AN OPEN FOUR-MANIFOLD HAVING NO INSTANTON

N OPEN FOUR-MNIFOLD HVING NO INSNON MSKI SUKMOO bstract. aubes proved that all compact oriented four-manifolds admit non-flat instantons. We show that there exists a non-compact oriented four-manifold having no non-flat instanton. 1. introduction aubes [15] proved that all compact oriented Riemannian 4-manifolds admit non-flat instantons. o be precise, if X is a compact oriented Riemannian 4-manifold then there exists a principal SU(2)-bundle E on X which admits a non-flat anti-self-dual (SD) connection. (aubes [15] considered self-dual connections. But recently people usually study anti-self-dual ones. So we consider anti-self-dual connections in this paper.) he purpose of this paper is to show that an analogue of this striing existence theorem does not hold for general non-compact 4-manifolds. Let (CP 2 ) Z be the connected sum of the infinite copies of the complex projective plane CP 2 indexed by integers. (he precise definition of this infinite connected sum will be given in Section 2.1.) (CP 2 ) Z is a non-compact oriented 4-manifold. heorem 1.1. here exists a complete Riemannian metric g on (CP 2 ) Z satisfying the following. If is a g-sd connection on a principal SU(2)-bundle over (CP 2 ) Z satisfying (1) F 2 gdvol g < +, X then is flat. Here F is the curvature of. g and dvol g are the norm and the volume form with respect to the metric g. connection is said to be g-sd if it satisfies g F = F where g is the Hodge star with respect to g. For a more general and precise statement, see heorem 2.1. s far as I now, this is the first example of oriented Riemannian 4-manifolds which cannot admit any non-flat instanton. Remar 1.2. I thin that the following question is still open: Is there an oriented Riemannian 4-manifold which does not have any non-flat SD connection (not necessarily Date: pril 21, 2010. 2000 Mathematics Subject Classification. 53C07. Key words and phrases. Yang-Mills theory, instanton, open four-manifold, infinite connected sum. 1

2 M. SUKMOO satisfying the finite energy condition (1))? We studied infinite energy SD connections and their infinite dimensional moduli spaces in [11], [17], [18]. naive idea toward the proof of heorem 1.1 is as follows. Let g be a Riemannian metric on (CP 2 ) Z. For each integer n 0, let M(n, g) be the moduli space of SU(2) g-sd connections on (CP 2 ) Z satisfying F (CP 2 ) Z 2 gdvol g = 8π 2 n. We have b 1 ((CP 2 ) Z ) = 0 and, formally, b + ((CP 2 ) Z ) = +. herefore, if we formally apply the usual virtual dimension formula [5, Section 4.2.5] to M(n, g), we get dim M(n, g) = 8n 3(1 b 1 ((CP 2 ) Z ) + b + ((CP 2 ) Z )) = 8n =. his suggests the following observation: If we can achieve the transversality of the moduli spaces M(n, g) by choosing the metric g sufficiently generic, then all M(n, g) (n 1) become empty. (M(0, g) is the moduli space of flat SU(2) connections, and it does not depend on the choice of a Riemannian metric.) cnowledgement. I wish to than Professor Kenji Fuaya most sincerely for his help and encouragement. I was supported by Grant-in-id for Young Scientists (B) (21740048). 2. Infinite connected sum 2.1. Construction. Let Y be a simply-connected compact oriented 4-manifold. Let x 1, x 2 Y be two distinct points, and set Ŷ := Y \ {x 1, x 2 }. Choose a Riemannian metric h on Ŷ which becomes a tubular metric on the end (i.e. around x 1 and x 2 ). his means that there is a compact set K Ŷ such that Ŷ \K = Y Y + with Y = (, 1) S 3 and Y + = (1, + ) S 3. Here = means that they are isomorphic as oriented Riemannian manifolds. (S 3 = S 3 (1) = {x R 4 x = 1} is endowed with the Riemannian metric induced by the standard Euclidean metric on R 4.) We can suppose that there is a smooth function p : Ŷ R satisfying the following conditions: p(k) = [ 1, 1]. p is equal to the projection to (, 1) on Y = (, 1) S 3, and p is equal to the projection to (1, + ) on Y + = (1, + ) S 3. For > 2, we set Y := p 1 ( + 1, 1) = ( + 1, 1) S 3 K (1, 1) S 3. (Later we will choose large.) Let Y (n) be the copies of Y indexed by integers n Z. We denote K (n), Y (n), Y (n) +, p (n), as the copies of K, Y, Y +, p, Y. (K (n), Y (n), Y (n) +, Y (n) Y (n) and p (n) : Y (n) R.) Y (n) We define X = Y Z by where we identify Y (n) (2) Y (n) X := n Z Y (n) /, Y (n) + with Y (n+1) Y (n+1) by Y (n) + = (1, 1) S 3 (t, θ) (t, θ) ( + 1, 1) S 3 = Y (n+1) Y (n+1).

N OPEN FOUR-MNIFOLD HVING NO INSNON 3 We define q : X R by setting q(x) := n + p (n) (x) on Y (n). his is compatible with the above identification (2). he identification (2) is an orientation preserving isometry. Hence X has an orientation and a Riemannian metric which coincide with the given ones over Y (n). We denote the Riemannian metric on X (given by this procedure) by g 0. g 0 depends on the Riemannian metric h on Y and the parameter. Since Y is simply-connected, X is also simply-connected. he homology groups of X are given as follows: H 0 (X) = Z, H 1 (X) = 0, H 2 (X) = H 2 (Y ) Z, H 3 (X) = Z, H 4 (X) = 0. H 2 (X) is of infinite ran if b 2 (Y ) 1. For every n Z, the inclusion Y (n) Y (n) + X induces an isomorphism H 3 (Y (n) Y (n) + ) = H 3 (X). he fundamental class of the crosssection S 3 Y (n) Y (n) + = (1, 1) S 3 becomes a generator of H 3 (X). 2.2. Statement of the main theorem. heorem 1.1 in Section 1 follows from the following theorem. heorem 2.1. Suppose b (Y ) = 0 and b + (Y ) 1. If is sufficiently large, then there exists a complete Riemannian metric g on X = Y Z satisfying the following conditions (a) and (b). (a) g is equal to the periodic metric g 0 (defined in Section 2.1) outside a compact set. (b) If is a g-sd connection on a principal SU(2) bundle E on X satisfying (3) F 2 gdvol g <, then is flat. he proof of this theorem will be given in Section 9. X Remar 2.2. (i) If a Riemannian metric g on X satisfies the condition (a), then it is complete. (ii) From the condition (a), the above (3) is equivalent to F 2 g 0 dvol g0 <. X (iii) Since X is non-compact, all principal SU(2)-bundles on it are isomorphic to the product bundle X SU(2). Hence we can assume that the principal SU(2)-bundle E in the condition (b) is equal to the product bundle X SU(2). 2.3. Ideas of the proof of heorem 2.1. In this subsection we explain the ideas of the proof of heorem 2.1. Here we ignore several technical issues. Hence the real proof is different from the following argument in many points. Let g be a Riemannian metric on X which is equal to g 0 outside a compact set. Let E = X SU(2) be the product principal SU(2)-bundle over X. If a g-sd connection

4 M. SUKMOO on E satisfies F X 2 1 gdvol g <, then we can show that 8π X F 2 gdvol 2 g is a nonnegative integer. For each integer n 0, we define M(n, g) as the moduli space of g-sd connections on E satisfying 1 8π X F 2 gdvol 2 g = n. ae [] M(n, g). We want to study a local structure of M(n, g) around []. Set D := d + d+ g : Ω1 (ade) (Ω 0 Ω + g )(ade). Here d is the formal adjoint of d : Ω 0 (ade) Ω 1 (ade) with respect to g 0, and d + g is the g-self-dual part of d : Ω 1 (ade) Ω 2 (ade). (Indeed we need to use appropriate weighted Sobolev spaces, and the definition of D should be modified with the weight. But here we ignore these points.) he equation d a = 0 for a Ω1 (ade) is the Coulomb gauge condition, and the equation d +g a = 0 is the linearization of the SD equation F +g ( + a) = 0. herefore we expect that we can get an information on the local structure of M(n, g) from the study of the operator D. he most important point of the proof is to show the following three properties of D. (In other words, we need to choose an appropriate functional analysis setup in order to establish these properties.) (i) he ernel of D is finite dimensional. (ii) he image of D is closed in (Ω 0 Ω + g )(ade). (iii) he coernel of D is infinite dimensional. hen the local model (i.e. the Kuranishi description) of M(n, g) around [] is given by the zero set of a map f : KerD CoerD. (Rigorously speaing, the map f is defined only in a small neighborhood of the origin.) From the conditions (i) and (iii), this is a map from the finite dimensional space to the infinite dimensional one. herefore (we can hope that) if we perturb the map f appropriately, then the zero set disappear. he parameter g gives sufficient perturbation, and we can prove that M(n, g) is empty for n 1 and generic g. Organization of the paper: In Section 3.1, we review the basic facts on anti-selfduality and conformal structure. In the above arguments we considered the moduli spaces M(n, g) parametrized by Riemannian metrics g. But SD equation depends only on conformal structures, and hence technically it is better to parameterize SD moduli spaces by conformal structures. Section 3.1 is a preparation for this consideration. In Section 3.2, we prepare some estimates relating to the Laplacians. In Section 4 we study the decay behavior of instantons over X, and show that they decay sufficiently fast. his is important in showing that all instantons can be captured by the functional analysis setups constructed in Sections 6 and 8.3. Section 5 is a preparation for Section 6. In Section 6 we study a (modified version of) operator D = d + d+ d and establish the above mentioned properties (i), (ii), (iii). In Section 7 we show that there is no non-flat reducible instantons on E. Here the condition b (Y ) = 0 is essentially used.

N OPEN FOUR-MNIFOLD HVING NO INSNON 5 Sections 8.1 and 8.2 are preparations for the perturbation argument in Section 8.3. In Section 8.3 we establish a transversality by using Freed-Uhlenbec s metric perturbation. Here we use the results established in Sections 6 and 7. Combining the results in Sections 4 and 8.3, we prove heorem 2.1 in Section 9. 3. Some preliminaries 3.1. nti-self-duality and conformal structure. In this subsection we review some well-nown facts on the relation between anti-self-duality and conformal structure. Specialists of the gauge theory don t need to read the details of the arguments in this subsection. he references are Donaldson-Sullivan [6, pp. 185-187] and Donaldson-Kronheimer [5, pp. 7-8]. We start with a linear algebra. Let V be an oriented real 4-dimensional linear space. We fix an inner product g 0 on V. he orientation and inner product give a natural isomorphism Λ 4 (V ) = R, and we define a quadratic form Q : Λ 2 (V ) Λ 2 (V ) R by Q(ξ, η) := ξ η Λ 4 (V ) = R. he dimensions of maximal positive subspaces and maximal negative subspaces with respect to Q are both 3 Let g and g be two inner products on V. hey are said to be conformally equivalent if there is c > 0 such that g 2 = cg 1. Let Conf(V ) be the set of all conformal equivalence classes of inner-products on V. Conf(V ) naturally admits a smooth manifold structure. We define Conf (V ) as the set of all 3-dimensional subspaces U Λ 2 (V ) satisfying Q(ω, ω) < 0 for all non-zero ω U. Conf (V ) depends on the orientation of V, but it is independent of the choice of the inner product g 0. Conf (V ) is an open set of the Grassmann manifold Gr 3 (Λ 2 (V )), and hence it is also a smooth manifold. Let Λ + be the space of ω Λ 2 (V ) which is self-dual with respect to g 0, and Λ be the space of ω Λ 2 (V ) which is anti-self-dual with respect to g 0. We define Conf (V ) be the set of linear map µ : Λ Λ + satisfying µ < 1 (i.e. µ(ω) < ω for all non-zero ω Λ where the norm is defined by g 0 ). his is also a smooth manifold as an open set of Hom(Λ, Λ + ). he map Conf (V ) Conf (V ), µ {ω + µ(ω) ω Λ } is a diffeomorphism. Hence Conf (V ) is contractible. (In particular it is connected.) Lemma 3.1. he map (4) Conf(V ) Conf (V ), [g] {ω Λ 2 (V ) ω is anti-self-dual with respect to g}, is a diffeomorphism. Proof. For SL(V ) and [g] Conf(V ) we define [g] Conf(V ) by setting (g)(u, v) := g( 1 u, 1 v). In this manner SL(V ) transitively acts on Conf(V ), and the isotropy subgroup at [g 0 ] is equal to SO(V ) = SO(V, g 0 ). Hence Conf(V ) = SL(V )/SO(V ). On the other hand, the Lie group SO(Λ 2 (V ), Q) ( = SO(3, 3)) naturally acts on Conf (V ). his

6 M. SUKMOO action is transitive. (For U Conf (V ) set U := {ω Λ 2 (V ) Q(ω, η) = 0 ( η U)}. Λ 2 (V ) = U U. Q is negative definite on U and positive definite on U. By choosing orthonormal bases on U and U with respect to Q, we can construct SO(Λ 2 (V ), Q) satisfying (Λ ) = U.) Let SO(Λ 2 (V ), Q) 0 be the identity component of SO(Λ 2 (V ), Q). Since Conf (V ) is connected, SO(Λ 2 (V ), Q) 0 also transitively acts on Conf (V ). he isotropy group of this action at Λ Conf (V ) is equal to SO(Λ + ) SO(Λ ). (It is easy to see that if SO(Λ 2 (V ), Q) 0 fixes Λ then it also fixes Λ +. Hence O(Λ + ) O(Λ ). Since Conf (V ) is contractible, the isotropy subgroup must be connected. herefore SO(Λ + ) SO(Λ ).) hus Conf (V ) = SO(Λ 2 (V ), Q) 0 /SO(Λ + ) SO(Λ ) SL(V ) naturally acts on Λ 2 (V ), and it preserves the quadratic form Q. Hence we have a homomorphism f : SL(V ) SO(Λ 2 (V ), Q) 0. direct calculation shows that it induces an isomorphism between their Lie algebras. Hence the homomorphism f : SL(V ) SO(Λ 2 (V ), Q) 0 is a (surjective) covering map. f 1 (SO(Λ + ) SO(Λ )) is equal to SO(V ). (It is easy to see that SO(V ) f 1 (SO(Λ + ) SO(Λ )) and their dimensions are both 6. SL(V ) is connected and SL(V )/f 1 (SO(Λ + ) SO(Λ )) = SO(Λ 2 (V ), Q) 0 /SO(Λ + ) SO(Λ ) = Conf (V ) is contractible. Hence f 1 (SO(Λ + ) SO(Λ )) must be connected. herefore it is equal to SO(V ).) hus SL(V )/SO(V ) = SO(Λ 2 (V ), Q) 0 /SO(Λ + ) SO(Λ ). his gives a diffeomorphism Conf(V ) = Conf (V ), and this diffeomorphism coincides with the above map (4). Let M be an oriented 4-manifold (not necessarily compact), and g 0 be a smooth Riemannian metric on M. wo Riemannian metrics g and g on M are said to be conformally equivalent if there is a positive function φ : M R satisfying g = φg. Let Conf(M) be the set of all conformal equivalence classes of C -Riemannian metrics on M. Let Λ + and Λ be the sub-bundles of Λ 2 := Λ 2 ( M) consisting of self-dual and antiself-dual 2-forms with respect to g 0. For [g] Conf(M) we define a sub-bundle Λ g Λ 2 as the set of anti-self-dual 2-forms with respect to g. here is a C -bundle map µ g : Λ Λ + such that (µ g ) x < 1 (x M) and that Λ g is equal to the graph {ω + µ g (ω) ω Λ }. Here (µ g ) x < 1 (x M) means that µ g (ω) < ω for all non-zero ω Λ. ( is the norm defined by g 0.) From the previous argument, we get the following result. Corollary 3.2. he map is bijective. Conf(M) {µ : Λ Λ + : C -bundle map µ x < 1 (x M)}, [g] µ g, 3.2. Eigenvalues of the Laplacians on differential forms over S 3. We will sometimes need estimates relating to lower bounds on the eigenvalues of the Laplacians on S 3. Here the 3-sphere S 3 is endowed with the Riemannian metric induced by the inclusion

N OPEN FOUR-MNIFOLD HVING NO INSNON 7 S 3 = {x R 4 x 2 1 + x 2 2 + x 2 3 + x 2 4 = 1} R 4. (R 4 has the standard Euclidean metric.) he formal adjoint of d : Ω i Ω i+1 is denoted by d : Ω i+1 Ω i. Lemma 3.3. he first non-zero eigenvalue of the Laplacian = d d acting on functions over S 3 is 3. Proof. See Saai [13, p. 272, Proposition 3.13]. Lemma 3.4. Let Ker(d ) Ω 1 be the space of 1-forms a over S 3 satisfying d a = 0. hen the first eigenvalue of the Laplacian = d d + dd acting on Ker(d ) is 4. Proof. See Donaldson-Kronheimer [5, p. 310, Lemma (7.3.4)]. s a corollary we get the following. (his is given in [5, p. 310, Lemma (7.3.4)].) Corollary 3.5. (i) Let a be a smooth 1-form over S 3 satisfying d a = 0. hen a 2 dvol 1 da 2 dvol. S 4 3 S 3 (ii) For any smooth 1-form a on S 3, we have a da 1 da 2 dvol. S 2 3 S 3 Proof. (i) da 2 = a, a 4 a 2. (ii) here is a smooth function f on S 3 such that b := a df satisfies d b = 0. hen a da = b db b 2 db 2 1 2 db 2 = 1 2 da 2. 4. Decay estimate of instantons 4.1. Classification of adapted connections. Let us go bac to the situation of Section 2.1. Y is a simply connected, compact oriented 4-manifold, and X = Y Z is the connected sum of the infinite copies of Y indexed by Z. Since X is non-compact, every principal SU(2)-bundle on it is isomorphic to the product bundle E = X SU(2). Following Donaldson [4, Definition 3.5], we mae the following definition. Definition 4.1. n adapted connection on E is a connection on E which is flat outside a compact set. (hat is, there is a compact set L X such that F = 0 over X \L.) wo adapted connections 1 and 2 on E are said to be equivalent as adapted connections if there is a gauge transformation u : E E such that u( 1 ) is equal to 2 outside a compact set. For m Z, let u m : X SU(2) be a smooth map such that (ρ m ) : H 3 (X) H 3 (SU(2)) satisfies (ρ m ) ([S 3 ]) = m[su(2)]. (Here [S 3 ] is the fundamental class of the cross-section S 3 Y (n) Y (n) +, and it is a generator of H 3 (X) = Z. See Remar 4.2 below.) his means that the restriction of u m to the cross-section S 3 Y (n) Y (n) + becomes a map of degree m from S 3 to SU(2) (for every n Z).

8 M. SUKMOO Remar 4.2. he cross-section S 3 Y (n) Y (n) + is endowed with the orientation so that the identification Y (n) Y (n) + = (1, 1) S 3 is orientation preserving. (he interval (1, 1) has the standard orientation.) he orientation on the Lie group SU(2) is chosen as follows: Let θ Ω 1 su(2) be the left invariant 1-form (on SU(2)) valued in the Lie algebra su(2) satisfying θ(x) = X for all X su(2) = 1 SU(2). (In the standard notation, we can write θ = g 1 dg for g SU(2).) We choose the orientation on SU(2) so that (5) 1 tr (θ dθ + 23 ) 8π 2 θ3 = 1 tr(θ 3 ) = 1. SU(2) 24π 2 SU(2) Since E is the product bundle, u m becomes a gauge transformation of E. Let ρ be the product flat connection on E = X SU(2), and set ρ m := u 1 m (ρ). Let (m) be a connection on E which is equal to ρ over q 1 (, 1) and equal to ρ m over q 1 (1, + ). (m) is an adapted connection on E. For t > 1 we have 1 (6) tr(f ((m)) 2 ) = 1 ( u 8π 2 8π 2 m tr(θ dθ + 2 ) 3 θ3 ) = m. X q 1 (t) Here we have used (5) and deg(u m q 1 (t) : q 1 (t) SU(2)) = m. Proposition 4.3. For m 1 m 2, (m 1 ) and (m 2 ) are not equivalent as adapted connections. If is an adapted connection on E, then is equivalent to (m) as an adapted connection where m = 1 trf 8π. 2 2 X (n important point for us is that there are only countably many equivalence classes of adapted connections.) Proof. he first statement follows from the equation (6). Let be an adapted connection on E. here is M > 0 such that is flat on q 1 (, M] and q 1 [M, ). We choose M > 1 so that q 1 (M) = S 3 Y (n) Y (n) + and q 1 ( M) = S 3 Y ( n) Y ( n) for some n > 0. Since q 1 (, M] and q 1 [M, ) are simply connected, there are gauge transformations u on q 1 (, M] and u on q 1 [M, ) such that u() = ρ and u () = ρ. We can extend u all over X. Hence we can suppose that u = 1 and that is equal to ρ over q 1 (, M]. Set m := deg(u q 1 (M) : q 1 (M) SU(2)). he degree of the map (u 1 m u ) q 1 (M) : q 1 (M) SU(2) is zero. hen there is a gauge transformation u of E such that u = u 1 m u on q 1 [M, + ) and u = 1 on q 1 (, M 1). hen u () is equal to ρ over q 1 (, M] and equal to u 1 m (ρ) over q 1 [M, + ). Hence u () is equal to (m) outside a compact set. We have m = 1 8π 2 X trf ((m)) 2 = 1 8π 2 X trf 2.

N OPEN FOUR-MNIFOLD HVING NO INSNON 9 4.2. Preliminaries for the decay estimate. We need the following. (his is a special case of [9, Proposition 3.1, Remar 3.2].) Proposition 4.4. Let Z be a simply-connected compact Riemannian 4-manifold with (or without) boundary, and W Z be a compact subset with W Z =. hen there are positive numbers ε 1 (W, Z) and C 1, (W, Z) ( 0) satisfying the following: Let be an SD connection on the product principal SU(2)-bundle over Z satisfying F L 2 (Z) ε 1 (W, Z). hen can be represented by a connection matrix à over a neighborhood of W satisfying à C 1, F L C (W ) 2 (Z), for all 0. Proof. See Fuaya [9, Proposition 3.1, Remar 3.2]. Lemma 4.5. Let L > 2. here exist positive numbers ε 2 and C 2, ( 0) independent of L satisfying the following. If is an SD connection on the product principal SU(2)- bundle G over (0, L) S 3 satisfying F L 2 ((0,L) S 3 ) ε 2, then can be represented by a connection matrix à over a neighborhood of [1, L 1] S3 satisfying (7) Ã(t, θ) C 2, F L 2 ((t 1,t+1) S 3 ), for (t, θ) [1, L 1] S 3 and 0. Proof. Proposition 4.4 implies the following. here exist positive numbers ε 2 and C 2, ( 0) such that if B is an SD connection on the product principal SU(2)-bundle over [0, 1] S 3 satisfying F B L 2 ([0,1] S 3 ) ε 2 then B can be represented by a connection matrix B over a neighborhood of [1/4, 3/4] S 3 satisfying B(x) C 2, F B L 2 ([0,1] S 3 ) (x [1/4, 3/4] S 3, 0). Let ε 2 be a small positive number with ε 2 < ε 2. We will fix ε 2 later. Suppose that is an SD connection on the product principal SU(2)-bundle G over (0, L) S 3 satisfying F L 2 ((0,L) S 3 ) ε 2. For 2 n [4L 4], set I n := [n/4, n/4 + 1/2] and J n := [n/4 1/4, n/4 + 3/4]. We have I n J n. For each n, there is a local trivialization h n of G over a neighborhood of I n S 3 such that the connection matrix n := h n () satisfies n (x) C 2, F L 2 (J n S 3 ) (x I n S 3, 0). Set g n := h n+1 h 1 n : (I n I n+1 ) S 3 SU(2). hen g n ( n ) = n+1 (i.e. dg n = g n n n+1 g n ). In particular dg n 4C 2,0ε 2. Fix a reference point x 0 S 3. By multiplying some constant gauge transformations on h n s, we can assume that g n (n/4 + 1/4, x 0 ) = 1. hen g n 1 const ε 2 over (I n I n+1 ) S 3 where const is independent of L, n. Since the exponential map exp : su(2) SU(2) is locally diffeomorphic around 0 su(2), if ε 2 is

10 M. SUKMOO sufficiently small (but independent of L, n), we have u n := (exp) 1 g n : (I n I n+1 ) S 3 su(2). (Here we have fixed ε 2 > 0.) hen g n = e u n over (I n I n+1 ) S 3 with u n (x) C 2, F L 2 ((J n J n+1 ) S 3 ) (x (I n I n+1 ) S 3, 0). Let φ be a smooth function in R such that supp(dφ) (1/4, 1/2), φ(t) = 0 for t 1/4 and φ = 1 for t 1/2. Set φ n (t) := φ(t n/4). (supp(dφ n ) Interior(I n I n+1 ).) We define a trivialization h of G over the union of (I n I n+1 ) S 3 (2 n [4L 4]) by setting h := e φ nu n h n on (I n I n+1 ) S 3. hen h is smoothly defined over a neighborhood of [1, L 1] S 3, and the connection matrix à := h() satisfies (7). Let us go bac to the given manifolds Y and Y = p 1 ( + 1, 1). Lemma 4.6. Let > 4. here exist positive numbers ε 3 and C 3, ( 0) independent of satisfying the following. If is an SD connection on the product principal SU(2)- bundle over Y satisfying F L 2 (Y ) ε 3, then can be represented by a connection matrix à over Y 1 such that for x Y 1 and 0. Ã(x) C 3, F L 2 (p 1 (t 6,t+6) Y ) (t = p(x)), Proof. Set Z := p 1 [ 3, 3] and W := p 1 [ 5/2, 5/2] Z. We apply Proposition 4.4 to these Z and W : here is ε 3 > 0 (depending only on Z, W and hence independent of ) such that if F L 2 (Z) ε 3 then can be represented by a connection matrix 1 over a neighborhood of W such that 1 (x) const F L 2 (Z) (x W, 0). On the other hand, by applying Lemma 4.5 to the tubes p 1 ( +1, 1) = ( +1, 1) S 3 and p 1 (1, 1) = (1, 1) S 3, if F L 2 (Y ) ε 2 (the positive constant introduced in Lemma 4.5) then can be represented by a connection matrix 2 over a neighborhood of p 1 [ + 2, 2] p 1 [2, 2] = [ + 2, 2] S 3 [2, 2] S 3 such that 2 (t, θ) C 2, F L 2 ((t 1,t+1) S 3 ) ((t, θ) [ +2, 2] S 3 [2, 2] S 3, 0). hen by patching 1 and 2 over p 1 ( 5/2, 2) and p 1 (2, 5/2) as in the proof of Lemma 4.5, we get the desired connection matrix Ã. 4.3. Exponential decay. In this subsection we study a decay estimate of instantons on the product principal SU(2)-bundle E = X SU(2). he results in this section will be used in Section 9. Our method is based on the arguments of Donaldson [4, Section 4.1] and Donaldson-Kronheimer [5, Section 7.3]. In this subsection we always suppose > 4. Let g be a Riemannian metric on X which is equal to g 0 (the Riemannian metric given in Section 2.1) outside a compact set. Let be a g-sd connection on E satisfying F 2 gdvol g <. X

For t R, set N OPEN FOUR-MNIFOLD HVING NO INSNON 11 J(t) := q 1 (t,+ ) F 2 gdvol g. For t 1 we have J(t) = F q 1 (t,+ ) 2 dvol where and dvol are the norm and volume form with respect to the periodic metric g 0. Recall that for each integer n we have q 1 (n + 1, (n + 1) 1) = Y (n) Y (n) + = (1, 1) S 3. Lemma 4.7. here is n 0 () > 0 such that for n n 0 () (he value 2 is not optimal.) J (t) 2J(t) (n + 2 t (n + 1) 2). Proof. In this proof we always suppose n + 2 t (n + 1) 2 and n 1. We have J (t) = F 2 dvol = 2 F ( t ) 2 L 2 (S 3 ) ( t := q 1 (t)). q 1 (t) Here we have used the fact F 2 = 2 F ( t ) 2. his is the consequence of the SD condition. From Lemma 4.5, we can assume that, for n 1, a connection matrix of over q 1 [n + 2, (n + 1) 2] is as small as we want with respect to the C 1 -norm (or any other C -norm). In particular we have F ( t ) L 2 1 for n 1. hen, by using [5, Proposition 4.4.11], we can suppose that t is represented by a connection matrix satisfying (8) t L 2 1 (S 3 ) const F ( t) L 2 (S 3 ). hen we can prove Sublemma 4.8. J(t) = tr( t d t + 2 S 3 3 t ) (=: θ( t )). 3 Proof. For m > n 1 and m + 2 s (m + 1) 2, (9) F 2 dvol θ( s ) θ( t ) mod 8π 2 Z. q 1 [t,s] We can suppose that the connection matrix s also satisfies (8). hen both of the left and right hand sides of the above equation (9) are sufficiently small. Hence F 2 dvol = θ( s ) θ( t ). q 1 [t,s] We have θ( s ) 0 as m +. hen we get the above result. From Corollary 3.5 (ii), tr( t d t ) 1 d t 2. S 2 3 S 3

12 M. SUKMOO Since d t = F ( t ) 2 t, d t 2 L 2 (S 3 ) F ( t) 2 L 2 + 2 F ( t ) L 2 2 t L 2 + 2 t 2 L 2. We have L 2 1(S 3 ) L 6 (S 3 ). Hence 2 t L 2 const t 2 L const F ( 2 t ) 2 1 L by (8). Hence 2 d t 2 L (1 + const F ( 2 t ) L 2) F ( t ) 2 L. In a similar way, we have 2 tr( 3 t ) const t 3 L const F ( 3 t ) 3 L. 2 S 3 hus we have ( ) 1 J(t) = θ( t ) 2 + const F ( t) L 2 F ( t ) 2 L. 2 Since J (t) = 2 F ( t ) 2 L and F ( 2 t ) L 2 1, we have J(t) F ( t ) 2 L = 1J (t). 2 2 Hence J (t) 2J. Corollary 4.9. For t n 0 () + 2, J(t) const, e 2(1 4/ )t. Here const, is a positive constant depending on and. Proof. First note that J(t) is monotone non-increasing. For n + 2 t (n + 1) 2 (n n 0 () =: n 0 ), we have J(t) e 2(t n 2) J(n + 2) by Lemma 4.7. Set a n := J(n + 2) (n n 0 ). a n+1 J((n + 1) 2) e 2( 4) a n. Hence a n e 2( 4)(n n0) a n0. For n + 2 t (n + 1) 2, J(t) e 2(t n 2) a n e 2(t 4n) e 4+2n 0 8n 0 a n0. Since t n + 2, we have t 4n (1 4/ )t + 8/. Hence J(t) const, e 2(1 4/ )t. For (n + 1) 2 < t < (n + 1) + 2, J(t) J((n + 1) 2) const, e 2(1 4/ )t. In the same way we can prove the following. Lemma 4.10. For t 1 we have F 2 dvol const, e 2(1 4/ )t. q 1 (, t) Corollary 4.11. here exists an adapted connection 0 on E satisfying for all integers 0. 0 ((x) 0 (x)) const,, e (1 4/ ) t (t = q(x)), Proof. For n 1, we have F ε L 2 (Y 3. (ε (n) ) 3 is a positive constant introduced in Lemma 4.6.) hen by Lemma 4.6, Corollary 4.9 and Lemma 4.10, can be represented by a connection matrix n on Y (n) 1 ( n 1) such that n (x) const,, e (1 4/ ) t (x Y (n) 1, t = q(x), 0). By patching these connection matrices over Y (n) 1 Y (n+1) 1 ( n 1) as in the proof of Lemma 4.5, can be represented by a connection matrix à on { t 1} such that Ã(x) const,, e (1 4/ ) t ( t 1, 0).

N OPEN FOUR-MNIFOLD HVING NO INSNON 13 o be more precise, there are t 0 1 and a trivialization h : E { t >t0 } { t > t 0 } SU(2) such that h() satisfies ρ(h() ρ) const,, e (1 4/ ) t ( t > t 0, 0), where ρ is the product connection. his means that h 1 (ρ) ( h 1 (ρ)) const,, e (1 4/ ) t ( t > t 0, 0). ae a connection 0 on E which is equal to h 1 (ρ) over { t t 0 + 1}. hen 0 is an adapted connection satisfying the desired property. 5. Preliminaries for linear theory In this section, we study differential operators over X. he results in this section will be used in Section 6. ll arguments in Sections 5.1 and 5.2 are essentially given in Donaldson [4, Chapters 3 and 4]. 5.1. Preliminary estimates over the tube. Let α be a real number with 0 < α < 1. In this subsection we study some differential operators over R S 3. We denote t as the parameter of the R-factor (i.e. the natural projection t : R S 3 R). Let d : Ω 1 R S 3 Ω 0 R S be the formal adjoint of the derivative d : Ω 0 3 R S Ω 1 3 R S over R S 3. We have 3 d = d where is the Hodge star over R S 3. We define a differential operator d,α : Ω 1 R S Ω 0 3 R S by setting d,α b := e 2αt d (e 2αt b) (b Ω 1 3 R S ). hen 3 d,α b = d b 2α (dt b). If f Ω 0 R S and b Ω 1 3 R S have compact supports, then 3 e 2αt df, b dvol = e 2αt f, d,α b dvol. R S 3 R S 3 Consider d + := 1(1 + )d : 2 Ω1 R S Ω + 3 R S, and set D α := d,α + d + : Ω 1 3 R S 3 Ω 0 R S Ω + 3 R S. 3 Let Λ i S (i 0) be the bundle of i-forms over S 3. Consider the pull-bac of Λ i 3 S by the 3 projection R S 3 S 3, and we also denote it as Λ i S for simplicity. We can identify the 3 bundle Λ 1 R S of 1-forms on R S 3 with the bundle Λ 0 3 S Λ 1 3 S by 3 Λ 0 S 3 Λ1 S 3 (b 0, β) b 0 dt + β Λ 1 R S 3. We also naturally identify the bundle Λ 0 R S with Λ 0 3 S. he bundle Λ + 3 R S of self-dual 3 forms can be identified with the bundle Λ 1 S by 3 Λ 1 S 3 β 1 2 (dt β + 3β) Λ + R S 3 ( 3 : the Hodge star on S 3 ). We define L : Γ(Λ 0 S Λ 1 3 S ) Γ(Λ 0 3 S Λ 1 3 S ) by setting ( ) ( 3 ) ( ) b 0 0 d 3 b 0 L :=, β d 3 3 d 3 β

14 M. SUKMOO where d 3 is the exterior derivative on S 3 and d 3 = 3 d 3 3. Let b = (b 0, β) Γ(Λ 0 S 3 Λ 1 S ) = Ω 1 3 R S (i.e. b = b 3 0 dt + β). hen D α b Ω 0 R S Ω 1 3 R S = Γ(Λ 0 3 S Λ 1 3 S ) is given by ( ) ( ( )) ( ) 3 D α b = b 0 2α 0 b 0 + L +. t β 0 0 β For u Ω i R S 3 (i 0), we define the Sobolev norm u L 2 ( 0) by (10) u 2 L := 2 j=0 We define the weighted Sobolev norm u L 2,α (11) u L 2,α R S 3 j u 2 dvol. by := e αt u L 2. he map L 2,α (R S3, Λ i R S ) u e αt u L 2 3 (R S3, Λ i R S ) is an isometry. 3 D α becomes a bounded linear map from L 2,α +1 (R S3, Λ 1 R S ) to L 2,α 3 (R S3, Λ 0 R S 3 Λ + R S 3 ). For b = b 0 dt + β Ω 1 R S 3 (12) e αt D α (e αt b) = t Set as above, we have ( ) ( b 0 + L + β L α := L + Recall that we have assumed 0 < α < 1. ( ) α 0. 0 α ( α 0 0 α )) ( ) b 0. β Lemma 5.1. Consider L α as an essentially self-adjoint elliptic differential operator acting on Ω 0 S 3 Ω 1 S 3 over S 3. If λ is an eigenvalue of L α, then λ α. Moreover if λ α, then λ > 1. Proof. We have Ω 0 S 3 Ω1 S 3 = (Ω0 S 3 d 3(Ω 0 S 3)) er d 3, where d 3 = 3 d 3 3 : Ω 1 S Ω 0 3 S. he subspaces Ω 0 3 S d 3 3 (Ω 0 S ) and er d 3 3 are both L α -invariant. For β er d 3, L α (0, β) = (0, 3 d 3 β αβ). Suppose that L α (0, β) = λ(0, β) and β is not zero. Since d 3β = 0 and H 1 (S 3 ) = 0, we have d 3 β 0. hen 3 d 3 β = (λ + α)β and λ + α 0. Since we have (Corollary 3.5 (ii)) β d 3 β 1 d 3 β 2 dvol S 2 3 S 3 and (λ + α)β d 3 β = d 3 β 2 dvol, we have hen λ 2 α > 1 > α. 2 λ + α.

N OPEN FOUR-MNIFOLD HVING NO INSNON 15 For (f, d 3 g) Ω 0 S d 3 3 (Ω 0 S ) (f and g are smooth functions on S 3 ), 3 ( ) ( ) L α f αf 3 g =, ( 3 = d d 3 g d 3 f αd 3 g 3d 3 is the Laplacian on functions over S 3 ). Suppose that L α (f, d 3 g) = λ(f, d 3 g) and (f, d 3 g) is not zero. hen 3 g = (α λ)f, d 3 f = (α + λ)d 3 g. Case 1: Suppose α + λ = 0. hen f is a constant, and 0 = 3 g dvol = 2α S 3 fdvol. S 3 Hence f 0. his implies 3 g 0 and hence d 3 g 0. his is a contradiction. Case 2: Suppose α + λ 0. hen 3 f = (λ 2 α 2 )f. Since the first non-zero eigenvalue of the Laplacian 3 is 3 (Lemma 3.3), λ 2 α 2 = 0 or λ 2 α 2 3. Since λ α, we have λ = α or λ 3 + α 2 3. Lemma 5.2. For a L 2,α 1 (R S 3, Λ 1 R S ), we have a 3 L 2,α 2 α 1 D α a L 2,α. Moreover a L 2,α const α D α a 1 L 2,α. Proof. We can suppose that a is smooth and compact supported. Set b := e αt a = b 0 dt + β where (b 0, β) Γ(R S 3, Λ 0 S 3 Λ 1 S 3 ). Let {φ λ } λ be a complete orthonormal basis of L 2 (S 3, Λ 0 S 3 Λ 1 S 3 ) consisting of eigen-functions of L α over S 3 with L α φ λ = λφ λ where λ runs over all eigenvalues of L α. From Lemma 5.1, we have λ α. Decompose (b 0, β) by {φ λ } as (b 0 (t, θ), β(t, θ)) = λ c λ (t)φ λ (θ). Since a is compact supported, the functions c λ are also compact supported. ( / t + L α )(b 0, β) = λ (c λ (t) + λc λ(t))φ λ. If ( / t + L α )(b 0, β) = (b 1, γ), then e αt D α (e αt b) = (b 1, 1(dt γ + 2 3γ)). Hence e αt D α (e αt b) = b 1 2 + γ 2 /2 ( / t + L α )(b 0, β) / 2. herefore e αt D α (e αt b) 2 dvol 1 c λ + λc λ 2 dt. R S 2 3 c λ + λc λ 2 = c λ 2 + λ(c 2 λ ) + λ 2 c 2 λ. Since λ α and the functions c λ are compact supported, c λ + λc λ 2 dt α 2 c λ 2 dt = α 2 b 2 L. 2 λ λ hen D α a L 2,α = e αt D α (e αt b) L 2 α b L 2 = α a L 2,α. 2 2 λ

16 M. SUKMOO Since e αt D α e αt = + t Lα is a translation invariant elliptic differential operator, for every n Z we have ( b 2 L 2 1 ((n,n+1) S3 ) const α b 2 L 2 ((n 1,n+2) S 3 ) + e αt D α (e αt b) ) 2. L 2 ((n 1,n+2) S 3 ) Here const α is independent of n. By summing up this estimate over n Z, we get ( b L 2 1 (R S 3 ) const α b L 2 (R S 3 ) + e αt D α (e αt b) ) L. 2 (R S 3 ) his shows a L 2,α 1 const α ( a L 2,α + D α a L 2,α) const α D α a L 2,α. Lemma 5.3. (i) Suppose α > 0. Let a be a smooth 1-form over the negative half tube (, 0) S 3 satisfying (,0) S 3 e 2αt a 2 dvol < +. Suppose D α a = 0. hen a, a const a,α e (1 α)t (t < 2). (ii) Suppose α < 0. Let a be a smooth 1-form over the positive half tube (0, + ) S 3 satisfying (0,+ ) S 3 e 2αt a 2 dvol < +, and suppose D α a = 0. hen a, a const a,α e (1+α)t (t > 2). Proof. We give the proof of the case (i) (α > 0). he case (ii) can be proved in the same way. Set b := e αt a = b 0 dt + β where (b 0, β) Γ(R S 3, Λ 0 S Λ 1 3 S ). hen 3 e αt D α (e αt b) = 0. Choose {φ λ } λ as in the proof of Lemma 5.2. Decompose (b 0, β) by {φ λ } as (b 0 (t, θ), β(t, θ)) = λ c λ(t)φ λ (θ). Since ( / t + L α )(b 0, β) = (c λ (t) + λc λ(t))φ λ = 0, we have c λ (t) = d λ e λt where d λ is a constant. For t < 0, b 2 dvol 3 = c λ 2 = d λ 2 e 2λt d λ 2 e 2λt. {t} S 3 λ λ Since the L 2 -norm of b over (, 0) S 3 is finite, we have d λ = 0 for λ 0. Set B := e 2 b 2 dvol 3 = e 2 d λ e λ 2 <. { 1} S 3 λ<0 From Lemma 5.1, negative eigenvalues λ satisfy λ < 1. Hence for t < 1 b 2 dvol 3 = d λ e λ 2 e 2λ(t+1) d λ e λ 2 e 2(t+1) = Be 2t. {t} S 3 λ<0 λ<0 hen for t < 2, (t 1,t+1) S 3 b 2 dvol B t+1 t 1 e 2s ds Be 2(t+1). Since e αt D α (e αt b) = 0 (and this is a translation invariant equation), the elliptic regularity implies b, b const α B e t (t < 2). (Indeed we can choose const α independent of α. But it is unimportant for us.) Since a = e αt b, we have a, a const a,αe (1 α)t (t < 2).

N OPEN FOUR-MNIFOLD HVING NO INSNON 17 5.2. Preliminary results over Ŷ. Recall that Y is a simply connected closed oriented 4-manifold and that Ŷ = Y \ {x 1, x 2 }. Ŷ has cylinderical ends, and we have p : Ŷ R. For a section of u of Λ i (i 0) over Ŷ, we define the Sobolev norm u L ( 0) as 2 in (10). We define the weighted Sobolev norm by u L 2,α := e αt u L 2,α where t = p(x) (x Ŷ ). Recall 0 < α < 1. For a 1-form a over Ŷ we set Dα a := d,α a + d + a = e 2αt d (e 2αt a) + d + a. Lemma 5.4. Let a be a 1-form over Ŷ with a L 2,α 1 <. If D α a = 0, then a = 0. Proof. We give the proof of the case α > 0. he case α < 0 can be proved in the same way. We divide the proof into three steps. Step 1: We will show that the above assumption implies da = 0. First we want to show a, da L 2. We have a 2 dvol e 2αt a 2 dvol <, da 2 dvol e 2αt da 2 dvol <. t>0 t>0 Lemma 5.3 implies that the L 2 -norms of a and da over Y = (, 1) S 3 are finite. Hence a, da L 2. For R > 1, let β R be a smooth function over Ŷ such that β R = 1 over p 1 ( R, R), β R = 0 over p 1 (, 2R) p 1 (2R, ) and dβ R 2/R. 0 = d(β R a da) = β R da da + dβ R a da. Since d + a = 0, we have da da = da 2 dvol and hence β R da 2 dvol = dβ R a da 2 R a L da 2 L 2. t>0 Let R +. hen da 2 dvol = 0. Hence da = 0. Step 2: We have t>0 a const a,α e αt (t > 1), a const a,α e (1 α)t (t < 1). he latter estimate comes from Lemma 5.3. he former one comes from the elliptic regularity and the following estimate: For t > 1, a 2 dvol e 2αt e 2αp(x) a(x) 2 dvol(x) a 2 L e 2αt. 2,α p 1 (t,+ ) p 1 (t,+ ) Step 3: From Step 1 and HdR 1 (Ŷ ) = 0, there is a smooth function f on Ŷ satisfying a = df. From Step 2, the limits f(+ ) := lim t + f(t, θ) and f( ) := lim t f(t, θ) exist and independent of θ S 3. In particular f is bounded. We can assume f(+ ) = 0. hen for t > 1 f(t, θ) = t f (s, θ)ds. s

18 M. SUKMOO Since f/ s a const a,α e αt for t > 1 (Step 2), (13) f const a,α e αt (t > 1). Let β R be the cut-off function used in Step 1. Since e 2αt d (e 2αt a) = d,α a = 0, 0 = e 2αt β R f, d,α a dvol = e 2αt d(β R f), a dvol = e 2αt f dβ R, a dvol + e 2αt β R a 2 dvol. Hence (14) e 2αt β R a 2 dvol 2 R supp(dβ R ) e 2αt f a dvol. We have supp(dβ R ) p 1 ( 2R, R) p 1 (R, 2R). Since f and a are bounded, e 2αt f a dvol 0 (R + ). p 1 ( 2R, R) On the other hand, by the above (13) 2 e 2αt f a dvol const a,α e αt a dvol R p 1 (R,2R) R p 1 (R,2R) const a,α vol((r, 2R) S3 ) e R 2αt a 2 dvol const a,α a L 2,α / R. p 1 (R,2R) his goes to 0 as R +. From (14), e 2αt a 2 = 0. hus a = 0. Lemma 5.5. For a L 2,α 1 (Ŷ, Λ1 ), Proof. Set U := p 1 ( 2, 2) a L 2,α 1 (Ŷ ) const α D α a L 2,α (Ŷ ). Ŷ. By using Lemma 5.2, for all a L2,α 1 (Ŷ ) (15) a L 2,α 1 (Ŷ ) const α( a L 2 (U) + Dα a L 2,α (Ŷ )). We want to show a L 2 (U) const α D α a L 2,α (Ŷ ). Suppose on the contrary there exist a sequence a n (n 1) in L 2,α 1 (Ŷ, Λ1 ) such that 1 = a n L 2 (U) > n Dα a n L 2,α (Ŷ ). From the above (15), {a n } is bounded in L 2,α 1 (Ŷ ). Hence, if we tae a subsequence (also denoted by a n ), the sequence a n wealy converges to some a in L 2,α 1 (Ŷ ). We have Dα a = 0. Hence Lemma 5.4 implies a = 0. By Rellich s lemma, a n strongly converges to 0 in L 2 (U). (Note that U is pre-compact.) his contradicts a n L 2 (U) = 1.

N OPEN FOUR-MNIFOLD HVING NO INSNON 19 5.3. Preliminary results over X = Y Z. Recall that X = Y Z has the periodic metric g 0 which is compatible with the given metric h over every Y (n) (n Z), and that g 0 depends on the parameter > 2. We define the Sobolev norm L 2 over X as in (10) by using the metric g 0 and its Levi-Civita connection. We define the weighted Sobolev norm by u L 2,α := e αt u L 2,α where t = q(x) (x X). For a 1-form a over X we set D α a := d,α a + d + a = e 2αt d (e 2αt a) + d + a. Lemma 5.6. here exists α > 2 such that if α then for any a L 2,α 1 (X, Λ 1 ) we have a L 2,α 1 (X) const α D α a L 2,α (X). he important point is that α depends only on α. Proof. Let β (n) be a smooth function on X such that 0 β (n) 1, suppβ (n) Y (n) = q 1 ((n 1) +1, (n+1) 1), β (n) = 1 over q 1 ((n 1/2), (n+1/2) ) and dβ (n) 3/. Since t = q(x) = p (n) (x) + n over Y (n), by applying Lemma 5.5 to β (n) a, we get β (n) a e = L 2,α 1 (X) eαn αp (n)(x) β (n) a L 2 1 (Y (n) ) const α e αn e αp (n)(x) D α (β (n) a) hen If 1, then = const α D α (β (n) a) L 2,α (X) const α a 2 L 2,α 1 (X) β (n) a 2 L 2,α 1 (X) n Z const α 2 n Z L 2 (Y (n) ) a + const L 2,α (Y (n) ) α D α a. L 2,α (Y (n) ) a 2 + const L 2,α (Y (n) ) α n Z const α 2 a 2 L 2,α (X) + const α D α a 2 L 2,α (X). D α a 2 L 2,α (Y (n) ) a 2 L 2,α 1 (X) const α D α a 2 L 2,α (X). For a 1-form a on X we set Da := d a + d + a. Its formal adjoint D is given by D (u, ξ) = du + d ξ = du dξ for (u, ξ) Ω 0 Ω +. We consider D as an unbounded operator from L 2 (X, Λ 1 ) to L 2 (X, Λ 0 Λ + ). he additive Lie group Z naturally acts on X = Y Z. Set Y + := X/Z. We have b 1 (Y + ) = 1 and b + (Y + ) = b + (Y ). he operator D is preserved by the Z-action, and its

20 M. SUKMOO quotient is equal to the operator d + d + : Ω 1 Y + Ω 0 Y + Ω + Y + on Y +. hen we can apply tiyah s Γ-index theorem (tiyah [3], Roe [12, Chapter 13]) to D and get ind Z D = ind( d + d + : Ω 1 Y + Ω0 Y + Ω+ Y + ) = 1 + b 1 (Y + ) b + (Y + ) = b + (Y ). Here ind Z D is the Γ-index of D (Γ = Z). he above implies that if b + (Y ) 1 then KerD L 2 (X, Λ 0 Λ + ) is infinite dimensional. Suppose ρ = (u, ξ) L 2 (X, Λ 0 Λ + ) satisfies D ρ = du + d ξ = 0 as a distribution. By the elliptic regularity, ρ is smooth, and for each n Z ρ L 2 1 (q 1 ( (n 1/2),(n+1/2) )) const ρ L 2 (Y (n) ). Here const is independent of n Z. Hence ρ L 2 1 (X) const ρ L 2 (X) ρ L 2 1(X). In particular u, ξ L 2 1(X) and hence du, d ξ L 2 = 0. hen < +, and 0 = D ρ, du L 2 = du L 2. So du = 0. his means that u is constant. But u L 2. Hence u = 0. herefore d ξ = 0. hus we get the following result. Lemma 5.7. Suppose b + (Y ) 1. he space of ξ L 2 1(X, Λ + ) satisfying d ξ = 0 is infinite dimensional. ae and fix a smooth function : R R satisfying t = t for t 1. For 0 < α < 1, set W (x) := e α q(x) for x X. Hence W is a positive smooth function on X satisfying W (x) = e α q(x) for q(x) 1. For a section η of Λ i (i 0) we set η L (X) := W η L 2 (X). For a self-dual form η over X, we set d,w η := W 2 d(w 2 η). If a Ω 1 X and η Ω + X have compact supports, then X W 2 da, η dvol = X W 2 a, d,w η dvol. Lemma 5.8. Suppose b + (Y ) 1 and α > 0. hen the space of η L 1 (X, Λ + ) satisfying d,w η = 0 is infinite dimensional. Moreover it is closed in L (X, Λ + ). Proof. Suppose that ξ L 2 1(X, Λ + ) satisfies d ξ = 0. Set η := W 2 ξ. hen d,w η = 0 and η L 1 (X) = W 1 ξ L 2 1 (X) < from α > 0. hus Lemma 5.7 implies the first statement. In order to prove the closedness of Ker(d,W ) L 1 (X, Λ + ) in L (X, Λ + ), it is enough to show that if η L (X, Λ + ) satisfies d,w η = 0 (as a distribution) then η L 1 (X, Λ + ). η is smooth by the (local) elliptic regularity. he differential operator d,w on Y (n) (n > 0) are naturally isomorphic to each other. he same statement also hold for n < 0. Hence, by the elliptic regularity, W η L 2 1 (q 1 ((n 1/2),(n+1/2) )) const,α ( W η + L d,w (W η) 2 (Y (n) ) L 2 (Y (n) )) const,α W η. L 2 (Y (n) ) Here const,α are independent of n Z. hus η L 1 (X) const,α η L (X) <.

N OPEN FOUR-MNIFOLD HVING NO INSNON 21 Lemma 5.9. Suppose b + (Y ) 1 and α > 0. For any ε > 0 and any pre-compact open set U X, there is η L 1 (X, Λ + ) such that η = 0 over U and d,w η L (X) < ε η L (X). Proof. First we prove the following statement: For any ε > 0 and any pre-compact open set U X there exists η L 1 (X, Λ + ) satisfying d,w η = 0 and η L (U) < ε η L (X). Suppose that this statement does not hold. hen there are ε > 0 and a pre-compact open set U X such that all η Ker(d,W ) L 1 (X, Λ + ) satisfies η L (U) ε η L (X). Ker(d,W ) is an infinite dimensional closed subspace in L (X, Λ + ) (Lemma 5.8). Let {η n } n 1 be a complete orthonormal basis of Ker(d,W ) with respect to the inner product of L (X, Λ + ). hey satisfies η n L (U) ε. he sequence η n wealy converges to 0 in L (X), and hence η n U wealy converges to 0 in L (U). hen, by the elliptic regularity and Rellich s lemma, a subsequence of η n U strongly converges to 0 in L (U). But this contradicts η n L (U) ε. Next tae a pre-compact open set V X satisfying the following: U V and there exists a smooth function β such that 0 β 1, β = 0 on U, β = 1 on X \ V, supp(dβ) V, and dβ ε. By the previous argument there exists η L 1 (X, Λ + ) satisfying d,w η = 0 and η L (V ) < (1/3) η L (X). hen βη L (X) > (2/3) η L (X). Since d,w (βη) = (dβ η) is supported in V, d,w (βη) L (X) ε η L (V ) < (ε/3) η L (X) < (ε/2) βη L (X). Hence βη L 1 (X, Λ + ) satisfies βη = 0 over U and d,w (βη) L (X) < ε βη L (X). 6. Linear theory In this section we always assume 0 < α < 1 and (16) max( α, α ). Here α and α are the positive constants introduced in Lemma 5.6. (Recall that they depend only on α.) he purpose of this section is to prove several basic properties of the linear operators D and D introduced below. he constants introduced in this section often depend on several parameters (α,, 0,, µ). But we usually don t explicitly write their dependence on parameters unless it causes a confusion.

22 M. SUKMOO 6.1. he image of D is closed. Let E = X SU(2) be the product principal SU(2)- bundle over X, and 0 be an adapted connection on E (see Definition 4.1). Let W = e α q(x) be the weight function on X introduced in Section 5.3. For a section u of Λ i (ade) (i 0), we define the Sobolev norm u L 2 by using the periodic metric g 0 and the connection 0. We define the weighted Sobolev norm by u L := W u L 2. Let Λ + and Λ be the bundles of self-dual and anti-self-dual forms (with respect to the metric g 0 ) on X, and µ : Λ Λ + be a smooth bundle map. We assume µ x < 1 for all x X (i.e. µ(ω) < ω for all non-zero ω Λ where the norm is defined by the metric g 0 ). Moreover we assume that µ is compact supported. Hence µ corresponds to a conformal structure on X which coincides with [g 0 ] outside a compact set (see Section 3.1). We define = 0 as the space of L 3 -connections (with respect to 0 ) on E: := { 0 + a a L 3 (X, Λ 1 (ade))}. (Recall that the connection 0 is used in the definition of the weighted Sobolev space L 3 (X, Λ 1 (ade)).) We will need the following multiplication rule: If 3 and l, then L L l L l, i.e. for f 1 L and f 2 L l ( 3, l 0) (17) f 1 f 2 L l const f 1 L f 2 L l In particular, for = 0 + a, we have F () = F ( 0 ) + d 0 a + a a L 2. For b Ω 1 (ade) over X, we set D b := d,w b + (d+ µd )b = W 2 d (W 2 b) + (d + µd )b. Here d b = d ( b) and d ± = 1(1± )d 2. ( is the Hodge star defined by the metric g 0.) D is an elliptic differential operator since we assume µ x < 1 for all x X. Rigorously speaing, we should use the notation D µ,w instead of D. But here we use the above notation for simplicity. We have (18) D b = D 0 b + [a b] + [a b] + µ([a b] ). From this and the above (17), the map D : L +1 (X, Λ1 (ade)) L (X, (Λ 0 Λ + )(ade)) (0 3) becomes a bounded linear map. Let r be a positive integer such that q 1 ( r, r ) contains the supports of F ( 0 ) and µ. Set U := q 1 ( (r + 5/2), (r + 5/2) ). Lemma 6.1. (i) For any b L 2,w +1 (X, Λ1 (ade)) ( 0) we have (19) b L +1 (X) const ( b L 2 (U) + D 0 b L (X) ). Here const is a positive constant independent of b. (We will usually omit this ind of obvious remar below.).

N OPEN FOUR-MNIFOLD HVING NO INSNON 23 (ii) For any = 0 + a, there is a pre-compact open set U X (which depends on µ, α,, 0, ) such that for any b L +1 (X, Λ1 (ade)) (0 3) (20) b L +1 (X) const ( b L 2 (U ) + D b L (X) ). Proof. (i) We first consider the case = 0. From Lemma 5.6 and the condition (16), for any b 1 L 2,α 1 (X, Λ 1 ) and b 2 L 2, α 1 (X, Λ 1 ) (21) (22) b 1 L 2,α 1 b 2 L 2, α 1 const D α b 1 L 2,α, const D α b 2 L 2, α. Let b L 1 (X, Λ 1 (ade)). Let β be a smooth function on X such that β = 0 on t (r + 1/2) and β = 1 on t (r + 1) (t = q(x)). Recall that supp(µ) and supp(f 0 ) are contained in q 1 ( r, r ) and that W = e αt for t 1. By applying the above (21) to βa, we get (23) βb L 1 (X) const D 0 (βb) L (X) const ( b L 2 (U) + D 0 b L (X) ). Let β be a smooth function on X such that β = 0 on t (r + 1/2) and β = 1 on t (r + 1). By applying (22) to β b, we get (24) β b L 1 (X) const D 0 (β b) L (X) const ( b L 2 (U) + D 0 b L (X) ). From the elliptic regularity, b L 1 (q 1 ( (r+3/2),(r+3/2) )) const ( b L 2 (U) + D 0 b L 2 (U) ). his estimate and the above (23) and (24) imply (25) b L 1 (X) const ( b L 2 (U) + D 0 b L (X) ). Next let b L +1 (X, Λ1 (ade)). From the elliptic regularity, for any n Z b L +1 (q 1 ((n 1/2), (n+1/2) )) = W b L 2 +1 (q 1 ((n 1/2), (n+1/2) )) const ( W b + D L 2 (Y (n) ) 0 (W b) L 2 (Y (n) )) const ( b L (Y (n) he above two const are independent of n Z. herefore ) + D 0 b L (26) b L +1 (X) const ( b L (X) + D 0 b L (X) ). By using this estimate and the above (25), we can inductively prove (19). (ii) From (i) (27) b L +1 (X) C( b L 2 (U) + D 0 b L (X) ), (Y (n) )).

24 M. SUKMOO where the positive constant C depends on µ, α,, 0. ae ε > 0 so that Cε < 1. From (17), (18) and a L 3, there is a positive integer r > r (U := q 1 ( (r + 5/2), (r + 5/2) ) U) such that On the other hand herefore, from (27), Since Cε < 1, we get D b D 0 b L (X\U ) ε b L (X) (0 3). D b D 0 b L (U ) const b L 2 (U ). b L +1 (X) const ( b L 2 (U ) + D b L (X) ) + Cε b L (X). By the induction on, we get (20). b L +1 (X) const ( b L 2 (U ) + D b L (X) ). Proposition 6.2. Let. If b L (X, Λ 1 (ade)) satisfies D b = 0 as a distribution, then b L 4 (X). Let 0 3. he ernel of the map D : L +1 (X, Λ1 (ade)) L (X, (Λ 0 Λ + )(ade)) is of finite dimension, and the image D (L +1 (X, Λ1 (ade))) is closed in L (X, (Λ 0 Λ + )(ade)). Proof. he first regularity statement (D b = 0 b L 4 ) follows from Lemma 6.1 (ii). Let KerD be the space of b L 4 (X, Λ 1 (ade)) satisfying D b = 0. For any b er D, b L 4 (X) const b L 2 (U ) by Lemma 6.1 (ii). Here U is a pre-compact open set. hen the standard argument using Rellich s lemma shows the finite dimensionality of er D. Sublemma 6.3. If b L +1 (X, Λ1 (ade)) (0 3) is L -orthogonal to KerD (i.e. X W 2 b, β dvol = 0 for all β KerD ) then b L +1 (X) const D b L (X). Proof. It is enough to prove b L 2 (U ) const D b L (X). Since U is pre-compact, this follows from the standard argument using Lemma 6.1 (ii) and Rellich s lemma. Let H L +1 (X, Λ1 (ade)) be the L -orthogonal complement of er D. hen Sublemma 6.3 shows that image(d ) = D (H) is a closed subspace in L (X, Λ 1 (ade)). 6.2. he ernel of D is infinite dimensional. For µ : Λ Λ + we define µ : Λ + Λ by µ(ξ) η = ξ µ (η) (ξ Λ, η Λ + ). Let = 0 + a. For ω Ω 2 (ade), we set d,w ω = W 2 d ( W 2 ω). If b Ω 1 (ade) and ω Ω 2 (ade) have compact supports, then W 2 d X b, ω dvol = X W 2 b, d,w ω dvol. For ρ = (u, η) Ω0 (ade) Ω + (ade), we set D ρ := d u + d,w (1 + µ )η = d u W 2 d (W 2 (1 µ )η).

N OPEN FOUR-MNIFOLD HVING NO INSNON 25 D is an elliptic differential operator. If b Ω1 (ade) and ρ Ω 0 (ade) Ω + (ade) have compact supports, then W 2 D X b, ρ dvol = W 2 b, D X ρ dvol. We have (28) D (u, η) = D 0 (u, η) [a, u] [a (1 µ )η]. From the multiplication rule (17), D defines a bounded linear map D : L +1 (X, (Λ0 Λ + )(ade)) L (X, Λ 1 (ade)) for 0 3. Lemma 6.4. For any ρ L +1 (X, (Λ0 Λ + )(ade)) (0 3), ρ L +1 (X) const ( ρ L (X) + D ρ L (X) ). Hence if ρ L (X) satisfies D ρ = 0 as a distribution, then ρ L 4 (X). Proof. In the same way as in the proof of the estimate (26), we get ρ L +1 (X) const ( ρ L (X) + D 0 ρ L (X) ). By using the multiplication rule (17), we get the desired estimate. he regularity statement easily follows from the above estimate. Let KerD be the space of b L 4 (X, Λ 1 (ade)) satisfying D b = 0, and KerD be the space of ρ L 4 (X, (Λ 0 Λ + )(ade)) satisfying D ρ = 0. Lemma 6.5. Let and 0 3. (i) We have the following L -orthogonal decomposition: L (X, (Λ 0 Λ + )(ade)) = D (L +1 (X, Λ1 (ade))) KerD. (ii) If ρ L +1 (X, (Λ0 Λ + )(ade)) is L -orthogonal to the space KerD, then ρ L +1 (X) const D ρ L (X). Hence D (L +1 (X, (Λ0 Λ + )(ade))) is a closed subspace in L (X, Λ 1 (ade)). (iii) We have the following L -orthogonal decomposition: L (X, Λ 1 (ade)) = D (L +1 (X, (Λ0 Λ + )(ade))) KerD. Proof. (i) KerD is closed in L. From Proposition 6.2, D (L +1 ) is closed in L, and it is L -orthogonal to KerD. If ρ L ((Λ 0 Λ + )(ade)) is L -orthogonal to the space D (L 1 ), then D ρ = 0 as a distribution. Hence L = D (L 1 ) KerD. By this decomposition, for ρ L ((Λ 0 Λ + )(ade)), there are b L 1 and ρ KerD satisfying ρ = D b + ρ. By Lemma 6.4, ρ L 4 and hence D b = ρ ρ L. hen by Lemma 6.1 (ii), b L +1 (ii) By (i), there is b L. his shows L = D (L +1 ) KerD. +1 (X, Λ1 (ade)) satisfying ρ = D b. We can choose b so that it is L -orthogonal to KerD and that b L const D b L = const ρ L (by Sublemma 6.3). hen ρ 2 L = ρ, D b L = D ρ, b L D ρ L b L const D ρ L ρ L.