Ck ≤ Ce−δT . +This result gives a much stronger control on ∥ �ϕT − ϕT ∥ than the C0,α bound that +was the best previously known. It has been used in various places in the literature but +10 + +to our knowledge there was still no clear proof of this fact. These stronger estimates are +especially important to analysis on twisted connected sums, as they also imply a control +on g�ϕT − gϕT and all its derivatives in O(e−δT ), and similarly for the operators that +are naturally associated with it. Hence the study of the mapping properties of elliptic +operators on twisted connected sums is amenable to the techniques described in Sections +4 and 5, and in particular Theorem 2.6 and Theorem 2.8 also apply in this context. +(2.3.4) +For twisted connected sums where the cross-section is simply the product of a +2-torus with a K3 surface, its Betty numbers are b0(X) = 1, b1(X) = 2, b2(X) = 23 and +b3(X) = 44. Thus we may deduce for instance that the lower spectrum of the Laplacian of +�ϕT acting on co-closed 3-forms is made of an infinite number of eigenvalues which decay +at rate T −2. +Moreover for large enough T the number of eigenvalues below +π2s +T 2 is of +order 88√s. From the physical perspective, when we consider M-theory compactified on +(MT , �ϕT ) this translates into an infinite tower of scalar fields becoming light as T → ∞, in +line with the swampland distance conjectures. By the above we obtain not only the rate of +decay but also the density of this tower of states. Considering 0, 1 and 2-forms we would +similarly obtain other towers of Kaluza–Klein states becoming light. The geometric origin +of these towers of light states is found to be the topology of the cross-section, through the +singularities of the resolvent of the Laplacian acting in the cylindrical neck region. +(2.3.5) +Let us finish this section with an overview of our strategy. We prove Theorem +2.6 by an explicit construction scheme, similar to constructions by Kapouleas of minimal +surfaces in Euclidean space [21, 22] by which we were inspired. The idea is to use cutoffs +separate the analysis in three different domains: the neck region, which is close to a finite +cylinder [−T, T] × X, and two regions isometric to the domains {ρi ≤ T + 1} ⊂ Zi. One +challenge is that when the indicial operator P0 acting on the cylinder has real roots, it +is not invertible nor even Fredholm in the Sobolev or Hölder range that we would like to +consider. However, this failure is due to the asymptotic behaviour of solutions, and we +only need to work on a compact region of the cylinder. To deal with this issue, let us +denote W l,p +c +the subspace of W l,p constituted of sections with compact essential support, +and Cl,α +c +the space of compactly supported sections of class Cl,α. In Section 3, we prove +the following theorem, which is the main analytical ingredient of our construction: +Theorem 2.10. Let P0 : C∞(E0) → C∞(F 0) be an elliptic translation-invariant operator +of order k acting on the cylinder R × X. Let d be the maximal order of a real root of P0. +For p > 1, there exists a map Q0 : Lp +c(F 0) → W k,p +loc (E0) such that for any f ∈ Lp(F 0) +with compact essential support, P0Q0f = f. Moreover there exists a constant C > 0 such +that for any T ≥ 1 and any f ∈ Lp +c(F 0) with essential support contained in [−T, T] × X: +∥Q0f∥W k,p([−T,T]×X) ≤ CT d∥f∥Lp. +Similarly for α ∈ (0, 1), P0 admits a right inverse Q0 : C0,α +c +(F 0) → Ck,α +loc (E0) satisfying +the same estimates as above. +Remark 2.11. The existence of the map Q0 can be deduced from standard results as [29] +or [30] for instance. However, the explicit expression that we give for Q0 will be important +for our purpose, since a precise understanding of the asymptotic behavior of Q0f will play +a key role in the construction of Section 4. +With this theorem in hand, we can try to build approximate solutions to the equation +PT u = f by first taking a cutoff of f in the neck region, and considering an equation of +11 + +the type P0u0 = f0 with f0 supported in [−T, T] × X. This equation can be solved using +the above theorem, and taking another cutoff of the solution the equation PT u = f can +be replaced with an equation of the form PT u′ = f ′, where f ′ is appropriately small in +the neck region and can be written as a sum f1 + f2, where each fi is a section defined +in the domain {ρi ≤ T + 1} ⊂ Zi and satisfies good decay properties. This allows us to +use weighted analysis to study the equations Piui = fi in a range where the operators Pi +satisfy the Fredholm property. Similar ideas can be found for instance in [35]. +Unfortunately there are generally obstructions to solving Piui = fi in weighted spaces, +and the main difficulty is to understand how these obstructions interact. Using a pairing +defined in §3.2, we can keep track of the obstructions and express their vanishing (up +to an exponentially decaying error term) as a finite-dimensional linear system, which we +call the characteristic system of our gluing problem. The unknown of this system is an +element v ∈ E , which represents our degrees of freedom in solving the equation P0u = f0, +and the coefficients of the system are linearly determined by f. In §4.3, we prove that in +full generality the characteristic system admits a solution if and only if f is orthogonal to +the substitute cokernel. With the extra assumption that P has only one root, this allows +us to build an approximate solution of the equation PT u = f, and when T is large enough +we can prove Theorem 2.6 using an iteration process. However, our method could apply +more generally, as long as one can ensure that the characteristic system admits a solution +with reasonable bounds. +3 +Translation-invariant elliptic PDEs on cylinders +Throughout this section, we fix an oriented compact manifold X and denote Y = R×X. If +E → X is a vector bundle, we denote E → Y the pull-back of E by the projection Y → X +on the second factor. Given any connection ∇ on E, we can endow E by the pull-back +connection ∇. Parallel transport along the vector field +∂ +∂t naturally defines a translation +operator on E, and does not depend on the choice of connection on E. A section of E is +translation-invariant if and only if it is the pull-back of a section of E. +We assume that Y is equipped with a cylindrical metric gY = dt2+gX and endow E and +F with translation-invariant connections and metrics. We define Sobolev and Hölder norms +on Y with respect to this data. In §3.1, we introduce some background about analysis on +cylinders. In §3.2 we study the action of a general elliptic translation-invariant operator P +on polyhomogeneous sections and define a pairing between the spaces of polyhomogeneous +solutions of Pu = 0 and of P ∗v = 0. Last we prove Theorem 2.10, by a method which +gives an implicit expression for solutions of Pu = f. Using the aforementioned pairing, +we are able to analyse precisely the asymptotic behaviour of these solutions, which will be +a key ingredient of our construction in Section 4. +3.1 +Analysis on cylinders by separation of variables +(3.1.1) +On the cylinder Y = R × Y , we have natural isomorphisms identifying Lp(E) +with Lp(R, Lp(E)) for any p ≥ 1, which follow from Fubini’s theorem. Therefore we can +think of sections of translation-invariant vector bundles over Y as maps from R to an +appropriate Banach space of sections over X. If α ∈ (0, 1), the there is unfortunately no +natural isomorphism between C0,α(E) and C0,α(R, C0,α(E)); however there is a sequence +of continuous maps +C0,α(E) −→ C0(R, C0,α(E)) −→ C0(E) +(3.1) +12 + +which is enough for our purpose. The main tools that we will need for the analysis of PDEs +on a cylinder R×X are the convolution and the Fourier transform along the variable t ∈ R. +For completeness, we will recall their definitions and basic properties in this setup. +(3.1.2) +To define the convolution product, we consider three complex Banach spaces +and a continuous bilinear product A ⊗ B → C. Just as for scalar-valued map, Fubini’s +theorem implies that the expression +f ∗ g(t) = +� +R +f(t − τ)g(τ)dτ +(3.2) +induces a well-defined continuous map L1(R, A) ⊗ L1(R, B) → L1(R, C). +Considering +bounded functions it also defines a continuous map L1(R, A) ⊗ L∞(R, B) → L∞(R, C). +In order to define the convolution product for more general Lp-spaces, we nee to use a +duality argument. Let us assume that C is reflexive, so that for r > 1 the topological dual +of Lr(R, C) is Ls(R, C′) where s is the conjugate exponent of r and C′ the topological dual +of C (see Remark 3.2 below). In particular Lr(R, C) itself is reflexive. Let (p, q) ̸= (1, 1) +such that 1 +p + 1 +q = 1 + 1 +r. Then if f ∈ Lp(R, A), g ∈ Lq(R, B) and h ∈ Lr(R, C′), the +Young’s inequality for scalar-valued functions implies that the map +(t, τ) ∈ R2 −→ h(τ)(f(t − τ)g(τ)) ∈ C +(3.3) +is in L1, with norm bounded by some universal constant times ∥f∥Lp∥g∥Lq∥h∥Ls. Using +Ls(R, C′) ≃ Lr(R, C) this means that the convolution f ∗g defines an element of Lr(R, C) +and that Young’s inequality holds in this setting. To summarise: +Lemma 3.1. Let A, B, C be Banach spaces with a continuous bilinear map A ⊗ B → C, +and 1 ≤ p, q, r ≤ ∞ such that 1 + 1 +r = 1 +p + 1 +q. If (p, q) ̸= (1, 1) or (q, r) ̸= (∞, ∞) assume +moreover that C is reflexive. Then the convolution product defines a continuous bilinear +map +Lp(R, A) ⊗ Lq(R, B) → Lr(R, C). +Remark 3.2. When C is a reflexive Banach space, it satisfies the Radon–Nikodym property +[12, III. Corollary 13]. By [12, IV. Theorem 1], for any finite interval I ⊂ R the topological +dual of Lr(I, C) for r > 1 is Ls(I, C′) where C′ is s is the conjugate exponent of s. This +can be extended to I = R by considering the sequence of intervals (−n, n) for n → ∞. +In our applications we do not really need this result for general reflexive spaces as we +will only use it for C = Lr(X, F), for which we already know that the dual of Lr(R, C) is +Ls(R, C′) ≃ Ls(Y, F). +(3.1.3) +Next, we turn to the definition of the Fourier transform for Banach space valued +maps. For integrable functions this is not a problem as the usual expression +ˆf(λ) = +� +R +e−iλtf(t)dt +(3.4) +is well-defined for all λ ∈ R. Thus if f ∈ L1(R, A), its Fourier transform ˆf belongs to +C0(R, A) and this defines a continuous map. If moreover t �→ tlf(t) is L1 for some integer +l ≥ 1 then ˆf ∈ Cl(R, A), and if f is compactly supported the expression (3.4) exists for +all λ ∈ C and this defines an analytic map ˆf : C → A. +The notion of Schwartz function with values in A can be defined without problem, +and the Schwartz space S (R, A) is invariant under Fourier transform. The inverse of the +13 + +Fourier transform takes the usual expression on S (R, A). By duality, one can therefore +define the Fourier transform on the dual S ′(R, A) of S (R, A). For 1 < p ≤ ∞, the Fourier +transform on Lp(R, A) is defined through the embedding of Lp(R, A) → S ′(R, A′) coming +from integration. +Last, we want to define products of functions with values in different Banach spaces, +and prove that as for scalar functions the Fourier transform intertwines convolution and +product. Consider Banach spaces A, B, C and a continuous map A ⊗ B → C. This map +naturally induces a continuous map A ⊗ C′ → B′. Let f : R → A be a smooth function +such that f and all of its derivatives have at most polynomial growth, and let g ∈ Lp(R, B) +considered as an element of S ′(R, B′). Then the measurable function fg : R → C can +be seen as an element of S ′(R, C′) defined by fg(u) = g(fu) for all u ∈ S (R, C′). Here +fu ∈ (R, B′) is defined with the product A ⊗ C′ → B′ mentioned above. +With these definitions in hand, the Fourier transform and the convolution of Banach- +space-valued maps satisfy the same identity as scalar-valued maps. As in the latter case +this can be proven by working first with test functions and then extending it to distribu- +tions by duality. We state the result in the form that we will use as a lemma: +Lemma 3.3. Let A, B, C be Banach spaces with a continuous bilinear map A ⊗ B → C, +and 1 ≤ p, q, r ≤ ∞ such that 1 + 1 +r = 1 +p + 1 +q. If (p, q) ̸= (1, 1) or (q, r) ̸= (∞, ∞) assume +moreover that C is reflexive. Let f ∈ Lp(R, A) and g ∈ Lq(R, B) such that ˆf is smooth +with at most polynomial growth. Then � +f ∗ g = ˆfˆg. +(3.1.4) +After these preliminaries, let us now turn to the study of PDEs on cylinders. +Let E and F be translation-invariant vector bundles over Y = R × X, equipped with +translation-invariant metrics and connections. We will denote y = (t, x) the points in Y . +Moreover denote ∂t the covariant derivative along +∂ +∂t . +A differential operator P : C∞(E) → C∞(F) of order k is translation-invariant if it +takes the form: +P(x, ∂x, Dt) = +k +� +l=0 +Ak−l(x, ∂x)Dl +t +(3.5) +where Ak−l(x, ∂x) are differential operators C∞(E) → C∞(F), and we use ∂x as a loose +notation for the derivatives along X. Equations of the type: +P(x, ∂x, Dt)u(t, x) = f(t, x) +(3.6) +have been widely studied in various contexts. If P has order k, it is a standard fact that +it induces continuous maps +P : W k+l,p(E) → W l,p(F) and P : Ck+l,α(E) → Cl,α(F). +(3.7) +From now on we assume that P is elliptic. In that case, the usual interior elliptic estimates +combined with invariance under translations yield the following a priori estimates: +Proposition 3.4. Let P : C∞(E) → C∞(F) be a translation-invariant elliptic operator +of order k. +Let p > 1 and l ≥ 0 be an integer. Then there exists C > 0 such that if f ∈ W l,p(F) +and u ∈ L1 +loc(E) is a weak solution of Pu = f, then u ∈ W k+l,p(E) with the bound: +∥u∥W k+l,p ≤ C(∥f∥W l,p + ∥u∥Lp). +Let α ∈ (0, 1) and l ≥ 0 be an integer. Then there exists C > 0 such that if f ∈ Cl,α(F) +and u ∈ L1 +loc(E) is a weak solution of Pu = f, then u ∈ Ck+l,α(E) with the bound: +∥u∥Ck+l,α ≤ C(∥f∥Cl,α + ∥u∥C0). +14 + +We will also need the following variation of this proposition for finite cylinders, which +is proved in the same way: +Proposition 3.5. Let P : C∞(E) → C∞(F) be a translation-invariant elliptic operator +of order k. +Let p > 1 and l ≥ 0 be an integer. Then there exists C > 0 such that for any T ≥ 1 +the following holds. If f is a section of F over the finite cylinder (−T − 1, T + 1) × X and +u ∈ L1 +loc((−T −1, T +1)×X, E) is a solution of Pu = f, then u ∈ W k+l,p((−T, T)×X, E) +with the bound: +∥u∥W k+l,p((−T,T)×X) ≤ C +� +∥f∥W l,p((−T−1,T+1)×X) + ∥u∥Lp((−T−1,T+1)×X) +� +. +The same statement holds with Hölder norms. +(3.1.5) +In the remaining of this part, we will be concerned with equations of the type: +P(x, ∂x, Dt)u(t, x) = f(t, x) +(3.8) +where P is a translation-invariant elliptic operator. It is usually studied by taking its +Fourier transform in the variable t, which takes the form: +P(x, ∂x, λ)ˆu(x, λ) = ˆf(x, λ). +(3.9) +For any fixed λ ∈ C, the operator P(x, ∂x, λ) : C∞(E) → C∞(F) is an elliptic operator +of order k, and hence defines Fredholm maps on Sobolev and Hölder spaces of sections +over X. By the results of [1] these maps are analytic in the variable λ, and there exists a +discrete set CP ⊂ C such that the homogeneous equation +P(x, ∂x, λ)ˆu(x, λ) = 0 +(3.10) +has a non-trivial solution if and only if λ ∈ CP. +Moreover, CP is finite on any strip +{δ1 < Im λ < δ2} of C. The elements of CP are called the roots of P. The discrete set +DP = {Im λ, λ ∈ CP } is called the set of indicial roots of P. +Example 3.6. Consider the translation-invariant bundle ΛCT ∗Y of complex-valued differ- +ential forms. It splits as a direct sum: +ΛCT ∗Y = ΛCT ∗X ⊕ dt ∧ ΛCT ∗X +(3.11) +where ΛCT ∗X is the pull-back of the bundle of differential forms on X. The operators dY +and d∗ +Y take the form: +� +dY (α + dt ∧ β) = dXα + dt ∧ (∂tα − dXβ) +d∗ +Y (α + dt ∧ β) = d∗ +Xα − ∂tβ − dt ∧ d∗ +Xβ +(3.12) +Thus if we define J ∈ End(ΛCT ∗Y ) by Jη = dt ∧ η − ι ∂ +∂t η where ι denotes the interior +product, we can write the Fourier transform of the operator dY + d∗ +Y as +(dY + d∗ +Y )(λ)η = (dX + d∗ +X)α − dt ∧ (dX + d∗ +X)β + iλJη. +(3.13) +with η = α+dt∧β. The Laplacian ∆Y = dY d∗ +Y +d∗ +Y dY can be written as ∆Y = −∂2 +t +∆X, +so that its Fourier transform is ∆X + λ2. For both operators, the roots are exactly the +values ±i√λn, where λn ≥ 0 are the eigenvalues of the Laplacian ∆X. In particular the +only real root is λ0 = 0, and the corresponding translation-invariant solutions are of the +form α + dt ∧ β, where α and β are harmonic forms on X. +15 + +(3.1.6) +For λ ∈ C denote P(λ) as a short-hand for P(x, ∂x, λ). It can be seen as a +Fredholm map between Sobolev spaces, analytic in the variable λ. This implies that P(λ) +is invertible for λ /∈ CP [1, 2]. Its inverse R(λ) is called the resolvent of P(λ); for any +m ≤ k + l it can be considered as a bounded operator form W l,p(F) to W m,p(E) (which is +compact when m < k + l). We will denote ∥R(λ)∥l,m the operator norm of the resolvent +seen as a map W l,p(F) → W m,p(E). By the results of [1] the resolvent is meromorphic in +λ ∈ C, with poles exactly at the roots of P. That is, around any λ0 ∈ CP we can write: +R(λ) = R−d(λ0) +(λ − λ0)d + · · · + R−1(λ0) +λ − λ0 ++ +∞ +� +n=0 +An(λ − λ0)n +(3.14) +where R−l are bounded operators W l,p(F) → W m,p(E) and the series has positive radius +of convergence. The largest positive integer d such that R−d(λ0) ̸= 0 is called the order +of λ0. The notions of root, pole and order do not depend on the Sobolev or Hölder spaces +we choose to work in. +The following bounds on the resolvent R(λ) and its derivative R′(λ) = dR +dλ (λ) are crucial +for our purpose, and follow from the more general [1, Theorem 5.4]: +Theorem 3.7. Let p > 1, l ≥ 0 and P be a translation-invariant elliptic operator. Then +the following holds: +(i) The resolvent R(λ) has no poles in a double sector {arg(±λ) ≤ δ, +|λ| ≥ N} and in +this domain there is a constant C > 0 such that: +k +� +j=0 +���λk−jR(λ) +��� +l,l+j ≤ C. +(ii) Furthermore, as |λ| → ∞ along the real axis: +k +� +j=0 +���λk−jR′(λ) +��� +l,l+j = O +�1 +λ +� +. +This statement also holds if we take Hölder norms and denote ∥ · ∥l,m the operator +norm of the resolvent seen as a map from Cl,α to Cm,α for some α ∈ (0, 1). +(3.1.7) +The last result that we want to mention here is the following well-known propo- +sition (see [24] for an original reference), which can be seen as a particular case of Theorem +2.10. When P has not roots along the real axis the following holds. +Proposition 3.8. Let p > 1, l ≥ 0, α ∈ (0, 1) and assume that P has no real roots. Then +the maps W k+l,p(E) → W l,p(F) and Ck+l,α(E) → Cl,α(F) induced by P admit bounded +inverses. +As this will serve as a model argument in our proof of Theorem 2.10 in §3.3, we shall +give a proof of this proposition. +Proof. Consider the resolvent R(λ) as a compact operator Lp(F) → Lp(E) if we work in +the Sobolev range, and C0,α(F) → C0,α(E) in the Hölder range. Since P has no real roots +this defines a smooth function R → Hom(Lp(F), Lp(E)) (or R → Hom(C0,α(F), C0,α(E))), +and by Theorem 3.7 it is bounded. As R(λ)P(λ) is a constant map, we can differentiate +this relation and use the fact that P(λ) is polynomial in the variable λ, to prove by an +16 + +easy induction that all the derivatives of R(λ) have at most polynomial growth. Hence we +can define its inverse Fourier transform Q. +Let us show that in fact Q is L1. Indeed, by Theorem 3.7 the resolvent satisfies +���λkR(λ) +��� + +���λk+1R′(λ) +��� ≤ C +(3.15) +and as k ≥ 1 is follows that R(λ) and R′(λ) are L2 as functions of λ. Hence Q(t) and tQ(t) +are L2 as functions of t, and since t → +1 +1+|t| is L2 we find that Q is L1 by Cauchy-Schwarz +inequality. +Let f ∈ Lp +l (F). +Then as P has no real roots the unique solution of the Fourier- +transformed equation P(λ)ˆu(λ) = ˆf(λ) is ˆu(λ) = R(λ) ˆf(λ). As Lp(F) is isomorphic to +Lp(R, Lp(F)) and Lp(F) is reflexive we may apply Lemma 3.3, and find that u = Q ∗ f +is an Lp solution of Pu = f. +Moreover there is a constant C independent of f such +that ∥u∥Lp ≤ C∥Q∥L1∥f∥Lp by the generalised Young’s inequality of Lemma 3.1. Using +the a priori estimates of Proposition 3.4, it follows that u ∈ W k+l,p(E) and ∥u∥W k+l,p ≤ +C′∥f∥W l,p for some universal constant C′ > 0. +When f ∈ Cl,α(F) one needs to be slightly more careful in the argumentation. We can +see f as an element of C0,α(E), which continuously embeds into L∞(R, C0,α(F)). By the +same argumentation as above u = Q ∗ f is the unique weak solution of Pu = f, but this +time Lemma 3.3 only implies that u ∈ L∞(R, C0,α(E)). Nevertheless, this is enough to +prove that u ∈ L1 +loc(E), so that by Proposition 3.4 u ∈ Ck+l,α(E) and its norm is controlled +by ∥f∥Cl,α and ∥u∥C0. As ∥u∥C0 is itself controlled by its norm in L∞(R, C0,α(E)) since +C0,α(E) continuously embeds into C0(E), we obtain an estimate ∥u∥Ck+l,α ≤ C′′∥f∥Cl,α +as required. +When P has real roots the statement of Proposition 3.8 does not hold anymore and the +maps induced by P in Sobolev or Hölder spaces are not even Fredholm. They still have +finite-dimensional kernel but their cokernel have infinite dimension. In order to understand +the mapping properties of P in more details, the difficulty is to make sense of the inverse +Fourier transform of the singular part of the resolvent. +3.2 +Polyhomogeneous sections +(3.2.1) +In this part, we prove that the action of P on polyhomogeneous sections admits +a right inverse and introduce a pairing which will play an important role in Section 4. A +section of E → Y is called exponential is it is of the form u(x, t) = eiλtp(x, t) where λ ∈ C +is called the rate of u and p is polynomial in the variable t. A polyhomogeneous section +is a finite sum of exponential sections (for possibly different values of λ). +To understand the action of P on polyhomogeneous sections, we fix λ0 ∈ C and define: +Pλ0(x, ∂x, Dt) = e−iλ0tP(x, ∂x, Dt)eiλ0t +(3.16) +which is a translation-invariant operator on Y . Explicitly Pλ0 has for expression: +Pλ0(Dt) = +� +n≥0 +1 +n! +∂nP +∂λn (λ0)Dn +t . +(3.17) +We consider Pλ0 as an operator mapping the space W k,p(E)[t] into Lp(F)[t], that is we +consider the action on sections of E → Y that are polynomial in the variable t and have +W k,p coefficients. Our goal is to show that Pλ0 admits a right inverse Qλ0. +17 + +Consider the resolvent R(λ) as an operator Lp(F) → W k,p(E). If λ0 is a root of P, it +is a pole of R and we denote d(λ0) its degree. By convention we set d(λ0) = 0 if λ0 is not +a root of P. In general we may expand R(λ) near λ0 as: +R(λ) = R−d(λ0)(λ0) +(λ − λ0)d(λ0) + · · · + R−1(λ0) +λ − λ0 ++ R0(λ0) + +� +m≥1 +Rm(λ − λ0)m. +(3.18) +where for m ≥ −d(λ0), Rm : Lp(X, F) → W k,p(X, E) are bounded operators. +The +relations R(λ)P(λ) = IdW k,p(E) and P(λ)R(λ) = IdLp(F ) that hold away form the roots of +P imply: +� +m+n=0 +1 +n!Rm(λ0)∂nP +∂λn (λ0) = IdW k,p(E), +� +m+n=0 +1 +n! +∂nP +∂λn (λ0)Rm(λ0) = IdLp(F ) +(3.19) +and for any non-zero l ∈ Z: +� +m+n=l +1 +n!Rm(λ0)∂nP +∂λn (λ0) = 0 = +� +m+n=l +1 +n! +∂nP +∂λn (λ0)Rm(λ0) +(3.20) +Example 3.9. One can easily see from Example 3.6 that λ0 = 0 is a root of order 1 of the +operator dY +d∗ +Y and of order 2 for the operator ∆Y . The singular parts of their resolvent +can be computed with the above relations. For the operator dY + d∗ +Y , relations (3.20) for +l = −1 imply that Rd+d∗ +−1 +(0) vanishes on the orthogonal space to harmonic forms and maps +into the space of harmonic forms. Relations (3.19) imply that: +iRd+d∗ +−1 +(0)Jη = η = iJRd+d∗ +−1 +(0)η +(3.21) +for any translation-invariant harmonic form on Y . As J2 = −1 we obtain: +Rd+d∗ +−1 +(0) = iJ ◦ ph = iph ◦ J +(3.22) +where ph is the L2-orthogonal projection onto the space of harmonic forms. As the Lapla- +cian ∆Y is the square of the operator dY + d∗ this implies that: +R∆ +−2(0) = (Rd+d∗ +−1 +(0))2 = (iJ)2p2 +h = ph. +(3.23) +On the other hand as the Fourier transform of ∆Y is an analytic function of the variable +λ2 it is easy to see that R∆ +−1(0) = 0. +(3.2.2) +With these notations in hand, let D−1 +t +be the endomorphism of Lp(F)[t] mapping +(it)j +j! v to (it)j+1 +(j+1)! v for any v ∈ Lp(F). It is easily seen that this is a right inverse of Dt. Let +us define the operator Qλ0 : Lp(F)[t] → W k,p(E)[t] by: +Qλ0(Dt, D−1 +t ) = +� +m≥−d(λ0) +Rm(λ0)Dm +t . +(3.24) +It maps polynomials of order m to polynomials of order at most m + d(λ0). Moreover +relations (3.19) and (3.20) imply the following: +Lemma 3.10. The map Qλ0 : Lp(F)[t] → W k,p(E)[t] is a right inverse of Pλ0. +Of course there is nothing special about the choice of Sobolev range W k,p, and the ar- +gument above carries out verbatim to show that the map Ck,α(E)[t] → C0,α(F)[t] induced +by the operator Pλ0 admits a right inverse. +18 + +(3.2.3) +Let us now turn our attention to the kernel of Pλ0. It is non-trivial if and only +if λ0 is a root of P, which amounts to saying that the homogeneous equation Pλ0u = 0 +admits a non-trivial translation-invariant solution. Moreover, the kernel of Pλ0 acting on +polynomial sections in the variable t is always finite-dimensional, the degree of its elements +is bounded above by the order of the root λ0, minus one [1]. In particular if λ0 has order +one the only polynomial solutions of Pλ0u = 0 are translation-invariant. +For any root λ0 of P, let us denote Eλ0 the (finite-dimensional) space of exponential +solutions of Pu = 0 of rate λ0, and E ∗ +λ0 the space of exponential solutions of P ∗v = 0 of +rate λ0 (note that P(λ)∗ = P ∗(λ) for any λ ∈ C). As we are mainly interested in the real +roots of P, we denote λ1, ..., λm the real roots and: +E = +m +� +j=1 +Eλj, +E ∗ = +m +� +j=1 +E ∗ +λj. +(3.25) +We shall now define a pairing E × E ∗ → C and derive its basic properties. Let χ : +R → R be a smooth function such that χ ≡ 0 in a neighbourhood of −∞ and χ ≡ 1 in a +neighbourhood of +∞. We define a sesquilinear pairing (·, ·) : E ×E ∗ → C by the integral: +(u, v) = +� +R +⟨P(Dt) [χ(t)u(t)] , v(t)⟩ dt. +(3.26) +where here ⟨·, ·⟩ is the L2-product on the compact manifold X. This is well-defined as +P(Dt) [χ(t)u(t))] is compactly supported for any u ∈ E . Further it does not depend on +the choice of function χ. Indeed if ˜χ is another smooth function that satisfies the same +assumptions, define χτ = (1 − τ)χ + τ ˜χ for τ ∈ [0, 1]. +As ∂χτ +∂τ (t)u(t) is a compactly +supported section of E, we can integrate by parts to obtain: +d +dτ +� +R +⟨P(Dt) [χτ(t)u(t)] , v(t)⟩ dt = +� +R +� +P(Dt) +�∂χτ +∂τ (t)u(t) +� +, v(t) +� +dt = 0 +(3.27) +as P ∗(DT )v(t) = 0. Therefore the pairing does not depend on the choice of χ. +An important consequence of this observation is that Eλi is orthogonal to E ∗ +λj for the +pairing (·, ·) unless i = j. Indeed, let u ∈ Eλi and v ∈ E ∗ +λj, and replace χ(t) by χ(t − τ) in +the definition of the pairing, for τ ∈ R. Then we can compute by a change of variables: +� +R +⟨P(Dt) [χ(t − τ)u(t)] , v(t)⟩ dt = ei(λi−λj)τ + +(u, v) + +� +l≥1 +al(u, v)τ l + + +(3.28) +where the coefficients al(u, v) are independent of τ, and only finitely many of them are +non-zero. As this has to be equal to (u, v) for all τ ∈ R, this implies al = 0 for l ≥ 1 and +(u, v) = 0 when i ̸= j. +(3.2.4) +The key property of the pairing (·, ·) is the following: +Lemma 3.11. The pairing (·, ·) is non-degenerate. +Proof. By the above remarks it suffices to show that the restriction of (·, ·) to Eλj × E ∗ +λj +is non-degenerate. Consider first v ∈ ker P ∗(λj), so that ˜v(t, x) = eiλjtv(x) is an element +of E ∗ +λj. +Considering v as an element of L2(F)[t], we define u(t, x) = Qλjv. +This is a +polynomial of order at most d(λj) in the variable t, and it satisfies: +Pλj(Dt)u(t) = v. +(3.29) +19 + +Differentiating this expression in the t variable it follows that: +Pλj(Dt)[Dtu(t)] = 0 +(3.30) +so that ˜u(t, x) = eiλjtDtu(t, x) is in Eλj. +Let us now pick a function χ as above and +compute: +� +R +� +P(Dt) [χ(t)˜u(t)] , eiλjtv +� +dt = +� +R +� +Pλj(Dt) [χ(t)Dtu(t)] , v +� +dt +(3.31) += 1 +i +� +d +dt +� +Pλj(Dt) [χ(t)u(t)] , v +� +dt − 1 +i +� � +Pλj(Dt) +�χ′(t)u(t) +� , v +� +dt +(3.32) += 1 +i ⟨v, v⟩ − 0 +(3.33) +which holds because P(Dt)[χ(t)u(t)] ≡ v as t goes to +∞ and P(Dt)[χ(t)u(t)] = 0 as t +goes to −∞. Thus we have (˜u, ˜v) = −i∥v∥2 +L2 which is non-zero when v ̸= 0. +In general, let v(t, x) be an element of E ∗ +λj of degree m. Then eiλjtDm +t e−iλjtv(t, x) is a +non-zero element of E ∗ +λj of degree zero. Thus by the above argument there exists u(t, x) +in Eλj such that (u, eiλjtDm +t e−iλjtv) ̸= 0. Moreover one can easily check that: +(u, eiλjtDm +t e−iλjtv) = (eiλjtDm +t e−iλjtu, v) +(3.34) +and eiλjtDm +t e−iλjtu ∈ Eλj. Hence the pairing (·, ·) is non-degenerate. +Example 3.12. The space of translation-invariant solutions of the operator dY + d∗ +Y acting +on ΛCT ∗Y is: +Ed+d∗ = E ∗ +d+d∗ = {α + dt ∧ β, α, β ∈ C∞(ΛCT ∗X), ∆Xα = ∆Xβ = 0} +(3.35) +If α + dt ∧ β, α′ + dt ∧ β′ ∈ E we can compute their pairing: +(α + dt ∧ β, α′ + dt ∧ β′) = +� +⟨(dY + d∗ +Y )(χ(τ)α + dt ∧ β), α′ + dt ∧ β′⟩dτ +(3.36) += +� +χ′(τ)⟨dt ∧ α − β, α′ + dt ∧ β′⟩dτ +(3.37) += ⟨α, β′⟩ − ⟨β, α′⟩ +(3.38) +which is clearly non-degenerate. +For the Laplacian ∆Y acting on q-forms, the spaces Eq and E ∗ +q are both isomorphic to +the space q-forms that can be written as η0 +tη1, where ηi = αi +dt∧βi with αi ∈ Ωq +C(X) +and βi ∈ Ωq−1 +C +(X) harmonic. In the same way one can easily derive: +(η0 + tη1, η′ +0 + tη′ +1) = ⟨α0, α′ +1⟩ + ⟨β0, β′ +1⟩ − ⟨α1, α′ +0⟩ − ⟨β1, β′ +0⟩. +(3.39) +3.3 +Existence of solutions +(3.3.1) +In this part we prove Theorem 2.10, following similar lines to the proof of Propo- +sition 3.8. We fix p > 1 and work with Sobolev spaces. We will indicate at the end how +to modify the proof to treat the case of Hölder norms. +Let us consider a translation-invariant elliptic differential operator P : C∞(E) → +C∞(F) of order k and denote λ1, . . . , λm its real roots. For 1 ≥ j ≥ m, let d(λj) be the +20 + +order of the root λj. Considering the resolvent as a family of (compact) operators from +Lp(F) to Lp(E), we have a decomposition of the form: +R(λ) = Rr(λ) + +m +� +j=1 +d(λj) +� +l=1 +R−l(λj) +(λ − λj)l +(3.40) +where the regular part of the resolvent Rr(λ) is an analytic function from a neighborhood +of the real line in C to Hom(Lp(F), Lp(E)). +Will will denote the second term of the +left-hand-side of equation (3.40) by Rs(λ); this is the singular part of the resolvent. +Consider a section f ∈ Lp +c(F) as an Lp-function from R to Lp(F), which has compact +essential support. We want to find a solution of equation (3.8) through the study of the +Fourier transformed equation (3.9). As the resolvent has poles we need to make sense of +the expression ˆu(λ) = R(λ) ˆf(λ), or rather of its inverse Fourier transform. +As in the proof of Proposition 3.8, the identity P(λ)R(λ) = IdLp(F ) and the estimates +of Theorem 3.7 imply that the resolvent and all its derivatives have at most polynomial +growth at infinity. +Since this also true of the singular part of the resolvent, which is +bounded at infinity as well as all of its derivatives, then the same holds for the regular +part of the resolvent. On the other hand, from Theorem 3.7 we have a bound: +���λkR(λ) +��� + +���λk+1R′(λ) +��� = O(1) +(3.41) +as |λ| → ∞. Further this bound clearly also holds for the singular part of the resolvent. +Therefore there exists a constant C > 0 such that for all λ ∈ R we have: +���λkRr(λ) +��� + +���λk+1R′ +r(λ) +��� ≤ C +(3.42) +This implies again that Rr(λ) and R′ +r(λ) are L2, which means that the inverse Fourier +transform Qr : R → Hom(Lp(F), Lp(E)) of Rr is L1, by the same argumentat as in the +proof of Proposition 3.8. Thus we may define ur = Qr ∗f, which is an element of Lp(E) by +Lemma 3.1. Further it satisfies ˆur(λ) = Rr(λ) ˆf(λ) by Lemma 3.3. Let us point out here +that as f has compact support, its Fourier transform is analytic in the variable λ ∈ C. As +Rr(λ) has no poles in a complex strip of the form {| Im λ| < δ} for some δ > 0, ˆur(λ) is +analytic in for λ varying in a neighborhood of the real line in C. +We now deal with the singular part of the resolvent. Our main problem is that we +cannot make sense of the inverse Fourier transform of Rs(λ) ˆf(λ). +Nevertheless, it is +natural to define the following: +us(t) = +m +� +j=1 +d(λj) +� +l=1 +ileiλjt +� t +−∞ +(t − τ)l−1 +(l − 1)! e−iλjτR−l(λj)f(τ)dτ. +(3.43) +Note that the integrals are well-defined because f has compact essential support. If we +denote Hl,λ(t) = eiλjt iltl−1 +(l−1)!H(t), then a more compact way to write the definition of us is: +us = +m +� +j=1 +d(λj) +� +l=1 +Hl,λj ∗ (R−l(λj)f). +(3.44) +In general us is not Lp, but it is still in Lp +loc, as can be deduced by taking appropriate +cutoffs of the functions Hl,λj and using Lemma 3.1. We will shortly provide more precise +estimates, but we first want to prove that u = ur + us satisfies Pu = f. +21 + +In order to do this, let us first compute P(Dt)ur(t), considering Dt as a weak derivative +wherever appropriate. +Taking the Fourier transform, we may compute P(λ)ur(λ) for +λ ∈ R\{λ1, ..., λm} as follows. As P(λ)R(λ) = IdLp(F ), we have: +P(λ)ˆur(λ) = ˆf(λ) − P(λ)Rs(λ) ˆf(λ). +(3.45) +For each root λj, we can expand P(λ) in Taylor series around λj to compute: +P(λ) +d(λj) +� +l=1 +R−l(λj) +(λ − λj)l = +� +n +d(λj) +� +l=1 +(λ − λj)n−l +n! +∂nP +∂λn (λj)R−l(λj). +(3.46) +By relations (3.20), the expansion of the sum in powers of λ − λj is polynomial, that is +the sum of the terms containing negative powers of λ − λj vanishes. This yields: +P(λ)ˆur(λ) = ˆf(λ) − +m +� +j=1 +� +l≥1, n−l≥0 +(λ − λj)n−l +n! +∂nP +∂λn (λj)R−l(λj) +(3.47) +which holds for λ ̸= λj. As both sides of the equality are analytic in the variable λ, this is +in fact true for all λ contained in a neighborhood of the real line in C. We can therefore +take the inverse Fourier transform to obtain: +P(Dt)ur(t) = f(t) − +m +� +j=1 +� +l≥1, n−l≥0 +eiλjtDn−l +t +� 1 +n! +∂nP +∂λn (λj)R−l(λj)e−iλjtf(t) +� +. +(3.48) +Next, we compute P(Dt)us. In order to do this, let us remark that for n < l we have +the following identity: +eiλtDn +t e−iλtHl,λ = (Dt − λ)nHl,λ = Hl−n,λ +(3.49) +and for n = l, we have: +eiλtDl +te−iλtHl,λ = δ(0) +(3.50) +where δ here is a Dirac mass centered at t = 0. Writing P(Dt) = eiλjtPλj(Dt)e−iλjt where +Pλj(Dt) is the operator defined in §3.2, we have +P +d(λj) +� +l=1 +Hl,λj ∗ (R−l(λj)f) = +� +n≥0 +d(λj) +� +l=1 +eiλjtDn +t e−iλjtHl,λj ∗ +� 1 +n! +∂nP +∂λn (λj)R−l(λj)f +� +. (3.51) +If we spilt the sum in two, we see that by the identity (3.50) we see that the sum of the +terms for which n ≥ l is equal to: +� +n≥l +d(λj) +� +l=1 +eiλjtDn−l +t +� 1 +n! +∂nP +∂λn (λj)R−l(λj)e−iλjtf(t) +� +(3.52) +On the other hand, the sum of the terms for which n < l can be computed using (3.49), +and in fact this sum vanishes by (3.20): +� +n 0 such that: +∥us∥Lp([−T−1,T+1]×X) ≤ CT d∥f∥Lp. +(3.56) +Combining this bound with the interior estimates of Proposition 3.5, this finishes the proof +of Theorem 2.10, for the case of Sobolev norms. +(3.3.2) +Let us indicate how to modify this construction to deal with Hölder norms. Let +f be a compactly supported section of class C0,α. As in the proof of Proposition 3.8, +the fact the regular part of the resolvent implies that ur which we define as above, is a +well-defined element at least in L∞(R, C0,α(E)), with L∞-norm controlled by the Lp-norm +of f. In fact ur is even continuous as f has compact support. +Similarly we can define us by (3.43) which gives an element of L∞ +loc(R, C0,α(E)), and +as before it is even continuous since f has compact support. The proof that u = ur + us +is a solution of Pu = f carries out verbatim. Elliptic regularity implies that u ∈ Ck,α +loc (E). +The proof of the estimate +∥u∥Ck,α([−T,T]×X) ≤ CT d∥f∥C0,α +(3.57) +for T ≥ 1 and some uniform constant C > 0 follows the same line as above. +(3.3.3) +In the remaining of this part shall comment on the asymptotic behavior of the +solutions we constructed above. For the sake of definiteness we work with Sobolev norms +but it will be clear that this can be adapted for Hölder norms. Let f ∈ Lp +c(F) and let +u, us and ur be defined as above. Assume that the essential support of f is contained in +[−T, T] × X. Then in fact outside of this compact set we have: +Pus = 0 = Pur. +(3.58) +As Pu = 0 in this domain it suffices to show that Pus = 0. This is a consequence of (3.52) +which yields: +Pus = +� +n≥l +d(λj) +� +l=1 +eiλjtDn−l +t +� 1 +n! +∂nP +∂λn (λj)R−l(λj)e−iλjtf(t) +� +. +(3.59) +For |t| > T the expression under brackets identically vanishes, which proves our claim. +An important consequence of this fact is that ur has exponential decay as |t| → ∞ +in the sense that the W k,p-norm of eδρur is finite for some δ > 0, where ρ denotes an +23 + +arbitrary smooth function on Y equal to |t| when |t| ≥ 1. This can be seen as a particular +case of Lockhart–McOwen theory (see §4.1). On the other hand it is easy to see from its +definition that us vanishes identically in the domain {t < −T}, and, more interestingly, +ur is equal to the restriction of a polyhomogeneous solution of Pu = 0 in the domain +{t > T}. Indeed for t > T (3.43) reads: +us(t) = +m +� +j=1 +d(λj) +� +l=1 +eiλjt +� T +−T +(t − τ)l−1 +(l − 1)! e−iλjτR−l(λj)f(τ)dτ +(3.60) +which is manifestly polyhomogeneous. Let us denote it uf ∈ E . We may use the pairing +(·, ·) introduced in §3.2 to implicitly define uf by duality: +Lemma 3.13. With the above notations, (uf, v) = ⟨f, v⟩ for any v ∈ E ∗. +Proof. Let χ be a smooth function such that χ ≡ 1 in (−∞, 0] and χ ≡ 0 in [1, ∞), and +let χτ(t) = χ(t − τ). Then for any τ > T we have the equality ⟨χτPu, v⟩ = ⟨f, v⟩. On the +other hand, let us prove that ⟨Pχτu, v⟩ = 0 for any τ ∈ R. If τ, τ ′ ∈ R, we have: +⟨Pχτu, v⟩ − ⟨Pχτ ′u, v⟩ = ⟨P(χτ − χτ ′)u, v⟩ +(3.61) += ⟨(χτ − χτ ′)u, P ∗v⟩ = 0 +(3.62) +where the integration by parts is justified because χτ −χτ ′ has compact support. Therefore +the value of ⟨Pχτu, v⟩ = 0 does not depend on τ. Hence we may send τ to −∞, and as +u(t) has exponential decay as t → −∞ we obtain ⟨Pχτu, v⟩ = 0. +It follows that ⟨f, v⟩ = − limτ→∞⟨[P, χτ]u, v⟩. Given the exponential decay of ur(t) +and its k first derivatives (in the sense explained above) as t → ∞, this yields: +⟨f, v⟩ = − lim +τ→∞⟨[P, χτ]us, v⟩ = lim +τ→∞⟨[P, 1 − χτ]us, v⟩ = (uf, v) +(3.63) +as claimed. +Example 3.14. Consider the case of the Laplacian ∆Y acting on q-forms. The singular +part of the resolvent is λ−2ph where ph is the projection on the space of harmonic forms. +Thus if η is a q-form on Y supported in [−T, T] × X and we denote ξ(τ) the L2-projection +of ητ onto the space of harmonic forms, and write ξ(τ) = α(τ) + dt ∧ β(τ) we have by +definition: +uη(t) = i2 +� T +−T +(t − τ)ξ(τ)dτ +(3.64) += +� T +−T +τα(τ) + dt ∧ τβ(τ)dτ − t +� T +−T +α(τ) + dt ∧ β(τ)dτ. +(3.65) +Thus it follows that for any v(t) = α0 + dt ∧ β0 + t(α1 + dt ∧ β1) ∈ Eq we can use the +formula of Example 3.12 to derive: +(uη, v) = +� T +−T +τ(⟨α(τ), α1⟩ + ⟨β(τ), β1⟩) + ⟨α(τ), α0⟩ + ⟨β(τ), β0⟩dτ = ⟨η, v⟩. +(3.66) +4 +Construction of solutions +In this section we explain the main construction of this paper. In §4.1 we review the +mapping properties of adapted operators on EAC manifolds. +In §4.2 we explain our +24 + +method to construct approximate solutions of the equation PT u = f and show that it can +be reduced to a finite-dimensional linear system, which we call the characteristic system. +Last, in §4.3 we prove that this system admits a solution if and only if f is orthogonal to +the substitute cokernel defined in §2.2. Under the restricting assumption that the indicial +operator of the gluing problem has only one real root, this enables us to prove Theorem +2.6. We also discuss more general conditions that would ensure that this result holds true. +4.1 +Analysis on EAC manifolds +(4.1.1) +The mapping properties of adapted operators on EAC manifolds have been stud- +ied by Lockhart–McOwen in [29], and we will give a brief review of their theory. The right +functional spaces to consider in this situation are weighted Sobolev and Hölder spaces. Let +(Z, g) be an EAC manifold asymptotic to a cylinder Y = R × X at infinity, and (E, h, ∇) +be an adapted bundle, and pick a cylindrical coordinate function ρ : Z → R>0 as defined +in §2.1. If u is a smooth compactly supported section of E, we can define its W l,p +ν -norm +(p ≥ 1, k ≥ 0, ν ∈ R) as follows: +∥u∥W l,p +ν += +l +� +j=0 +∥eνρ∇ju∥Lp. +(4.1) +Note that for ν = 0 this is just the usual W l,p norm. The weighted Sobolev space W l,p +ν (E) +can be defined as the completion of C∞ +c (E) with respect to the W l,p +ν -norm. +For Hölder norms, pick r > 0 smaller than the injectivity radius of (Z, g), and define: +∥u∥Cl,α +ν += +� +j≥k +∥eνρ∇ju∥C0 + sup +z∈Z +sup +d(z,z′) 1, l ≥ 0 and α ∈ (0, 1). Then the following holds. +(i) The maps W k+l,p +ν +(E) → W l,p +ν (F) and Ck+l,α +ν +(E) → Cl,α +ν (F) induced by P are Fred- +holm if and only if ν /∈ DP0. +In that case, the image of P is the L2-orthogonal +complement of ker P ∗ ∩ C∞ +−ν(F). +25 + +(ii) If ν′ < ν are not indicial roots of P, then the index change is given by +indν′(P) − indν(P) = +� +ν′ 1} with the cylinder (1, ∞) × X. +From now on, let us assume that 0 is an indicial root of P0, and let: +σ = min{µ, +min +ν∈DP0\{0} |ν|} +(4.4) +Take any δ ∈ (0, σ). Recall that we defined K as the kernel of P acting on sections with +sub-exponential growth, and K0 the kernel of P acting on decaying sections. In particular +K is the kernel of P acting on W k,p +δ +(E) and K0 the kernel of the action of P on W k,p +−δ (E). +In §2.2 we defined a map κ : K → F such that any element v ∈ K is asymptotic to κ(v). +In particular, K0 is the kernel of κ. Similarly we defined K ∗, K ∗ +0 and κ∗. Let us point +out that the index change formula in Theorem 4.1 implies the following equality: +dim im κ + dim im κ∗ = dim F. +(4.5) +(4.1.3) +We want to study equations of the type Pu = f when f has exponential decay, +say f ∈ Lp +δ. By Theorem 4.1, the obstructions to solve this equation for u ∈ W k,p +δ +lie in +K ∗, whereas the obstructions to solve it in W k,p +−δ lie in K ∗ +0 . Here, we will use the pairing +defined in §3.2 to give a precise description of the form of the obstructions and of the +asymptotic behaviour of solutions in W k,p +−δ . +Let v ∈ K ∗ be asymptotic to κ∗(v) = v0 ∈ E ∗ and consider u ∈ C∞(E) asymptotic to +u0 ∈ E , such that u − u0 and all their derivatives are exponentially decaying as ρ → ∞. +Then the L2 product ⟨Pu, v⟩ is well-defined as Pu decays exponentially. It turns out that +its value only depends on the asymptotic data. More precisely we claim that: +Lemma 4.2. With the above notations, ⟨Pu, v⟩ = (u0, v0). +Proof. Let χ : R → R be a smooth function such that χ ≡ 0 in (−∞, 0] and χ ≡ 1 in +[1, ∞), and let χτ(t) = χ(t − τ) for τ ∈ R. Then for any τ ≥ 1 we have: +⟨Pu, v⟩ = ⟨Pχτ(ρ)u, v⟩ + ⟨P(1 − χτ(ρ))u, v⟩ +(4.6) += ⟨Pχτ(ρ)u, v⟩ + ⟨(1 − χτ(ρ))u, P ∗v⟩ +(4.7) += ⟨Pχτ(ρ)u, v⟩ +(4.8) +since P ∗v = 0. Thus ⟨Pu, v⟩ = limτ→∞⟨Pχτ(ρ)u, v⟩. As u−u0, v −v0 and the coefficients +of P − P0 decay exponentially as ρ → ∞, as well as all derivatives, this implies: +⟨Pu, v⟩ = lim +τ→∞⟨P0χτu0, v0⟩ = (u0, v0) +(4.9) +since ⟨P0χτu0, v0⟩ = (u0, v0) for all τ. +As a consequence of the previous lemma applied to u ∈ K and v ∈ K ∗, im κ and +im κ∗ are orthogonal for the pairing (·, ·). Together with equality (4.5), this implies that +im κ is exactly the orthogonal space of im κ∗. +26 + +Let us denote K ∗ ++ the subspace of K orthogonal to K ∗ +0 for the L2-product, so that +κ∗ induces an isomorphism between K ∗ ++ and im κ∗. Moreover choose an arbitrary com- +plement F0 of im κ in E . Denote m = dim im κ∗. Pick smooth sections h1, . . . , hm which +are asymptotic to a basis of F0 at infinity, with the difference and all their derivatives +asymptotically decaying, and denote F ⊂ C∞(E) the vector space they span. By Lemma +4.2 we may choose a basis g1, . . . , gm of K ∗ ++ such that ⟨Phi, gj = δij⟩ for all 1 ≤ i, j ≤ m. +Let f ∈ Lp +δ be a section of F, and w be the L2-projection of f onto K ∗ +0 . Let us write: +f = f ′ + +m +� +j=1 +⟨f, gj⟩Phj + w. +(4.10) +where f ′ ∈ Lp +δ is by construction orthogonal to the obstruction space K ∗. As |⟨f, g⟩| ≤ +C∥f∥Lp +δ∥g∥Lq +−δ for any g ∈ K ∗, where q is the conjugate exponent of p and C > 0 is some +constant, we have ∥f ′∥Lp +δ ≤ C′∥f∥Lp +δ for some universal constant C′ > 0. By Theorem 4.1 +there exists u′ such that Pu′ = f ′ and ∥u′∥W k,p +δ +≤ C′′∥f ′∥Lp +δ. This proves the following: +Proposition 4.3. Let any 0 < δ < σ. Then there exists a constant C > 0 such that +the following holds. Let f ∈ Lp +δ(F), and denote w its L2 projection onto K0. Then there +exists a section u′ ∈ W k,p +δ +(E) with the estimate ∥u′∥W k,p +δ +≤ C∥f∥Lp +δ and such that +P + +u′ + +m +� +j=1 +⟨f, gj⟩hj + + = f − w. +4.2 +Characteristic system +(4.2.1) +In the same setup as Section 2, we now consider the gluing problem of two +adapted operators P1, P2 on EAC manifolds Z1, Z2. For the present discussion we need not +to put any restrictions on the real roots of the indicial operator P0. By definition, there is a +compact K1 ⊂ Z1 and an orientation-preserving diffeomorphism φ1 : (0, ∞)×X → Z1\K1 +and we picked a positive cylindrical coordinate function ρ1 on Z1 such that ρ1(φ1(t, x)) = t +when t ≥ 1 and ρ1 < 1 everywhere else in Z1. As in Section 2 we fix a cutoff function +χ : R → [0, 1] such that χ ≡ 0 in (−∞, − 1 +2] and χ ≡ 1 in [1 +2, ∞). For τ ∈ R we keep our +usual notation χτ(t) = χ(t−τ). It will be convenient to introduce a family ζ1 +τ : Z1 → [0, 1] +of cutoffs functions for the construction. For τ ≥ 0 define: +ζ1 +τ (z) = +� +0 +if z ∈ K1 +χ(t − τ − 1 +2) +if z = φ1(t, x), (t, x) ∈ (0, ∞) × X . +(4.11) +We similarly define a family of cutoff functions on Z2, denoted ζ2 +τ for τ ≥ 0. Consider now +the compact manifold MT obtained by gluing the compact domains {ρ1 ≤ T + 2} ⊂ Z1 +and {ρ2 ≤ T + 2} ⊂ Z2 along the annuli {T ≤ ρi ≤ T + 2}. We can define a family of +cutoffs functions ζτ : MT → [0, 1] for 0 ≤ τ ≤ T by patching together ζ1 +τ with ζ2 +τ in the +following way: +ζτ ≡ +� +ζ1 +τ +if ρT ≤ 0 +ζ2 +τ +if ρT ≥ 0 . +(4.12) +Note that the support of ζτ is diffeomorphic to the finite cylinder [−T −1+τ, T +1−τ]×X. +We now turn to the gluing problem of two adapted operators Pi : C∞(Ei) → C∞(Fi) +as described in §2.1. Our goal is to prove that we can construct solutions of the equation +27 + +PT u = f for f taking values in a complement of the substitute cokernel introduced in +§2.2. We shall do this by considering three regions in MT : the neck region {|ρT | ≤ T} for +which our main tool is Theorem 2.10, and the two regions compact regions {ρT ≤ 0} and +{ρT ≥ 0}, for which we will use weighted analysis on Z1 and Z2 in the form of Proposition +4.3. The crucial point of the construction is to understand the interactions between these +three regions, especially in terms of the obstructions to solving the equation Piu = f on +each Zi. Using the pairing (·, ·) defined in §3.2 in order to implicitly keep track of these +obstructions, we will be able to essentially reduce this problem to a finite-dimensional +linear system. +(4.2.2) +From now on we fix p > 1 and work with Sobolev spaces W l,p. Adapting the +construction for Hölder spaces will be straightforward. Let f ∈ Lp(FT ) be an arbitrary +section. The we may identify the section ζ1f with a section of the translation-invariant +vector bundle F 0 over the cylinder Y = R×X, which we denote f0. Moreover, the essential +support of f0 is contained in the finite cylinder [−T, T] × X. Note that the Sobolev norm +of sections supported in the neck region of MT and the Sobolev norm in the finite cylinder +[−T, T] × X are equivalent. In particular we have an estimate +∥f0∥Lp ≤ C∥f∥Lp +(4.13) +By Theorem 2.10, the operator P0 admits a right inverse Q0 : Lp +c(F 0) → W k,p +loc (E0). Thus +we can define u0 = Q0f0, which satisfies P0u0 = f0. Using cutoff function ζ0 to identify +ζ0u0 with a section of ET → MT one has: +f − PT ζ0u0 = f − [PT , ζ0]u0 − ζPT u0 +(4.14) += (1 − ζ1)f − [PT , ζ0]u0 − ζ0(PT − P0)u0. +(4.15) +Note that the section (1−ζ1)f −[PT , ζ0]u0 is supported in the compact region {|ρT | ≥ T}. +Moreover the operator ζ0(PT −P0) vanishes in the region {|ρT | ≤ 1 +2} so that we may write: +f − PT ζ0u0 = f1 + f2 +(4.16) +where f1 = χ(ρT )(f − PT ζ0u0) can be identified with a section of F1 supported in {ρ1 ≤ +T + 1} ⊂ Z1, and f2 = (1 − χ(ρT ))(f − PT ζ0u0) can be identified with a section of F2 +over {ρ2 ≤ T + 1} ⊂ Z2. Both sections are Lp, and in fact as the coefficients of Pi − P0 +and all their derivatives have exponential decay as ρi → ∞, the Lp-norm of f controls the +Lp +ǫ-norms of f1 and f2, for any 0 < δ < σ. More precisely, the following estimates hold: +Lemma 4.4. Let d be the maximal order of the real roots of P0. Then with the above +notations, for i = 1, 2 there exists a constant C > 0 such that +∥fi∥Lp +δ ≤ CT d∥f∥Lp. +Proof. Let us prove the estimate for f1, which can be written as: +f1 = (1 − χ(ρT ))((1 − ζ1)f − [PT , ζ0]u0 − ζ0(PT − P0)u0). +(4.17) +The term (1 − χ(ρT ))(1 − ζ1)f is supported in the compact region {ρ1 ≤ 2} ⊂ Z1 and +therefore satisfies: +∥(1 − χ(ρT ))(1 − ζ1)f∥Lp +δ ≤ e2δ∥f∥Lp +(4.18) +since the function (1−χ(ρT ))(1−ζ1) is bounded by 1. On the other hand, the second term +(1 − χ(ρT ))[PT , ζ0]u0 is supporter in {ρ1 ≤ 1}, and the W k,p-norm of u0 in the cylinder +28 + +[−T −1, T +1]×X is bounded by CT d∥f∥ for some constant C. As ζ and all its derivatives +are uniformly bounded independently from T, this yields an estimate: +∥(1 − χ(ρT ))[PT , ζ0]u0∥Lp ≤ C′T d∥f∥Lp. +(4.19) +For the last term (1−χ(ρT ))ζ0(PT −P0)u0, we can use the bound on the W k,p-norm of u0 +and the exponential decay of the coefficients of P1 − P0 and all their derivatives to obtain +a similar bound: +∥(1 − χ(ρT ))ζ0(PT − P0)u0∥Lp ≤ C′′T d∥f∥Lp. +(4.20) +These three bounds prove the lemma. +(4.2.3) +Next we want to understand the obstructions to solving Piui = fi with ui ∈ +W k,p +δ +(Ei). The key result is the following: +Lemma 4.5. Choose arbitrary norms on K ∗ +1 and K ∗ +2 and let any 0 < δ < σ. Then for +T → ∞ the following holds. If g1 ∈ K ∗ +1 and g1,T (t) = κ∗ +1[g1](t + T + 1) then: +⟨f1, g1⟩ = ⟨f, (1 − χT+1(ρ1))g1⟩ − ⟨(1 − χ)f0, g1,T ⟩0 + O +� +e−δT ∥f∥Lp∥g1∥ +� +. +If g2 ∈ K ∗ +2 and g2,T = κ∗ +2[g2](t − T − 1) then: +⟨f2, g2⟩ = ⟨f, (1 − χT+1(ρ2))g2⟩ + ⟨(1 − χ)f0, g2,T ⟩0 + O +� +e−δT ∥f∥Lp∥g2∥ +� +. +Proof. Notice first that for any τ ≤ T − 2 we have: +⟨f1, g1⟩ = ⟨(1 − χ(ρT ))f, g1⟩ − ⟨(1 − χ(ρT ))Pζ0u0, g1⟩ +(4.21) += ⟨(1 − χ(ρT ))f, g1⟩ − ⟨(1 − χ(ρT ))Pζτu0, g1⟩ +(4.22) +since (ζτ − ζ0)u0 has support in {ρ1 ≤ T − 1} and P ∗ +1 g1 = 0. Given the decay of the +coefficients of P1 − P0 we have: +⟨(1 − χ(ρT ))PζT−2u0, g1⟩ = ⟨(1 − χ)P0χ−2u0, g1,T ⟩0 + O +� +e−δT ∥f∥Lp∥g∥ +� +. +(4.23) +Moreover we have 1 − χ(ρT ) = (1 − χT+1(ρ1)) with the usual identifications. Thus the +equality ⟨(1 − χ(ρT ))f, g1⟩ = ⟨f, (1 − χT+1(ρ1))g1⟩ clearly holds. +It remains to compute the value of ⟨(1 − χ)P0χ−2u0, g1,T ⟩0. By the same argument we +used many times before, for any τ ≥ −2 we have: +⟨(1 − χ)P0χ−2u0, g1,T ⟩0 = ⟨(1 − χ)P0χτu0, g1,T ⟩0. +(4.24) +But now we can write Pχτu0 = χτf0 + [P0, χτ]u0. As u0 and its derivatives of order less +than k have exponential decay at infinity, we can send τ → −∞ and obtain: +⟨(1 − χ)P0χ−2u0, g1,T ⟩0 = +lim +τ→−∞⟨(1 − χ)χτf0, g1,T ⟩0 +(4.25) += ⟨(1 − χ)f0, g1,T ⟩0 +(4.26) +This proves the first equality of Lemma 4.5. +For the second equality we can prove as above that: +⟨f2, g2⟩ = ⟨f, χT+1(ρ2)g2⟩ − lim +τ→∞⟨χP0(1 − χτ)u0, g2,T ⟩0 + O +� +e−δT ∥f∥Lp∥g2∥ +� +. +(4.27) +29 + +Then for τ large enough we have: +⟨χP0(1 − χτ)u0, g2,T ⟩0 = ⟨χf0, g2,T ⟩0 + ⟨χ[P0, 1 − χτ]u0, g2,T ⟩0 +(4.28) +−→ ⟨χf0, g2,T ⟩0 − (uf0, g2,T ) +(4.29) +as τ → ∞, where uf0 ∈ E is the polyhomogeneous solution defined in §3.3. By Lemma +3.13, the last term is equal to: +(uf0, g2,T ) = ⟨f0, g2,T ⟩0. +(4.30) +The second equality follows. +In the next section, it will be useful to use a variation of the above lemma for arbitrary +solutions of the equation P0u = f0. Thus let v ∈ E and define u′ +0 = Q0f0 + v, and as +above write: +f − Pζ0u′ +0 = f ′ +1 + f ′ +2 +(4.31) +where f ′ +i ∈ Lp +δ(Fi). As a corollary of Lemma 4.5, we can describe the obstructions to +solving Piu = f ′ +i as follows. +Corollary 4.6. Choose arbitrary norms on E , K ∗ +1 +and K ∗ +2 . +Then if g1 ∈ K ∗ +1 +and +g1,T (t) = κ∗ +1[g1](t + T + 1) it holds: +⟨f ′ +1, g1⟩ = ⟨f, (1−χT+1(ρ1))g1⟩−⟨(1−χ)f0, g1,T ⟩0−(v, g1,T )+O +� +e−δT (∥f∥Lp + ∥v∥)∥g1∥ +� +. +If g2 ∈ K ∗ +2 and g2,T = κ∗ +2[g2](t − T − 1) then: +⟨f ′ +2, g2⟩ = ⟨f, (1−χT+1(ρ2))g2⟩+⟨(1−χ)f0, g2,T ⟩0+(v, g2,T )+O +� +e−δT (∥f∥Lp + ∥v∥)∥g2∥ +� +. +(4.2.4) +So far, everything we did works in full generality without need to impose any +conditions on the real roots of the indicial operator P0. We now outline the construction +which we will perform in the next part and emphasise where the restricting assumptions +that we need to take come from. The general idea of our construction is to identify a +subspace of Lp(FT ) on which we can find approximate solutions of the equation Pu = f +with estimates, with a control on the error of the form ∥f − Pu∥Lp ≤ Ce−δT ∥f∥Lp for T +large enough. Once we can achieve this, we will simply use an iterative process to build +exact solutions, by taking successive projections onto this good subspace. +By taking cutoffs as above, we can solve the equation P0u = f0 on the cylinder, with +a general solution of the form u = Q0f0 + v for some arbitrary v ∈ E . With the above +notations, it remains to consider the equations Piui = f ′ +i on the EAC manifolds Z1 and +Z2. The idea is to choose v appropriately so that all the obstructions to finding decaying +solutions ui ∈ W k,p +δ +(Ei) vanish, up to exponentially decaying terms. If this can be done +then we just need to take cutoffs of these solutions to build an approximate solution, up +to an exponentially decaying term. Using Corollary 4.6 we have essentially reduced our +linear PDE problem to the following finite-dimensional linear system, where the unknown +is v ∈ E : +� +(v, g1,T ) = ⟨f, (1 − χT+1(ρ1))g1⟩ − ⟨(1 − χ)f0, g1,T ⟩0, +∀g1 ∈ K ∗ +1 +(v, g2,T ) = −⟨f, (1 − χT+1(ρ2))g2⟩ + ⟨(1 − χ)f0, g2,T ⟩0, +∀g2 ∈ K ∗ +2 +(4.32) +and we use the notations: +g1,T (t) = κ∗ +1[g1](t + T + 1), +and g2,T = κ∗ +2[g2](t − T − 1). +(4.33) +30 + +We call this system the characteristic system of our gluing problem. There are obvious +obstructions to finding a solution to this system. Indeed we need at least to impose f to +be orthogonal to all the sections of the form (1 − χT+1(ρi))gi with gi ∈ Ki,0 for the L2- +product, as in this case gi,T = 0. Actually, a more careful examination of the characteristic +system shows that we need f to be orthogonal to the full substitute cokernel. Indeed, a +pair (g1, g2) ∈ K ∗ +1 × K ∗ +2 +is matching at T if and only if g1,T = g2,T with the above +notations. Thus if there exists v ∈ E solving the system we must have +⟨f, (1 − χT+1(ρ1))g1 + (1 − χT+1(ρ2))g2⟩ = 0 +(4.34) +for any pair of solutions matching at T. +As a consequence, the substitute cokernel K ∗ +T naturally arises as a space of obstructions +to constructing approximate solutions of PT u = f by our method, as it is necessary to +require f to be orthogonal to K ∗ +T for the linear system (4.32) to admit a solution. In +fact, we will see that this is also a sufficient condition (Lemma 4.10). Unfortunately, the +coefficients of this system vary analytically with T, and therefore the rank of the system +might drop at some points. Further, the system is generally underdetermined, with an +obvious kernel formed by the subspace of E orthogonal to all g1,T and g2,T , for gi varying in +K ∗ +i . Hence, even if the characteristic system admits a solution v whenever f is orthogonal +to the substitute cokernel we might not be able to obtain reasonable estimates on the norm +of v, especially near the values of T at which the rank of the system drops. We shall prove +that these difficulties can be avoided in the case where the indicial operator P0 has only +one root, which will be sufficient for our applications. +4.3 +Main construction +(4.3.1) +Let us first consider the case where P0 has a single root λ0 of order 1, before +generalising to any order. In that case, the elements of E are of the form eiλ0tu(x) with u +translation-invariant section of E0, and similarly for E ∗. As a consequence, the matching +condition (2.18) does not really depend on T, up to overall factors of e±iλ0(T+1). +In +particular, the dimensions of KT and K ∗ +T are independent from T: +dim KT = dim K0,1 + dim K0,2 + dim(im κ1 ∩ im κ2) +(4.35) +and similarly for the substitute cokernel K ∗ +T . This implies that we can bound uniformly +the L2-orthogonal projection onto KT and K ∗ +T : +Lemma 4.7. Let p > 1 and l ≥ 0. Then for T large enough the norm of the L2-orthogonal +projection of W l,p(FT ) onto K ∗ +T is bounded from above by a uniform constant C1 > 0. +Similarly the norm of the L2-orthogonal projection of W l,p(ET ) onto KT is bounded +from above by a uniform constant C′ +1 > 0. +Proof. This can be proved by fixing basis for K0,1, K0,2 and im κ1 ∩im κ2 and considering +the corresponding basis of KT . Using the Gram–Schmidt orthonormalization process one +can deduce an explicit expression for the L2-projection, from which the lemma easily +follows. +(4.3.2) +Let us now choose an arbitrary complement E ∗ +1 of im κ∗ +1 ∩ im κ∗ +2 in im κ∗ +1, and a +complement E ∗ +2 of im κ∗ +1 ∩ im κ∗ +2 in im κ∗ +2. Thus we have a direct sum decomposition: +im κ∗ +1 + im κ∗ +2 = im κ∗ +1 ∩ im κ∗ +2 ⊕ E ∗ +1 ⊕ E ∗ +2 +(4.36) +31 + +in E ∗. Pick moreover a complement E ′ of im κ1 ∩ im κ2 in E , so that the pairing +E ′ × (im κ∗ +1 + im κ∗ +2) → C +(4.37) +induced by (·, ·) is non-degenerate. For i = 1, 2 define Ei = im κi ∩ E ′. Then the pairings: +E1 × E ∗ +2 → C and E2 × E ∗ +1 → C +(4.38) +induced by (·, ·) are non-degenerate. +Indeed if u ∈ E1 is orthogonal to E ∗ +2 then it is +orthogonal to im κ∗ +1 + im κ∗ +2 and therefore belongs to im κ1 ∩ im κ2, which means that +u = 0 as it is an element of E ′. On the other hand if v ∈ E ∗ +2 is orthogonal to E1, then it is +orthogonal to im κ1 and to im κ2, which means that it belongs to im κ∗ +1 ∩ im κ∗ +2 and thus +v = 0 by definition of E ∗ +2 . Therefore, if we define E0 as the orthogonal space of E ∗ +1 ⊕ E ∗ +2 in +E ′ for the above pairing, we have a direct sum decomposition: +E ′ = E0 ⊕ E1 ⊕ E2. +(4.39) +This implies that the pairing +E0 × (im κ∗ +1 ∩ im κ∗ +2) → C +(4.40) +is non-degenerate. These conventions will be useful to put the system (4.32) in a more +tractable form, and for definiteness we prefer to work in a complement of im κ1 ∩ im κ2 as +this is the kernel of this system. This allows us to prove the following: +Proposition 4.8. Let p > 1 and 0 < δ < σ. Then there exists constants C2, C3 > 0 such +that for T large enough the following holds. +If f ∈ Lp(FT ) is L2-orthogonal to K ∗ +T , then there exists u ∈ W k,p(ET ) L2-orthogonal +to KT such that: +∥f − Pu∥Lp ≤ C2e−δT ∥f∥Lp +and ∥u∥W k,p ≤ C3T∥f∥Lp. +Proof. For i = 1, 2 let us fix subspaces Fi ⊂ C∞(Ei) as in Proposition 4.3. The orthogonal +space K ∗ +i,+ of Ki,0 in Ki is isomorphic to im κ∗ +i , so that the decomposition im κ∗ +i = +im κ∗ +1∩im κ∗ +2⊕E ∗ +i induce a corresponding decomposition K ∗ +i,+ = K ∗ +i,m⊕K ∗ +i,⊥ (the subscript +m stands for matching). +If f ∈ Lp(FT ) is orthogonal to the substitute cokernel, we can use the above decompo- +sitions of E amd E ∗ to put the system (4.32) in the form: + + + + + + + +(v0, κ∗ +1[g0]) = e−iλ(T+1)⟨f, (1 − χT+1(ρ1))g0⟩ − ⟨(1 − χ)f0, κ∗ +1[g0]⟩0, +∀g0 ∈ K ∗ +1,m +(v1, κ∗ +1[g1]) = e−iλ(T+1)⟨f, (1 − χT+1(ρ1))g1⟩ − ⟨(1 − χ)f0, κ∗ +1[g1]⟩0, +∀g1 ∈ K ∗ +1,⊥ +(v2, κ∗ +2[g2]) = −eiλ(T+1)⟨f, (1 − χT+1(ρ2))g2⟩ + ⟨(1 − χ)f0, κ∗ +2[g2]⟩0, +∀g2 ∈ K ∗ +2,⊥ +(4.41) +where we decompose any element v ∈ E ′ as v = v0 + v1 + v2 ∈ E0 ⊕ E1 ⊕ E2 and the factors +e±iλ(T+1) come from κ∗ +1,T = eiλ(T+1)κ∗ +1 and κ2,T = e−iλ(T+1)κ∗ +2. Non-degeneracy of the +pairings (4.38) and (4.40) implies that this is of the form: +Av = bT (f) ∈ RN +(4.42) +where N = dim E ′ = dim im κ∗ +1 + dim im κ∗ +2 and A ∈ Hom(E ′, RN) is an invertible linear +map which does not depend on T. Thus there is a unique solution v = A−1b(f), and if we +fix norms on E ′ and RN we have a uniform bound: +∥v∥ ≤ C∥bT (f)∥. +(4.43) +32 + +As the elements of K ∗ +1 and K ∗ +2 are bounded in C0 norm, each of the sections (1−χT+1)gi +have Lq-norm bounded by CT +1 +q ∥gi∥ where q is the conjugate exponent of p, and thus we +can deduce that the norm of bT (f) satisfies a bound of the form: +∥bT (f)∥ ≤ C′T +1 +q ∥f∥Lp +(4.44) +Thus ∥v∥ ≤ C′′T +1 +q ∥f∥Lp for a constant independent of T. +Following the idea outlined in the previous part, let us write: +f − PT (ζ0Q0f0 + ζ0v) = f1 + f2 +(4.45) +with f1 ∈ Lp(F1) and f2 ∈ Lp(F2), each of the sections fi being supported in the domain +{ρi ≤ T + 2} ⊂ Zi. By Theorem 2.10, we have estimates: +∥ζ0Q0f0∥W k,p ≤ CT∥f0∥ ≤ C′T∥f∥Lp. +(4.46) +Further, as v is uniformly bounded and ζ0v has support in a domain equivalent to a finite +cylinder [−T − 1, T + 1] × X we have a bound: +∥ζ0v∥W k,p ≤ CT +1 +p ∥v∥ ≤ C′T∥f∥Lp. +(4.47) +As in the proof of Lemma 4.4, we can use the uniform bound on v to prove that the +weighted norms of f1 and f2 satisfy bounds: +∥fi∥Lp +δ ≤ CT∥f∥Lp. +(4.48) +We now consider the equations P1u1 = f1 on Z1 and P2u2 = f2 on Z2. By Proposition +4.3, there exists wi ∈ K ��� +i,0, hi ∈ Fi and ui ∈ W k,p +δ +(Ei) such that: +P(ui + hi) = fi − wi. +(4.49) +Moreover, our choice of v implies uniform bounds of the form: +∥ui∥W k,p +δ +≤ C∥fi∥Lp +δ ≤ C′T∥f∥Lp, +∥hi∥ + ∥wi∥ ≤ C′′e−δT ∥f∥Lp +(4.50) +for some uniform constants C′ and C′′. Taking cutoffs we can write: +fi − PT (χT+1(ρi)ui + χT+1(ρi)hi) = χT+1(ρi)wi + ri +(4.51) +where ri is an error term of the form +ri = (Pi − PT )(χT+1(ρi)ui + χT+1(ρi)hi) + [Pi, χT+1(ρi)](ui + hi). +(4.52) +As the coefficients of Pi − PT and their derivatives have exponential decay with T, ui has +exponential decay at infinity and given the bound on hi, it follows that for any ǫ > 0 we +can bound the errors terms by: +∥ri∥Lp ≤ Ce−(δ−ǫ)T ∥f∥Lp +(4.53) +for some uniform constant. Denote +u = ζ0u0 + χT+1(ρ1)(u1 + h1) + χT+1(ρ2)(u2 + h2). +(4.54) +Then f−PTu = χT+1(ρ1)w1+χT+2(ρ2)w2+r1+r2 satisfies ∥f−PT u∥Lp ≤ Ce−(δ−ǫ)T ∥f∥Lp, +and ∥u∥W l,p ≤ CT∥f∥Lp, for some constant C. By Lemma 4.7, we can decompose u = +u′ + w where w ∈ KT and u′ is orthogonal to the substitute kernel. Moreover we have +bounds: +∥u′∥W k,p ≤ (1 + C′ +1)∥u∥W k,p ≤ C′T∥f∥Lp, +∥w∥W k,p ≤ C1∥u∥W k,p ≤ C′′T∥f∥Lp +(4.55) +and u′ satisfies +f − PT u′ = f − PT u + PT w +(4.56) +and as w ∈ KT , ∥PT w∥Lp ≤ C−δT ∥w∥W k,p ≤ Ce−(δ−ǫ)T ∥f∥Lp. +33 + +(4.3.3) +We now have all the tools to prove Theorem 2.6 in the case where λ0 has order +1. Let f ∈ Lp(FT ) be an arbitrary section. By Lemma 4.7, there exists ˜f ∈ Lp(FT ) and +w0 ∈ K ∗ +T such that f = ˜f + w, ˜f is orthogonal to K ∗ +T and with bounds: +∥ ˜f∥Lp ≤ (1 + C1)∥f∥Lp, +∥w0∥Lp ≤ C1∥f∥Lp. +(4.57) +Moreover, by Proposition 4.8 there exists u0 ∈ W k,p(ET ) orthogonal to K ∗ +T and f1 ∈ +Lp(FT ) such that: +˜f = Pu0 + f1 +(4.58) +with bounds: +∥u0∥W k,p ≤ C3T∥ ˜f∥Lp ≤ (1 + C1)C3T∥f∥Lp +(4.59) +and +∥f1∥Lp ≤ C2e−δT ∥ ˜f∥Lp ≤ (1 + C1)C2e−δT ∥f∥Lp. +(4.60) +Choose T large enough such that η = (1 + C1)C2e−δT < 1, and denote f0 = f. Then +inductively we can construct sequences {fn, n ≥ 0} in Lp(FT ), {un, n ≥ 0} in the L2- +orthogonal complement of KT in W k,p(ET ) and {wn, n ≥ 0} in K ∗ +T such that for all +n ≥ 0 we have: +fn − fn+1 = Pun + wn +(4.61) +with the bounds: +∥fn∥Lp ≤ ηn∥f∥, +∥un∥W k,p ≤ ηn(1 + C1)C3T∥f∥Lp, +and ∥wn∥Lp ≤ ηnC1∥f∥Lp. (4.62) +As W k,p(ET ) is complete and η < 1, the series � un converges. Let u = �∞ +n=0 un. As +each term of the series is orthogonal to KT , u belongs to the orthogonal space to KT in +W k,p(ET ) as this is a closed subspace. In the same way the series � wn converges to an +element w ∈ K ∗ +T . It follows from the above bounds that we have: +∥u∥W k,p ≤ (1 + C1)C2 +1 − η +T∥f∥Lp, +∥w∥Lp ≤ +C1 +1 − η∥f∥Lp +(4.63) +Further the map W k,p(ET ) → Lp(FT ) is continuous and therefore we can sum over n in +equality (4.61) to obtain f = Pu + w. +This proves the existence part in Theorem 2.6 for f in the Lp range. In particular the +index of P satisfies the inequality: +ind(P) ≥ dim KT − dim K ∗ +T . +(4.64) +As the map W k,p(ET ) → Lp(FT ) induced by P is Fredholm, the uniqueness of u ∈ +W k,p(ET ) orthogonal to KT and w ∈ K ∗ +T satisfying f = Pu + w is equivalent to proving +that inequality (4.64) is in fact an equality. But the same reasoning applied to P ∗ yields: +ind(P ∗) ≥ dim K ∗ +T − dim KT . +(4.65) +Since ind(P ∗) = − ind(P), uniqueness in Theorem 2.6 follows. +To complete the proof of the theorem in the Sobolev range, it remains to remark that if +one assumes further that f ∈ W l,p(FT ) the the a priori estimate of Proposition 3.4 implies +that +∥u∥W k+l,p ≤ C(∥f∥W l,p + ∥u∥Lp) ≤ C∥f∥W l,p + C′T∥f∥Lp +(4.66) +for some constant C′ > 0. +Remark 4.9. One of the advantages of treating the case of a root of order 1 first is that +we proved that the Sobolev constant does not grow more than linearly with T, whereas in +the general case it is more complicated to find the optimal rate of growth of the constant. +This will be useful in our applications in Section 5 to derive the rate of decay of the low +eigenvalues of the Laplacian. +34 + +(4.3.4) +Let us now go back to the general case before indicating how to modify our +construction to treat the case of a single root of any order. We now prove our previous +claim, that without any restrictions on the number of real roots of P0 the characteristic +system admits a solution if and only if f is orthogonal to the substitute cokernel: +Lemma 4.10. For any T ≥ 1 and any f ∈ Lp(FT ) orthogonal to K ∗ +T the characteristic +system (4.32) admits a solution v ∈ E . +Proof. Let us use the following notations for u1 ∈ K1 and g1 ∈ K ∗ +1 : +κ1,T [u1](t, x) = κ1[u1](t + T + 1, x), +κ∗ +1,T [u1](t, x) = κ∗ +1[u1](t + T + 1, x) +(4.67) +and for u2 ∈ K2 and g2 ∈ K ∗ +2 : +κ2,T [u2](t, x) = κ2[u2](t − T − 1, x), +κ∗ +2,T [u2](t, x) = κ∗ +2[u2](t − T − 1, x). +(4.68) +As the pairing (·, ·) is invariant by translation, it is still true that im κi,T is the orthogonal +space to im κ∗ +i . Thus we may proceed exactly as above, choosing a complement E ′ +T of +im κ1,T ∩ im κ2,T in E , and complements E ∗ +i,T of im κ∗ +1,T ∩ im κ∗ +2,T in κ∗ +i,T . +Once these +arbitrary choices are made we can decompose: +E ′ +T = E0,T ⊕ E1,T ⊕ E2,T +(4.69) +where Ei,T = im κi,T ∩ E ′ +T and E0,T is the orthogonal space of E1,T ⊕ E2,T in E ′ +T . Hence the +non-degenerate pairing E ′ +T × (im κ1,T + im κ2,T ) → C induced by (·, ·) decomposes as the +orthogonal sum of the non-degenerate pairings: +E0,T × (im κ1,T ∩ im κ2,T ) → C, +E1,T × E ∗ +2,T → C and E2,T × E ∗ +1,T → C. +(4.70) +As in the proof of Proposition 4.8, for i = 1, 2 the orthogonal space K ∗ +i,+ of Ki,0 in Ki is +isomorphic to im κ∗ +i,T , so that the decomposition im κ∗ +i,T = im κ∗ +1,T ∩ im κ∗ +2,T ⊕ E ∗ +i,T induces +a corresponding decomposition K ∗ +i,+ = K ∗ +i,T,m ⊕K ∗ +i,T,⊥. Thus if f ∈ Lp(MT ) is orthogonal +to the substitute cokernel K ∗ +T the characteristic system can be written as: + + + + + + + +(v0, κ∗ +1,T [g0]) = ⟨f, (1 − χT+1(ρ1))g0⟩ − ⟨(1 − χ)f0, κ∗ +1[g0]⟩0, +∀g0 ∈ K ∗ +1,T,m +(v1, κ∗ +1,T [g1]) = ⟨f, (1 − χT+1(ρ1))g1⟩ − ⟨(1 − χ)f0, κ∗ +1[g1]⟩0, +∀g1 ∈ K ∗ +1,T,⊥ +(v2, κ∗ +2,T [g2]) = −⟨f, (1 − χT+1(ρ2))g2⟩ + ⟨(1 − χ)f0, κ∗ +2[g2]⟩0, +∀g2 ∈ K ∗ +2,T,⊥ +(4.71) +where v = v0+v1+v2 ∈ E0,T ⊕E1,T ⊕E2,T . Given the non-degeneracy of the above pairings +this system is manifestly invertible. +Despite the fact that we can solve the characteristic system whenever f is orthogonal +to the substitute cokernel KT , this does not imply that we can find a solution v ∈ E +with bounds of the form ∥v∥ ≤ C(T)∥f∥Lp with a good control on C(T), which was a +key argument in the previous construction. This is due to the fact that the characteristic +system is in general underdetermined, and only becomes determined after a choice of +arbitrary complements E ′ +T of im κ1,T ∩im κ2,T in E and E ∗ +i,T of im κ∗ +1,T ∩im κ∗ +2,T in im κ∗ +i,T , +using the notations introduced in the above proof. +In the case where P0 has a single +root of order 1, this was not problematic as we could simply make any arbitrary choice +independently from T, but in general we cannot make such a consistent choice. This is +especially true at values of T where the rank of the characteristic system drops. +As we discussed in §2.2, the advantage of assuming that P0 has only one root is that in a +good basis the coefficients of the characteristic system are polynomial in T. Therefore the +35 + +rank of the system is constant whenever T is large enough and we can fix a complement +of its kernel to invert it with polynomial control on the norm. Thus if f ∈ Lp(FT ) is +orthogonal to the substitute cokernel we can find solutions of the characteristic system with +∥v∥ ≤ CT β∥f∥Lp for some exponent β > 0. In the same way all matching conditions can +be expressed as linear equations with coefficients polynomial in T and therefore the norm +of the L2-orthogonal projections onto KT and K ∗ +T do not grow more than polynomially. +Then we can use the same argument as above to prove Theorem 2.6 in the general case. +Moreover, we worked with Sobolev norms just to fix a convention but it is clear that we +can also do this with Hölder norms. +In fact, any assumptions ensuring that we can invert the characteristic system with less +than exponential growth on the norm after fixing complements of its image and kernel, +and that the norm of the projections onto KT and K ∗ +T do not grow too quickly would +yield the same result. Moreover in cases where the rank of the system is not constant +we may also obtain good bounds by staying away from the values of T where it drops. +However we do not need to consider more complicated cases for our applications. +5 +Spectral aspects +In this section, we want to interpret our results from a spectral perspective. Indeed, for +self-adjoint operators the approximate kernel can be regarded as a finite-dimensional space +associated with very low eigenvalues of the operator PT . In the case of the Laplacian for +instance, we shall see in §5.1 that the substitute kernel provides a good approximation for +the space of harmonic forms. +Orthogonally to the space of harmonic forms, we shall see that the L2-norm of the +inverse of ∆T has decay in T −2, and thus this is the rate of decay of the lowest eigenvalues. +In §5.2 we give precise estimates on the density of the eigenvalues with fastest decay rate +and prove Theorem 2.8. In Section 6, we will see that our analysis also applies to G2- +manifolds constructed by twisted connected sum. +5.1 +Approximate harmonic forms +(5.1.1) +In this part we are particularly interested in the Laplacian operator, and want +to show that the corresponding substitute kernel that we defined in §2.2 represents a space +of approximate harmonic forms. Before doing this, we review some standard properties +of the Laplacian on EAC manifolds (see for instance [30, 6.4]). Let (Z, g) be an oriented +EAC Riemannian manifold of rate µ > 0, ρ a cylindrical coordinate function on Z and +let Y = R × X be the cylinder it is asymptotic to. It follows from Lockhart-McOwen +theory that the space H q of bounded closed and co-closed q-forms is equal to the space +of bounded harmonic q-forms, and moreover there is a direct sum decomposition: +H q = H q +0 ⊕ H q +d ⊕ H q +d∗ +(5.1) +where H q +0 is the space of decaying harmonic q-forms, H q +d is the space bounded exact +harmonic q-forms and H q +d∗ the space of bounded co-exact harmonic q-forms. +On the +other hand, the map κq mapping a bounded harmonic q-form to the translation-invariant +harmonic q-form it is asymptotic to induces two maps +αq : H q → Hq(X), +βq : H q → Hq−1(X) +(5.2) +36 + +such that κq(η) = α0 + dt ∧ β0 where α0 and β0 are the harmonic representatives of αq(η) +and βq(η) respectively. The map αq fits into a commutative diagram: +H q +� +αq +�◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +Hq(Z) +� Hq(X) +where the vertical arrow maps any bounded closed and co-closed q-form in H q to its de +Rham cohomology class in Hq(Z), and the vertical map Hq(Z) → Hq(X) comes from +the long exact sequence of the pair (Z, X), where Z = Z ∪ X is considered as a compact +manifold with boundary X. By [4, Proposition 4.9], H q +0 is mapped isomorphically to the +image of the map Hq +c (Z) → Hq(Z) coming from the same exact sequence. In particular +this implies that H q +0 ⊕ H q +d ⊂ ker αq, and by considering Hodge duals it follows that +H q +0 ⊕ H q +d∗ ⊂ ker βq. As the kernel of the map κq is H q +0 this implies that we have in fact: +ker αq = H q +0 ⊕ H q +d , +ker βq = H q +0 ⊕ H q +d∗. +(5.3) +By [30, Proposition 6.18], the map H q +0 ⊕ H q +d∗ → Hq(Z) is an isomorphism, and in par- +ticular αq maps H q +d∗ isomorphically onto the image of the map Hq(Z) → Hq(X) coming +from the exact sequence of (Z, X). +(5.1.2) +After this background, let 0 ≤ q ≤ dim Z and denote σq the minimum of µ and +of the square roots of the lowest eigenvalues of the Laplacian acting on (q−1)- and q-forms +on X. Any bounded closed and co-closed q-form η on Z is asymptotic to a translation- +invariant form η0 = α0 + dt ∧ β0, up to terms in O(e−δρ) for any δ < σq. With the above +notations (α0, β0) are the harmonic representatives of (αq(η), βq(η)). It is a standard fact +that there exists a (q − 1)-from ξ on Z such that η − η0 = dξ in the domain {ρ > 1}, such +that |∇lξ| = O(e−δρ) at infinity, for any l ≥ 0 and ρ < σq. Indeed, in this region identified +with the cylinder (1, ∞) × X on can write η − η0 = α(t) + dt ∧ β(t) where α, β and all +their derivatives have the usual exponential decay, and the closedness of η and η0 implies: +dα(t) = 0 = ∂tα(t) − dβ(t) +(5.4) +for all t > 1, where d denotes the exterior differential on X. Hence we can for instance +define ξ in the region ρ > 1 by +ξ(t, x) = +� t ++∞ +β(τ, x)dτ, +∀(t, x) ∈ (1, ∞) × X +(5.5) +and take a cutoff to extend it to Z. The (q − 1)-form ξ allows us to build a 1-parameter +family of closed q-forms: +ηT = η − d(χT (ρ)ξ) +(5.6) +that interpolates between η when ρ < T − 1 +2 and η0 when ρ > T 1 +2, which all represent +the cohomology class of ρ in Hq(Z). Moreover the difference ηT − η and all its derivatives +satisfy uniform bounds in O(e−δT ) for any 0 < δ < σq. +Consider now the 1-parameter family of compact Riemannian manifolds (MT , gT ) ob- +tained by gluing two matching EAC manifolds (Z1, g1) and (Z2, g2) as explained in §2.1. +We are interested in finding estimates for the Green’s functions of the associated opera- +tors d + d∗ +T and ∆T . Strictly speaking these operators differ from the operators obtained +by gluing d + d∗ +1 with d + d∗ +2 and ∆1 with ∆2 in the gluing region {|ρT | ≤ 3 +2}, but the +37 + +coefficients of the difference and and all their derivatives have exponential decay with T, +so our results still apply. It is also convenient to slightly modify our definition of approxi- +mate kernel so that it is constituted of closed forms, Thus if (η1, η2) is a matching pair of +harmonic forms we can define a closed form on MT by: +ηT = + + + + + + + +η1,T +if ρT ≤ − 1 +2 +η2,T +if ρT ≥ 1 +2 +η0 +if |ρT | ≤ 1 +2 +(5.7) +where both η1 and η2 are asymptotic to η0 and η1,T and η2,T are closed forms constructed +as above. It follows that ηT is closed. We denote H q +T the finite-dimensional space of +q-forms constructed as above from a pair of matching q-forms. Again this differs from our +previous definition of substitute kernel and cokernel only up to terms that are bounded in +O(e−δT ) as well as all their derivatives for any δ < σq, so that our results an in particular +Theorem 2.6 still applies. As the elements of H q +T are closed there is a well-defined map: +H q +T → Hq(MT ) +(5.8) +sending every element to its de Rham cohomology class. The key point for us, which +partly motivates our general definition of substitute kernels and cokernels, is the following +theorem [32, Theorem 3.1]: +Theorem 5.1. For T large enough, the map H q +T → Hq(MT ) is an isomorphism. +We shall give a brief sketch proof of this theorem, referring to the original reference +for more details. It relies on a close examination of the Mayer-Vietoris sequence: +. . . → Hq−1(Z1) ⊕ Hq−1(Z2) → Hq−1(X) → Hq(MT ) → Hq(Z1) ⊕ Hq(Z2) → . . . +(5.9) +As the space of approximate harmonic q-forms H q +T is isomorphic to +H q +T ≃ Hq +c (Z1) ⊕ Hq +c (Z2) ⊕ im α1,q ∩ im α2,q ⊕ im β1,q ∩ im β2,q +(5.10) +and since ker βi,q ≃ Hq(Zi) it is clear that the restriction of H q +T → Hq(MT ) to the space +obtained by matching pairs in H q +i,0 ⊕ H q +i,d∗ yields an isomorphism: +Hq +c (Z1) ⊕ Hq +c (Z2) ⊕ im α1,q ∩ im α2,q ≃ im(Hq(MT ) → Hq(Z1) ⊕ Hq(Z2)). +(5.11) +Moreover the subspace of H q +T obtained by gluing matching pairs bounded exact harmonic +q-forms, which is isomorphic to im β1,q ∩ im β2,q, maps into the image of Hq−1(X) → +Hq(MT ). By Lemma 4.2, im β1,q∩im β2,q is the orthogonal space of im α1,q−1⊕im α2,q−1 for +the inner product induced by the L2-product on harmonic representatives, and therefore +im β1,q∩im β2,q has the same dimension as the kernel of Hq(MT ) → Hq(Z1)⊕Hq(Z2). Thus +it only remains to prove that the subspace of H q +T obtained by gluing matching pairs of +exact q-forms maps isomorphically on the kernel of the map Hq(MT ) → Hq(Z1)⊕Hq(Z2) +coming from the exact sequence. In [31, Theorem 3.1] it is proven that this is the case for +T large enough. +Alternatively, one could also argue using Theorem 2.6. As the spaces H q +T and Hq(MT ) +have same dimension by the above argument, and the Laplacian ∆T maps the orthogonal +space of H q +T in W 2,2(ΛqT ∗MT ) isomorphically onto a complement of H q +T in L2(ΛqT ∗MT ) +for T large enough, the map H q +T → Hq(MT ) must be an isomorphism for large T. Oth- +erwise it would not be injective and thus there would be a non-trivial exact form in H q +T , +implying that the image of the Laplacian would have codimension strictly less than bq(MT ) +in L2(ΛqT ∗MT ), which is absurd. +38 + +Remark 5.2. The above theorem remains true when considering the variant of our gluing +problem explained in Remark 2.3, where the matching condition between the two building +blocks is twisted by an isometry γ : X → X of the cross-section. +(5.1.3) +As a consequence, the L2-projection of the space H q +T of approximate harmonic +q-forms onto the space H q(MT ) of actual harmonic q-forms is an isomorphism for T large +enough. A natural question which arises is how close are the elements of H q +T from their +harmonic part. If η ∈ H q +T is decomposed in harmonic and exact parts η = ξ + dν then: +∥∆T dν∥L2 = ∥∆T (η − ξ)∥L2 = ∥∆T η∥L2 = O +� +e−δT ∥η∥L2 +� +(5.12) +for any δ < σq. By Theorem 2.6, there exists a (q − 1)-form η′ with ∆T η′ = ∆T dν and +satisfying a bound of the form +∥η′∥W 2,2 ≤ CT 2∥∆T dν∥L2 ≤ C′e−δT ∥η∥L2 +(5.13) +for some constant C′. As η′ −dν is harmonic it follows that ∥dν∥L2 ≤ ∥η′∥L2, which yields: +∥η − ξ∥L2 = O +� +e−δT ∥η∥L2 +� +(5.14) +for any δ < σq. Thus not only the L2-projection of H q +T onto H q(MT ) is an isomorphism, +but the norm of the projection is close to 1, up to O(e−δT ) terms. Once this inequality is +established in L2, the a priori estimates of Proposition 3.4 imply that +∥η − ξ∥W l,2 = O +� +e−δT ∥η∥L2 +� +(5.15) +for any l ≥ 0. Then the Sobolev embedding theorem (see Remark 2.5) yields estimates: +∥η − ξ∥W l,p = O +� +e−δT ∥η∥Lp +� +, +∥η − ξ∥Cl,α = O +� +e−δT ∥η∥C0 +� +(5.16) +for any l ≥ 0, p > 1, α ∈ (0, 1) and 0 < δ < σq. +With these remarks, Theorem 2.6 combined with the fact that the Laplacian ∆T is the +square of the operator d + d∗ +T whose only root has order 1 imply the following: +Corollary 5.3. Let l ≥ 0 and p > 1, and assume that 0 is an indicial root of the Laplacian +action on q-forms on Y , that is bq−1(X)+bq(X) > 0. Then there exist constants C, C′ > 0 +such that for large enough T and any η ∈ W l,p(ΛqT ∗MT ) orthogonal to H q(MT ), the +unique solution η′ ∈ W 2+l,p(ΛqT ∗MT ) of ∆η′ = η orthogonal to H q(MT ) satisfies: +∥η′∥W l+2,p ≤ C∥η∥W l,p + C′T 2∥η∥Lp. +Similar estimates hold with Hölder norms. +In particular, in the L2-range this means that when bq−1(X) + bq(X) > 0 the lowest +eigenvalue of ∆T acting on q-forms satisfies a bound of the form λ1(T) ≥ +C +T 2 as T → ∞. +In §5.2 we will be interested in the distribution of the eigenvalues that have the fastest +decay rate, that is T −2. +39 + +5.2 +Density of low eigenvalues +(5.2.1) +Let us now study the density of low eigenvalues Λq,inf(s), Λq,sup(s) of the Lapla- +cian ∆T acting on q-forms, as defined in §2.3. When bq−1(X) + bq(X) = 0, the Laplacian +acting on q-forms has no real roots, and thus it has no decaying eigenvalues and Theorem +2.8 is true in that case. Thus we assume that m = bq−1(X) + bq(X) > 0. We shall prove +Theorem 2.8 using a min-max principle. +The easiest part, which does not require the results of Section 4, is to find a lower +bound for Λq,inf(s). Let us denote 0 ≤ λ1(T) ≤ . . . ≤ λn(T) ≤ . . . the non-decreasing +sequence of eigenvalues of the Laplacian counted with multiplicity. Note that here the +lowest eigenvalues vanish if bq(MT ) ̸= 0. The n-th eigenvalue is determined by: +λn(T) = min +� +max +�∥∆T η∥L2 +∥η∥L2 +, η ∈ V \{0} +� +, V ⊂ W 2,2(ΛqT ∗MT ), dim V = n +� +(5.17) +Using this we claim: +Lemma 5.4. Let V ⊂ C2([−1, 1], R) be an n-dimensional space of functions such that +f(−1) = f(1) = f ′(−1) = f ′(1) = 0 for all f ∈ V . Let λ > 0 such that for all non-zero +f ∈ V we have: +� 1 +−1 +(f ′′)2(t)dt < λ2 +� 1 +−1 +f 2(t)dt +Then for T large enough λmn(T) ≤ +λ +T 2. +Proof. Any f ∈ V can be extended as a C1 function to R by setting f(t) = 0 for any +|t| ≥ 1. +Moreover with this extension f ∈ W 2,2(R) and f ′′ ∈ L2(R) vanishes outside +of [−1, 1] and is equal to the usual second derivative inside this interval. Let us choose +0 < τ < 1 small enough such that: +� 1 +−1 +(f ′′)2(t)dt < λ2(1 − τ)4 +� 1 +−1 +f 2(t)dt +(5.18) +for any f ∈ V . For T ≥ 1 let VT be the subspace of W 2,2(ΛqT ∗Y ) spanned by sections of +the form: +η(t, x) = f +� +t +(1 − τ)T +� +ν(x) +(5.19) +where f ∈ V and ν is a translation-invariant harmonic form on Y . In particular it has +dimension dim VT = mn. +As the elements of VT vanish outside of the finite cylinder +[−(t − τ)T, (1 − τ)T] × X, it can be identified with a subspace of W 2,2(ΛqT ∗MT ). In +this finite cylinder, the Laplacian ∆T and the metric gT approach ∆0 = ∆X − ∂2 +t and +g0 = gX + dt2 up to terms in O(e−δτT ) and similarly for all derivatives, for some δ > 0 +appropriately small. Therefore there exist constants C, C′ > 0 such that: +sup +η∈VT \{0} +∥∆T η∥L2 +∥η∥L2 +≤ (1 + Ce−δτT ) +(1 − τ)2T 2 +sup +f∈V \{0} +∥f ′′∥L2 +∥f∥L2 + C′e−δτT +sup +η∈VT \{0} +∥η∥W 2,2 +∥η∥L2 . +(5.20) +As the ratio between the W 2,2-norm and the L2-norm on VT does not grow more than +polynomially with T, the second term in the right-hand-side has exponential decay. On +the other hand, by (5.18) the first term is less than (λ−ǫ) +T 2 +for large enough T and small +enough ǫ > 0. This proves the lemma. +40 + +We can apply the above lemma to the spaces: +Vn = + + + +� +1≤|k|≤n +akeikπt, +� +1≤|k|≤n +(−1)kak = +� +1≤|k|≤n +(−1)kkak = 0 + + + +(5.21) +For n ≥ 2, the space Vn has dimension 2n − 2 and for any non-zero f ∈ Vn we have: +� 1 +−1 +(f ′′)2(t)dt < (nπ)2 +� 1 +−1 +f 2(t)dt +(5.22) +The above lemma yields the inequality: +Λq,inf(s) ≥ (2⌊√s⌋ − 2)(bq−1(X) + bq(X)), +∀s ≥ 1. +(5.23) +(5.2.2) +We now want an upper bound on Λq,sup(s). Let us denote: +GT : L2(ΛqT ∗MT ) ∩ H q(MT )⊥ → L2(ΛqT ∗MT ) ∩ H q(MT )⊥ +(5.24) +the composition of the inverse of the Laplacian acting on W 2,2(ΛqT ∗MT ) ∩ H q(MT )⊥ +with the compact embedding W 2,2 ֒→ L2. Then for n ≥ bq(MT ) the eigenvalues λn+1(T) +can be characterised by: +λ−1 +n+1(T) = min +� +max +�∥GT η∥L2 +∥η∥L2 +, η ∈ V \{0} +� +, V ⊂ L2(ΛqT ∗MT ), codim V = n +� +(5.25) +where moreover V ranges over closed subspaces orthogonal to H q(MT ). Since we devel- +oped a rather explicit construction of solutions to the equation ∆Tν = η, the idea is to +show that if we impose enough orthogonality conditions to η ∈ L2(ΛqT ∗MT ), and not only +orthogonality to the space of harmonic forms (or to the substitute kernel), we can give +explicit bounds for the norm of GT η. +Let us denote by N the sum of the dimensions of the spaces of harmonic forms with at +most polynomial growth on Z1 and Z2. Moreover, denote by E ⊂ L2([−1, 1]) the closed +subspace of functions f(t) = � akeikπt which satisfy: +a0 = 0, +� +|k|≥1 +(−1)k ak +k = +� +|k|≥1 +(−1)k ak +k2 = 0. +(5.26) +Thus E has codimension 3 in L2([−1, 1]). For f ∈ L2(R) with compact essential support +let us define: +Hf(t) = +� t +−∞ +(τ − t)f(τ)dτ. +(5.27) +The first two conditions in the definition of E imply that for any f ∈ E on has: +� 1 +−1 +f(τ)dτ = +� 1 +−1 +τf(τ)dτ = 0. +(5.28) +On the other hand, the last condition is a matter of scaling under change of variables. +Since we have +� t +−T +(τ − t)e +ikπτ +T dt = +T 2 +(kπ)2 e +ikπt +T ++ (−1)k +� +T(T + t) +ikπ +− +T 2 +(kπ)2 +� +(5.29) +41 + +if follows that for any f(t) = � akeikπt ∈ E, the function fT(t) = f +� t +T +� satisfies: +HfT(t) = T 2 +π2 +� +|k|≥1 +ak +k2 e +ikπt +T +(5.30) +for any −T ≤ t ≤ T. +Bearing this in mind, we shall find an upper bound on Λq,sup(s) with the help of the +following technical lemma: +Lemma 5.5. Let V ⊂ E be a closed subspace of codimension n, and let λ, ǫ > 0 such that +for all non-zero f ∈ V we have: +� 1 +−1 +(Hf)2(t)dt2 ≤ +1 +(λ + ǫ)2 +� 1 +−1 +f 2(t)dt. +Then for T large enough λm(n+3)+N+bq(MT )(T) ≥ +λ +T 2 . +Proof. The idea is to follow the construction of §4.3 to build a solution of ∆T ν = η, where +η is a q-form orthogonal to the space of harmonic forms, and showing that if we assume +sufficiently many orthogonality conditions we can give a precise bound on the constant C +such that ∥ν∥L2 ≤ CT 2∥η∥L2. To do this we need to introduce a parameter τ > 0 and +replace the cutoffs ζ0 and ζ1 (see §4.2) by ζτ and ζτ+1. +Let us denote ητ = ζτ+1η, considered as a q-form on the cylinder Y = R×X, supported +in the cylinder [−T + τ, T − τ] × X. We pick a basis η1, . . . , ηm of the space of translation- +invariant harmonic q-forms on Y , orthonormal for the L2-product on X. Then we may +write: +ητ(t, x) = �ητ(t, x) + +m +� +j=1 +fτ,j(t)ηj(x) +(5.31) +with �ητ orthogonal to any function of the form f(t)ηj(x) with f compactly supported +smooth function, and fτ,j ∈ L2([−T + τ, T − τ]). Moreover the solution ντ = Qητ of +∆0ν = ητ provided by Theorem 2.10 can be written as (see Example 3.14): +ντ = Qr ∗ ητ + +m +� +j=1 +Hfj,τ · ηj. +(5.32) +with Qr defined as in §3.3. Let us assume that each of the functions +t ∈ [−1, 1] �→ fj,τ((T − τ)t) +(5.33) +belongs to V ⊂ E. This imposes m(n + 3) orthogonality conditions on η. Given (5.28) +the L2-functions Hfj,τ vanish outside of [−T + τ, T − τ], and therefore ντ ∈ L2(ΛqT ∗Y ) +and from (5.30) it satisfies: +∥ντ∥L2 ≤ C∥ητ∥L2 + (T − τ)2 +λ + ǫ +∥ητ∥L2 ≤ +T 2 +λ + ǫ∥ητ∥L2 +(5.34) +for large enough T. Let us consider ζτ as a section of ΛqT ∗MT supported in the neck +region. As such, there exists a constant C > 0, which does not depend on τ, such that: +∥ζτντ∥L2 ≤ 1 + Ce−δτ +λ + ǫ +T 2∥η∥L2 +(5.35) +42 + +Following the method of §4.2, we can write: +η − ∆T (ζτντ) = η1 + η2 +(5.36) +with ηi ∈ L2 +δ′(ΛqT ∗Zi), for some small δ′ > 0 that we fix. Moreover, as in the proof of +Lemma 4.4 to show the bounds: +∥ηi∥L2 +δ′ ≤ Ceδ′τ∥η∥L2 + C′e−δτ∥η∥L2 ≤ C′′T 2e−δτ∥η∥L2 +(5.37) +for T large enough, where δ, δ′ and τ are fixed, and C′′ does not depend on any of these +choices. Up to O(e−δT ) terms, the vanishing of the obstructions to solving fi = ∆iνi with +νi ∈ W 2,2 +δ′ +can be expressed as the vanishing of N linear forms (this is to say that the +coefficients of the characteristic system are linear in η ∈ L2). Thus imposing N additional +orthogonality conditions on η, we can use the same argument as in Proposition 4.8 to show +that there exists ν′ ∈ W 2,2, η′ ∈ L2 such that η −∆T ν′ = η′ with ∥η′∥L2 ≤ CT 2e−δ′T ∥η∥L2 +for some constant C′ possibly depending on δ′ but not on τ or T. From (5.34) and (5.37) +we can moreover deduce a bound: +∥ν′∥L2 ≤ +� +1 + Ce−δτ +λ + ǫ ++ C′e−δτ +� +T 2∥η∥L2 +(5.38) +for large enough T. On the other hand, as η is by assumption orthogonal to the space of +harmonic forms, so is η′ and by Corollary 5.3 there exists ν′′ such that ∆Tν′′ = η′ with a +bound: +∥ν′′∥L2 ≤ CT 2∥η′∥L2 ≤ C′′T 4e−δ′T ∥η∥L2 +(5.39) +for some constant C′′ which does not depend on τ no T large enough. Thus if ν = ν′ + ν′′ +we have ∆T ν = η with a universal bound: +∥ν∥L2 ≤ +� +1 + Ce−δτ +λ + ǫ ++ C′e−δτ + C′′T 2e−δ′T +� +T 2 +(5.40) +for some constants C, C′, C′′ that may depend on the choices of δ, δ′ but not on τ or large +enough T. As ∥GT η∥L2 ≤ ∥ν∥L2 it follows that if τ and T are large enough we have: +∥GT η∥L2 ≤ T 2 +λ ∥η∥L2. +(5.41) +This inequality holds true provided η satisfies all the orthogonality conditions described +above, which define a closed subspace of codimension less or equal than m(n + 3) + N + +bq(MT ) in L2(ΛqT ∗MT ). The lemma follows. +To use this lemma we consider the subspaces V ′ +n ⊂ E defined by: +V ′ +n = +� +f(t) = +� +akeikπt ∈ E, ak = 0 ∀|k| ≤ n +� +. +(5.42) +The space V ′ +n has codimension 2n in E, and moreover for any f ∈ V ′ +n (5.30) implies: +� 1 +−1 +(Hf)2(t)dt2 ≤ +1 +(n + 1)4π4 +� 1 +−1 +f 2(t)dt +(5.43) +Hence we have an upper bound: +Λq,sup(s) ≤ 2(⌊√s⌋ + 2)(bq−1(X) + bq(X)) + N + bq(MT ). +(5.44) +Together with the bound on Λq,inf(s) this proves Theorem 2.8. We even have a slightly +stronger statement, that asymptotically both Λq,inf(s) and Λq,sup are in fact equal to: +2(bq−1(X) + bq(X))√s + O(1). +(5.45) +43 + +(5.2.3) +In order to prove the second assertion in Theorem 2.8, let us consider the subset +Wn ⊂ Vn defined by: +Wn = + + + +� +1≤|k|≤n +akeikπt ∈ Vn, +� +1≤|k|≤n +(−1)kk2ak = 0 + + + +(5.46) +seen as a subspace of C3([−1, 1], R). Any f ∈ Wn can e extended as a C2-function on R, +with f ′ ∈ Vn. Moreover if β is a harmonic (q − 1)-form then: +d(fβ) = f ′dt ∧ β. +(5.47) +Using this, we can deduce that the density of low eigenvalues of the Laplacian acting on +co-exact q-forms, which we define as in (2.3.2), satisfies: +Λe +q,inf(s) ≥ 2bq−1(X)√s − Nq +(5.48) +for some constant Nq ≥ 0. By Hodge duality, this implies the lower bound: +Λ∗ +q,inf(s) ≥ 2bq(X)√s − Ndim MT −q. +(5.49) +As we know that Λ∗ +q,sup(s) + Λe +q,sup(s) ≤ 2(bq−1(X) + bq(X))√s + O(1), this means that +when bq(X) ̸= 0 we do have: +Λ∗ +q,inf(s) ∼ 2bq(X)√s ∼ Λ∗ +q,sup(s). +(5.50) +If on the other hand bq(X) = 0 then Λ∗ +q,sup(s) = O(1), so that only finitely many eigen- +values of the Laplacian acting on co-exact q-forms may decay at rate T −2, the rest of the +low eigenvalues would decay at a slower rate. +(5.2.4) +It is natural to wonder whether stronger statements about the distribution of +low eigenvalues could be made, and in particular one could ask whether the spectrum +of the Laplacian splits with a lower part represented by the spectrum of the Laplacian +acting on S1 +2T × X, where the circle factor has length 2T. Even in the simple case of the +Laplacian acting on functions this does not hold. Considering the function ρT and applying +a min-max argument as above one can easily see that for large enough T the lowest non- +zero eigenvalue of the scalar Laplacian satisfies λ1(T) ≤ +6 +T 2 . In general, it seems that +the interactions between the two building blocks tend to shift the lower spectrum of ∆T +compared with the spectrum of the Laplacian acting on S1 +2T ×X. The precise way in which +this shift happens is likely to depend on the building blocks, and it may not be tractable +analytically to give sharp bounds for the sequence of low eigenvalues of the Laplacian +acting on forms. +To finish this section, let us discuss possible generalisations of Theorem 2.8. If we +consider more general formally self-adjoint operators PT that fit into the set-up of The- +orem 2.6, the substitute kernel of PT can be interpreted as a finite number of very low +eigenvalues, with decay rate exponential in T. The rest of the eigenvalues have at most +polynomial decay, and in fact one could easily see that this decay rate is in T −d (except +for a finite number that may have faster polynomial decay), where d is the maximal order +of the real roots of the indicial operator P0. Assuming that PT has non-negative eigen- +values and that P0 can be written as P0(λ) = D + λd, the proof given above carries out +without problem to show that the density of eigenvalues of PT contained in the interval +� +0, πdsT −d� +is equivalent to 2(dim ker D)s +1 +d as s → ∞. +‘ +44 + +6 +Improved estimates for twisted connected sums +In this section, we apply our results to the study of compact manifolds with holonomy G2 +constructed by twisted connected sum. We review the basics of G2-geometry in §6.1. In +§6.2 we prove Corollary 2.9, using quadratic estimates which we derive in §6.3. +6.1 +G2-manifolds +(6.1.1) +Let us introduce G2-structures by some linear algebra. Let V be an oriented real +vector space of dimension 7. A 3-form ϕ ∈ Λ3V ∗ is said to be positive if for any non-zero +v ∈ V we have: +ιvϕ ∧ ιvϕ ∧ ϕ > 0 +(6.1) +relatively to the choice of orientation, where ι denotes the interior product. Let Λ3 ++V ∗ +denote the set of positive forms on V . This is an open subset of Λ3V ∗ and the group of +oriented automorphisms GL+(V ) acts transitively on it. The stabiliser of any positive form +under this action is identified with the group G2, which is a simply-connected semi-simple +Lie group of dimension 14. +If Vol is a volume form on V and ϕ a positive 3-form, we have +ιuϕ ∧ ιvϕ ∧ ϕ = ⟨u, v⟩ Vol +(6.2) +for some inner product ⟨·, ·⟩, but there is an ambiguity in the scaling of ⟨·, ·⟩ and Vol. +This can be fixed by requiring |ϕ|2 = 7, and hence any positive form ϕ canonically defines +a volume form Volϕ and an inner product gϕ on V . With this choice, Volϕ is also the +volume form associated with gϕ. The maps ϕ �→ Volϕ and ϕ �→ gϕ are equivariant under +the action of GL+(V ), and in particular the stabilizer of any positive form also stabilises +the associated volume form and inner product. This shows that G2 ⊂ SO(7). There is +another equivariant map commonly denoted by Θ, which associates to ϕ its Hodge dual +for the associated inner product, that is Θ(ϕ) = ∗ϕϕ. This map is non-linear. +A G2-structure on an oriented manifold M of dimension 7 corresponds to the choice of +a 3-form ϕ such that ϕx ∈ Λ3 ++T ∗ +xM for all x ∈ M. As in the linear case a G2-structure ϕ +naturally induces a Riemannian metric gϕ, a volume form Volϕ and a 4-form Θ(ϕ) on M. +A G2-structure ϕ on M is called torsion-free is ∇ϕϕ ≡ 0 for the Levi-Civita connection +∇ϕ of gϕ. A manifold equipped with a torsion-free G2-structure is called a G2-manifold. +As was first shown in [14], the torsion-free condition for ϕ is equivalent to: +dϕ = 0 = dΘ(ϕ). +(6.3) +The non-linearity of this condition accounts for the difficulty of finding non-trivial G2- +metrics. The torsion-free condition on ϕ implies that the holonomy of gϕ is contained +in G2 ⊂ SO(7), up to conjugation in SO(7). +An interesting feature of metrics with +holonomy contained in G2 is that they are automatically Ricci-flat. Combined with the +Cheeger–Gromoll splitting theorem [9], this implies the following: +Proposition 6.1. A compact G2-manifold (M7, ϕ) has full holonomy G2 if and only if +π1(M) is finite. +(6.1.2) +As many interesting features of the geometry of G2-manifolds are best explained +at the level of linear algebra, we describe below some of the representations of G2. Let V +be an oriented 7-dimensional real vector space and ϕ be a positive form and identify the +stabiliser of ϕ with G2. +45 + +The representations V and V ∗ are irreducible, as G2 acts transitively on the unit sphere +in V . The representation Λ2V ∗ is not irreducible but decomposes as: +Λ2V ∗ = Λ2 +7 ⊕ Λ2 +14 +(6.4) +where Λ2 +14 is isomorphic to the Lie algebra of G2 and Λ2 +7 ≃ V ∗ is its orthogonal complement +in Λ2V ∗ identified with the Lie algebra of SO(7). +To decompose Λ3V ∗, the derivative of the action of GL+(V ) on ϕ yields an equivariant +map End(V ) → Λ3V ∗, and as the orbit of ϕ is open in Λ3V ∗ it follows that the map is +onto and its kernel is the Lie algebra of G2. On the other hand we have: +End(V ) ≃ V ∗ ⊗ V ∗ ≃ Λ2V ∗ ⊕ S2V ∗ ≃ Λ2 +7 ⊕ Λ2 +14 ⊕ Rgϕ ⊕ S2 +0V ∗ +(6.5) +where S2 +0V ∗ denote the space of trace-free symmetric bilinear forms, which has dimension +27. Hence we have an irreducible decomposition: +Λ3V ∗ = Λ3 +1 ⊕ Λ3 +7 ⊕ Λ3 +27. +(6.6) +As the Hodge star operator gives isomorphisms ΛkV ∗ ≃ Λ7−kV ∗ we obtain a full decom- +position of the exterior algebra of V ∗. +On a G2-manifold (M, ϕ), the decomposition ΛkV ∗ ≃ ⊕Λk +m induces a decomposition +of the space of k-forms Ωk(M) ≃ ⊕Ωk +m. It follows from the Weitzenböck formula that +the Laplacian operator leaves this decomposition invariant. Thus on compact manifolds +Hodge theory yields a decomposition of the cohomology groups Hk(M; R) ≃ ⊕Hk +m(M). +With these identifications, the torsion of ϕ (seen as the obstruction for the Levi-Civita +connection of gϕ to be compatible with ϕ) can be identified with a 1-form valued in +the bundle Λ2 +7M [14], which we denote by τ(ϕ). More precisely, if ∇′ is any connection +compatible with ϕ, and we write the Levi-Civita connection ∇ = ∇′ + A for some 1- +form A ∈ Ω1(Λ2T ∗M), then we can define τ(ϕ) as the orthogonal projection of A onto +T ∗M ⊗ Λ2 +7M (which does not depend on the choice of ∇′). In particular the connection +�∇ defined by: +�∇ = ∇ − τ(ϕ) +(6.7) +is compatible with ϕ. This observation will be useful later as form some purposes it is more +convenient to work with a connection compatible with ϕ rather than with the Levi-Civita +connection, when ϕ is not torsion-free. +(6.1.3) +Let X be a complex manifold equipped with a Calabi–Yau structure (ω, Ω), +where ω ∈ Ω2(X) is the Kähler form and Ω ∈ Ω3 +C(X) is the holomorphic volume form. +Then Y = R × X is naturally equipped with the G2-structure: +ϕ = dt ∧ ω + Re(Ω). +(6.8) +The associated G2-metric is the product metric gϕ = dt2+gX, where gX is the Calabi–Yau +metric on X. The Hodge dual of ϕ takes the form: +Θ(ϕ) = 1 +2ω2 − dt ∧ Im Ω. +(6.9) +In particular if X is compact then (Y, ϕ) is called a G2-cylinder. Moreover any translation- +invariant G2-metric on Y is obtained in this way. +A non-compact G2-manifold (Z, ϕ) is called an EAC G2-manifold of rate µ > 0 if it is an +EAC Riemannian manifold of rate µ, and there exists a translation-invariant G2-structure +ϕ0 on the asymptotic cylinder Y = R × X such that ϕ − ϕ0 and all their derivatives +are O(e−µρ) at infinity, for any cylindrical coordinate function ρ on Z. In particular this +implies that Θ(ϕ) − Θ(ϕ0) satisfies similar bounds. +46 + +6.2 +The twisted connected sum construction +(6.2.1) +All known constructions of compact manifolds with holonomy G2 are based on +an argument of Joyce [20, §10.3], which we outline here. The starting point is to consider +a compact manifold M7 equipped with a closed G2-structure ϕ with appropriately small +torsion, and look for a nearby torsion-free G2-structure �ϕ = ϕ+dη in the same cohomology +class. As the condition defining G2-structures is open, there exists a universal constant ǫ0 +such that if ξ ∈ Ω3(M) satisfies ∥ξ∥C0 ≤ ǫ0 then ϕ+ξ is also a G2-structure. In particular +Θ(ϕ + ξ) is well-defined and can be written as: +Θ(ϕ + ξ) = Θ(ϕ) + Lϕξ + F(ξ) +(6.10) +where Lϕ is the linearisation of Θ at ϕ and F a smooth function defined on a ball of radius +ǫ0 in Λ3T ∗M. In particular F satisfies a bound of the form: +|F(ξ) − F(ξ′)| ≤ C|ξ − ξ′|(|ξ| + |ξ′|), +|ξ|, |ξ′| ≤ ǫ0 +(6.11) +for some universal constant C [20, Proposition 10.3.5]. With these notations, there exists +a universal constant ǫ1, which does not depend on M or ϕ, such that the following holds +[20, Theorem 10.3.7]: +Theorem 6.2. Let (M, ϕ) be a compact manifold equipped with a closed G2-structure. +Suppose η is a 2-form on M such that ∥dη∥C0 ≤ ǫ1 and ψ a 4-form on M such that +dΘ(ϕ) = ∗dψ and ∥ψ∥C0 ≤ ǫ1. If (η, ψ) satisfy: +∆η + d +�� +1 + 1 +3⟨dη, ϕ⟩ +� +ψ +� ++ ∗dF(dη) = 0 +then �ϕ = ϕ + dη is a torsion-free G2-structure on M. +With the above theorem, one can therefore start with a compact manifold equipped +with a closed G2-structure with small torsion and apply a fixed point theorem in order +to deform it to a nearby torsion-free G2-structure in the same cohomology class. This +involves a good understanding of the linearised problem, as well as some control on the +non-linear part F in the form of quadratic estimates. +(6.2.2) +Let us now outline the twisted connected sum construction. The building blocks +are a pair of EAC G2-manifolds (Z1, ϕ1) and (Z2, ϕ2) of rate µ > 0, asymptotic to the +cylinder R × X. +Denote ϕ0,i the asymptotic translation-invariant model for ϕi. +The +G2-structures ϕ1 and ϕ2 are said to be matching if there exists an isometry γ of the +cross-section X such that the map +γ : R × X → R × X, +(t, x) �→ (−t, γ(x)) +(6.12) +satisfies γ∗ϕ0,2 = ϕ0,1. If γ0,i = dt ∧ ω0,i + Re Ω0,i for Calabi-Yau structure (ω0,i, Ω0,i) on +X then the matching condition amounts to: +γ∗ω0,2 = −ω0,1, +γ∗Ω0,2 = Ω0,1. +(6.13) +In all known examples [25, 11, 33] such matching pairs are trivial circle bundles over EAC +Calabi-Yau threefolds (or some quotient of it), and the cross-section is isometric to the +product of a K3 surface with a flat 2-torus (or a corresponding quotient). Moreover, the +map γ is designed so that the family of compact manifolds MT obtained by gluing Z1 and +47 + +Z2 along γ (see Remark 2.3) has finite fundamental group, in order to construct manifolds +with full holonomy G2. The details of the construction of such matching pairs go beyond +the scope of the present paper and do not affect our analysis, so we refer to the original +references for more details. +Let us denote by σ the minimum of µ and of the smallest non-trivial eigenvalue of the +Laplacian acting on 2- and 3-forms on X. As we have seen in (5.1.2), the closed forms ϕi +and Θ(ϕ) admit admit an expansion: +ϕi = ϕi,0 + dηi, +Θ(ϕi) = Θ(ϕi,0) + dξi +(6.14) +where ηi ∈ Ω2(Zi), ξi ∈ Ω3(Zi) and all their covariant derivatives have exponential decay +in O(e−δρi), where ρi are cylindrical coordinate functions on Zi and 0 < δ < σ. Thus +picking a cutoff function χ : R → [0, 1] such that χ ≡ 0 in (−∞, − 1 +2] and χ ≡ 1 in [1 +2, ∞) +we can build 1-parameter families of closed forms: +ϕi,T = ϕi − d(χ(ρi − T − 1)ηi), +Θi,T = Θ(ϕi) − d(χ(ρi − T − 1)ξi) +(6.15) +As ∥ϕ − ϕi,T ∥C0 has exponential decay with T, it follows that for T large enough ϕi,T is +a G2-structure, which is closed by construction. Thus by patching up ϕ1,T with ϕ2,T and +Θ1,T with Θ2,T as in (5.1.2), the 1-parameter family of compact manifolds MT obtained by +gluing Z1 and Z2 along γ is endowed with a family of closed G2-structures ϕT and a closed +4-forms ΘT . Let us denote ψT = Θ(ϕT ) − ΘT , so that dψT = dΘ(ϕT ). By construction, +we have estimates of the form: +∥ψT ∥Ck = O +� +e−δT � +(6.16) +for any k ≥ 0 and 0 < δ < σ, and similarly with Sobolev norms [25, Lemma 4.25]. +Thus for T large enough we are in the correct setup to apply Theorem 6.2 an seek +solutions η ∈ Ω2(MT ) with ∥dη∥C0 ≤ ǫ1 solving the equation: +∆T η + ∗d +�� +1 + 1 +3⟨dη, ϕT ⟩ +� +ψT +� ++ ∗dF(dη) = 0. +(6.17) +It follows from [20, Theorem 11.6.1] that for T large enough this equation admits a solution +ηT , so that �ϕT = ϕT + dηT is a torsion-free G2-structure on MT . However, the proof of +[25, Theorem 5.34] that there are estimates of the form ∥ �ϕT − ϕT ∥C0 = O(e−δT ) seems +to be incorrect as the asymptotic kernel considered for the linearised problem does not +have the dimension of the space of harmonic 2-forms on MT ; it has the dimension of +the kernel of the Laplacian acting on decaying 2-forms on Z1 plus the dimension of the +kernel of the Laplacian acting on 2-forms with less than exponential growth on Z2 in +general. In particular as the cross-section has non-zero first and second Betty numbers +this asymptotic kernel is never trivial, whereas there are examples of twisted connected +sums with b2(MT ) = 0. +Using Theorem 2.6 and Corollary 5.3 we are able to fix this +issue. Moreover, we can even obtain a uniform control on all the derivatives of �ϕT − ϕT +in O(e−δT ) for any 0 < δ < σ (Corollary 2.9). +(6.2.3) +Let us know explain how to set up a fixed-point argument to prove these esti- +mates. For T large enough let us consider the differential operator PT : C∞(Λ2T ∗MT ) → +C∞(Λ2T ∗MT ) defined by: +PT η = ∆Tη + 1 +3 ∗ d(⟨dη, ϕT ⟩ψT ). +(6.18) +48 + +Then PT maps into the L2-orthogonal space to harmonic 2-forms, and moreover give the +decay of ψT and all its derivatives we have for any k ≥ 0 and p > 1 a bound of the form +∥(PT − ∆T )η∥W k,p = O(e−δT ∥η∥W 2+k,p). Thus it follows from Corollary 5.3 that for T +large enough, the map: +H 2(MT )⊥ ∩ W 2+k,p(Λ2T ∗MT ) → H 2(MT )⊥ ∩ W k,p(Λ2T ∗MT ) +(6.19) +induced by PT admits a bounded inverse QT , with norm satisfying +∥QT ∥ ≤ CT 2 +(6.20) +for some constant C > 0. On the other hand, we also have a control: +∥ ∗ dψT ∥W k,p ≤ C′e−δT +(6.21) +for any δ < σ from (6.16). +It remains to obtain a good control on the non-linear part of (6.17). The key result +for our purpose are the following quadratic estimates: +Proposition 6.3. Let p ≥ 7 and k ≥ 1 be an integer. Then there exists constants ǫk,p > 0 +and Ck,p > 0 such that for T large enough, if ξ, ξ′ ∈ W k,p(Λ3T ∗MT ) satisfy the condition +∥ξ∥W k,p, ∥ξ′∥W k,p ≤ ǫk,p then: +∥F(ξ) − F(ξ′)∥W k,p ≤ Ck,p∥ξ − ξ′∥W k,p(∥ξ∥W k,p + ∥ξ′∥W k,p). +Remark 6.4. This condition in particular contains the fact that F(ξ) is well-defined in +a neighborhood of 0 in W k,p. Essentially we need p ≥ 7 and k ≥ 1 in order to have a +Sobolev embedding W k,p(Λ3T ∗MT ) ֒→ Ck−1(Λ3T ∗MT ). Recall moreover that the norm +of this embedding is uniformly bounded for T large enough. Thus if the W k,p-norm of ξ +is small enough then its C0-norm is small enough so that F(ξ) is indeed defined. +We shall prove the above quadratic estimates in the next part. +Once they are es- +tablished, Corollary 2.9 follows by applying a contraction-mapping argument to the map +η �→ PT η + ∗dF(dη) defined on an appropriately small ball in the orthogonal space to +H 2(MT ) in W 2+k,p(Λ2T ∗MT ), for some choice of p ≥ 7 and k ≥ 1. By (6.20) and (6.21), +for small enough ǫ and large enough T equation (6.17) has a unique solution ηT in a ball of +radius +ǫ +T 2 centered at 0, which moreover satisfies ∥ηT ∥W 2+k,p = O(e−δT ) for any 0 < δ < σ. +As the norm of the Sobolev embedding W 1+k,p ֒→ Ck is uniformly bounded (Remark 2.5) +this implies that �ϕT = ϕT + dηT satisfies ∥ �ϕT − ϕT ∥Ck = O(e−δT ) as T → ∞. +6.3 +Proof of the quadratic estimates +(6.3.1) +In order to prove Proposition 6.3 it is useful to work at some level of generality. +Essentially, what we need is to improve [20, Proposition 10.3.5] in order to control more +derivatives. The key property that allows us to obtain good estimates on the derivatives +of F is the equivariance of the map Θ under the action of GL+(7, R). Thus we consider +the following setting. Let M be an oriented n-dimensional manifold, which for now need +not be compact, or even complete, as we will mostly work locally. Let us denote P the +oriented frame bundle of M, considered as a GL+(n, R)-bundle. Let G ⊂ SO(n) be a +compact subgroup, and Q ⊂ P a G-structure on M, so that it induces a Riemannian +metric g on M. We moreover implicitly endow all representations of G with an invariant +inner product, and consider the corresponding metrics on associated bundles. +49 + +With the above data, let (ρ0, V0), (ρ1, V1) be representations of GL+(n, R) and denote +Ei = P×ρiVi the associated bundles on M. Assume that there exists an open set O0 ⊂ V0, +which is stable under ρ0, and let Υ : O0 → V1 be a (possibly non-linear) smooth map which +is equivariant under the action of GL+(n, R), that is: +Υ(ρ0(g)u) = ρ1(g)Υ(u), +∀g ∈ GL+(n, R). +(6.22) +In the case we are interested in, n = 7, the group G is identified with G2, and our +equivariant map is Υ = Θ. +In general, we can consider U0 = P ×ρ0 O0 as an open +subbundle of E0, and Υ induces a bundle map U0 → V1, which we still denote by Υ. Let +us remark that the differential map: +DΥ : O0 → End(V0, V1) +(6.23) +is also G-equivariant. Indeed, if we differentiate (6.22) with respect to u we obtain: +Dρ0(g)uΥ(ρ0(g) ˙u) = ρ1(g)DuΥ( ˙u), +∀g ∈ GL+(n, R) and ∀ ˙u ∈ V0. +(6.24) +Thus we have in fact a whole family of G-equivariant maps +DkΥ : O0 → V ∗ +0 ⊗ · · · ⊗ V ∗ +0 ⊗ V1 +(6.25) +and induced maps on the corresponding bundles. +Let ∇ be a connection on M which is compatible with Q. In particular, we can find +trivialisations of P with transition maps valued in G and local connection forms valued +in its Lie algebra g. If u is a local section of u then we can compute in local trivialisations: +∇Υ(u) = dΥ(u) + dρ1(A)Υ(u) +(6.26) += DuΥ(du) + DuΥ(dρ0(A)u) +(6.27) += DuΥ(∇u) +(6.28) +where we denote by A a local connection form, and differentiate (6.22) with respect to g +this time to obtain the identity: +dρ1(a)Υ(u) = DuΥ(dρ0(a)u), +∀a ∈ gln and u ∈ O0. +(6.29) +Given the equivariance of the maps DlΥ, we can deduce by iteration that: +∇kF(u) = +k +� +l=1 +� +j1+···+jl=k, ji≥1 +Cj1...jlDl +uΥ(∇j1u, ..., ∇jlu) +(6.30) +where Cj1...jl are universal combinatorial coefficients. Suppose now that u0 ∈ O0 is invari- +ant under the action of G. Then it induces a section of U which is parallel for ∇, and in +particular the above computations imply: +∇u0 = ∇Υ(u0) = ∇Du0Υ = 0. +(6.31) +As O0 is open in V0, there exists a universal constant ǫ0 > 0 such that if u is a section of +E0 with ∥u∥C0 ≤ ǫ0 then u0 + u is a section of U, where we measure the C0-norm with +respect to the bundle metric induced by Q. Thus if ∥u∥C0 ≤ ǫ0 we may write: +∇k(Υ(u0 + u) − Υ(u0) − Du0Υ(u)) = (Du0+uΥ − Du0Υ)(∇ku) ++ +k +� +l=2 +� +j1,...,jl +Cj1...jlDl +u0+uΥ(∇j1u, . . . , ∇jlu). +(6.32) +50 + +Recall that in local trivialisations we may just consider Dl +u0+uΥ as fixed maps defined on +a small ball of radius ǫ0 in V0. Thus for any k there exists a universal 0 < ǫk ≤ ǫ0 such +that if u, v are contained in the ball of radius ǫk, there are estimates of the form: +|Du0+uΥ − Du0Υ| ≤ C|u|, +|Dl +u0+uΥ| ≤ C′ +l +(6.33) +and +|Dl +u0+uΥ − Dl +u0+vΥ| ≤ C′′ +l |u − v| +(6.34) +for some universal constants possibly depending on the choice of ǫk. +(6.3.2) +Let us now denote R(u) = Υ(u0 + u) − Υ(u0) − Du0Υ(u), defined on a ball of +radius ǫ0 in V0. We can also consider R(u) as defined for any section of V0 with C0-norm +less than ǫ0. Note that it does depend on u0, which we consider either as a fixed element +of O0 or a parallel section of V0. We want local bounds for the quantity |∇k(R(u)−R(v))|. +Assuming that u, v have C0-norm less than ǫk as above, we can first estimate the term: +|(Du0+uΥ − Du0Υ)(∇ku) − (Du0+vΥ − Du0Υ)(∇kv)| ≤ |(Du0+uΥ − Du0+vΥ)(∇ku)| ++ |(Du0+vΥ − Du0Υ)(∇ku − ∇kv)|. +(6.35) +Using the previous inequalities we obtain an estimate of the form: +|(Du0+uΥ − Du0Υ)(∇ku) − (Du0+vΥ − Du0Υ)(∇kv)| ≤ C(|u − v| · |∇ku|) ++ C′(|v| · |∇ku − ∇kv|) +(6.36) +for some universal constants depending only on the choice of ǫk. Similarly, for l ≥ 2 we +have universal estimates of the form: +|Dl +u0+uΥ(∇j1u, . . . , ∇jlu) − Dl +u0+vΥ(∇j1v, . . . , ∇jlv)| ≤ C(|u − v| · |∇j1u| · · · |∇jlu|) ++ C1|∇j1u − ∇j1v| · |∇j2u| · · · |∇jlu| + · · · + Cl|∇j1v| · · · |∇jl−1v| · |∇jlu − ∇jlv| +(6.37) +for some constants which only depend on ǫk. Each term in the above sum only contains +factors of |∇ju| or |∇jv| with 1 ≤ j ≤ k − 1, so that the only terms containing factors +|∇ku|, |∇kv| or |∇ku − ∇kv| come from (6.36). Thus, if we impose not only |u|, |v| ≤ ǫk +but also � +0≤l≤k−1 |∇lu| ≤ ǫ′ +k and similarly for v, form some arbitrary choice of ǫ′ +k ≤ ǫk, +we have overall an estimate: +|∇k(R(u) − R(v))| ≤ C|u − v| · (|∇ku| + |∇kv|) + C′|∇ku − ∇kv| · (|u| + |v|) ++ C′′ +� +0≤l,l′≤k−1 +|∇lu − ∇lv| · (|∇l′u| + |∇l′v|) +(6.38) +for some universal constants C, C′, C′′ depending only on our choice of ǫ′ +k. +As a consequence, globally if u satisfies ∥u∥Ck−1 ≤ ǫ′ +k and similarly for v, and u, v are +in W k,p for some p > 1 we obtain: +∥∇k(R(u) − R(v))∥Lp ≤ C∥u − v∥Ck−1 · (∥u∥W k,p + ∥v∥W k,p) ++ C′(∥u∥Ck−1 + ∥v∥Ck−1)∥u − v∥W k,p +(6.39) +for some uniform constants. In particular if we are on a compact manifold and p ≥ n, +then the W k,p-norms controls the Ck−1 norm. +Thus there exists an ǫk,p such that if +51 + +∥u∥W k,p ≤ ǫk,p then ∥u∥Ck−1 ≤ ǫ′ +k, so that F(u) is well-defined and in W k,p. If moreover +v is another section satisfying ∥v∥W k,p ≤ ǫk,p then we have an estimate: +∥R(u) − R(v)∥W k,p ≤ Ck,p∥u − v∥W k,p(∥u∥W k,p + ∥v∥W k,p). +(6.40) +for some constant Ck,p. Note however that ǫk,p and Ck,p are not universal, they do depend +on the manifold (M, g) through the constant in the Sobolev embedding W k,p ֒→ Ck−1. +Moreover, it is important to note that to obtain such estimates we implicitly redefine the +W k,p and Ck-norms using the compatible connection ∇, and not the Levi-Civita connection +of the metric induced by Q on M. This yields equivalent definitions, but to use the above +estimates we also need to take into account the extra constants coming from the fact that +we are dealing with a different definition of the usual norms. +(6.3.3) +In the case we are interested in, the constant in the Sobolev embedding W k,p ֒→ +Ck−1 on (MT , ϕT , gT ) can be chosen to be independent of T ≥ 1 (see Remark 2.5). +However, we cannot directly apply the above argument as the Levi-Civita connection ∇T +is not compatible with the G2-structure ϕT . Nevertheless, we have seen in (6.1.2) that the +torsion of ϕT , which is represented here by dΘ(ϕT ), can be identified with a 1-form τ(ϕT ) +valued in the bundle Λ2 +7MT . Moreover the connection �∇T = ∇T − τ(ϕT ) is compatible +with ϕT . The torsion τ(ϕT ) satisfies +���∇l +T τ(ϕT ) +��� +C0 = O +� +e−δT � +(6.41) +for k ≥ 0 and any small enough δ > 0. 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