The anti-self-dual deformation complex and a conjecture of Singer

. Let .M 4 ; g/ be a smooth, closed, oriented anti-self-dual (ASD) four-man-ifold. .M 4 ; g/ is said to be unobstructed if the cokernel of the linearisation of the self-dual Weyl tensor is trivial. This condition can also be characterised as the vanishing of the second cohomology group of the ASD deformation complex


Introduction
Let M 4 be a smooth, closed, oriented four-manifold.Given a Riemannian metric g on M 4 , the bundle of two-forms ƒ 2 D ƒ 2 .M 4 / splits into the subbundles of self-dual and antiself-dual two-forms under the action of the Hodge ?-operator: By a result of Singer-Thorpe [37], the curvature operator RmW ƒ 2 !ƒ 2 has a canonical block decomposition of the form where where I is the identity and R is the scalar curvature.
The corresponding author is Matthew J. Gursky.
The notion of (anti-)self-duality is conformally invariant: if W C g D 0 for a metric g and z g D e f g, then W C z g D 0. This property will be crucial for the proof of our main result below.There are topological obstructions to the existence of ASD metrics.By the Hirzebruch signature formula, 48 2 .M 4 / D

Z
.jW C g j 2 jW g j 2 / dv g ; where .M 4 / is the signature of the intersection form on H 2 dR .M 4 /.In particular, we see that if .M 4 ; g/ is ASD, then .M 4 / Ä 0, with equality if and only if g is LCF.If .M 4 ; g/ is ASD with positive scalar curvature, then the intersection form is actually definite (see [32, Proposition 1]).
To see this, we first observe that the splitting of ƒ 2 .M 4 / induces a splitting on the space of harmonic two-forms; hence 4 /, then the Weitzenböck formula for the Hodge Laplacian 2 is given by where D g ij r i r j is the rough Laplacian.If .M 4 ; g/ is ASD and the scalar curvature R > 0, then (1.1) immediately implies !D 0.
Examples of ASD manifolds include locally conformally flat (LCF) manifolds since, in dimensions greater than three, LCF is equivalent to the vanishing of the Weyl tensor.In particular, S 4 endowed with the round metric g c is ASD.Non-simply connected examples include the product metric on S 3 S 1 , and more generally the metrics constructed via gluing on the connected sums S 3 S 1 # #S 3 S 1 .A non-LCF example is given by complex projective space with the Fubini-Study metric, and we take the opposite of the its natural orientation as a complex manifold, i.e., .CP 2 ; g FS /.
Roughly speaking, the constructions of ASD manifolds are based on either "twistor" or "analytic" methods.The former approach relies on the so-called Penrose correspondence, which will play no role in our work but is of profound importance in the study of ASD manifolds.Briefly, the unit sphere bundle Z of ƒ 2 C .M 4 / carries a canonical complex structure.As shown in [1], this complex structure is integrable if and only if the metric is ASD.Therefore, we can associate to any ASD manifold a complex manifold of (complex) dimension three, called the twistor space of .M 4 ; g/.This important observation allows one to use methods of complex geometry to study the existence and deformation theory of ASD conformal structures.
Analytic methods involve the construction of an ASD metric on the connected sum of two manifolds admitting ASD metrics via perturbative methods.As in other geometric gluing constructions, if .M 1 ; g 1 / and .M 2 ; g 2 / are ASD manifolds, one first constructs a metric h on the connected sum M 1 #M 2 which is "approximately" ASD, i.e., W C .h/ is small in some appropriately defined norm.This reduces the problem to the study of the mapping properties of the linearised operator, in order to perturb h to produce an actual ASD metric.To make this more precise, we now introduce the ASD deformation complex.
1.1.The ASD deformation complex.Let M.M 4 / be the space of smooth Riemannian metrics on M 4 , and R.M 4 / the bundle of algebraic curvature tensors.We can view W C as a mapping Let g 2 M.M 4 / be an ASD metric.We can identify the formal tangent space of M at g with sections of the bundle of symmetric two-tensors, S In fact, since g is ASD, by conformal invariance, D g .fg/D 0 for any f 2 C 1 .M 4 /; hence We also let D g W .W C / !.S 2 0 .T M 4 // denote the L 2 -formal adjoint of D g .Although the formula for D g is somewhat involved, the formula for D is much more compact (see [24,Proposition A.4]): where P denotes the Schouten tensor (see Section 2).Let K g W .T M 4 / !S 2 0 .T M 4 / denote the Killing operator, where ı g !D r k !k is the divergence of !.The kernel of K consists of those one-forms whose dual vector fields are conformal Killing.Moreover, by diffeomorphism and conformal invariance, Im K ker D.
The ASD deformation complex is given by Vanishing of the cohomology groups also provides information on the local structure of the moduli space of ASD conformal structures.Proposition 1.2 (see [24,40]).Suppose .M 4 ; g/ is ASD with Then the moduli space of anti-self-dual conformal structures near g is a smooth, finite-dimensional manifold of dimension dim H 1 ASD .M 4 ; g/.
This leads to the following definition.
Definition 1.3.Let .M 4 ; g/ be ASD.We say that .M 4 ; g/ is unobstructed if By the work of Floer [17] and Donaldson-Friedman [13], if .M 1 ; g 1 / and .M 2 ; g 2 / are unobstructed ASD manifolds, then the connected sum M 1 #M 2 admits an ASD metric.Thus, we are lead to the question: under what condition is an ASD manifold unobstructed?The following conjecture is often attributed to Singer 1) .Conjecture 1.4.Let .M 4 ; g/ be ASD.If the Yamabe invariant of .M 4 ; g/ is positive, then .M 4 ; g/ is unobstructed.
For ASD Einstein manifolds, the operators DD and D D were explicitly computed in [25] and [30].It follows from these calculations (and can also be seen by more or less direct calculation) that .S 4 ; g c / and .CP 2 ; g FS / are unobstructed.We remark that, for non-Einstein ASD manifolds, the formulas for these operators are fairly intractable.
The vanishing of H 2 ASD .M 4 ; OEg/ can sometimes be verified when the twistor space is explicitly known.For example, the LCF metrics k#S 3 S 1 (see [34,Theorem 8.2], [14]), and the ASD metrics on m#.CP 2 / constructed by LeBrun [33], are unobstructed.Our goal in this paper is to provide a criterion for the vanishing of H 2 ASD that only involves conformal invariants of the ASD manifold (and in particular does not depend on verifying any properties of the twistor space).Our main result is the following.Theorem 1.5.Suppose .M 4 ; g/ is ASD with Yamabe invariant Y .M 4 ; OEg/ > 0: 2 .M 4 / C 3 .M 4 / 1 24 2 Y .M 4 ; OEg/ 2 ; then .M 4 ; g/ is unobstructed. 1)See [16, p. 369], which also contains the index formula in (1.3) above.The authors thank Claude LeBrun for pointing out the apparent source of the conjecture in the literature.
The proof of Theorem 1.5 relies on two key ideas.The first is that any element U 2 ker D can be associated to a self-dual harmonic two-form z D z.U / 2 ƒ 2 C .A/ taking its values in the adjoint tractor bundle (see Section 2).Therefore, z satisfies a twisted version of the usual Weitzenböck formula for self-dual (real-valued) two-forms.This twisted version provides us with two identities for U (see Theorem 3.6).
The second key idea is to make a judicious choice of conformal representative in order to show that (1.4) implies the vanishing of U .As explained in Section 4, we choose a conformal metric whose scalar curvature satisfies a differential inequality involving the Schouten tensor (see Theorem 4.1), and is adapted to the curvature terms appearing in the Weitzenböck formula(s) for U .Curiously, to prove the existence of this metric, we consider a modification of the functional determinant of a conformally covariant elliptic operator first computed by Branson and Ørsted [4].A robust theory for the existence of critical points of this functional was developed by Chang and Yang [10], and we are able to show that their ideas also give existence for our modified functional.
If Y.M 4 ; OEg/ > 0, then by the estimate of Aubin, Therefore, (1.4) will fail (for topological reasons) if .2C 2 / is sufficiently negative.On the other hand, using Kobayashi's estimate [31] of the Yamabe invariant of connected sums, it is easy to construct examples of ASD metrics on connected sums of CP 2 or S 3 S 1 satisfying (1.4).
1.2.Organisation.In Section 2, we provide the necessary background on the tractor bundle and associated connection and metric.The main result (for our purposes) is Proposition 2.2, in which we associate to U 2 ker D a twisted SD harmonic two-form In Section 3, we compute two Weitzenböck formulas for U that follow from this correspondence.In Section 4, we give the proof of Theorem 1.5, and relegate the PDE aspects of our work to the appendix.
2. Background and the interpretation via tractor calculus 2.1.Some conventions for Riemannian geometry.For index calculations, we will use Penrose's abstract index notation, unless otherwise indicated.In this, we write E a and E a as alternative notations for, respectively, the cotangent and tangent bundles and the contraction !.v/ of a one-form ! with a tangent vector v a is written with a repeated index !a v a (mimicking the Einstein summation convention).Tensor bundles are denoted then by adorning the symbol E with appropriate indices and sometimes also indicating symmetries.For example, E .ab/ is the notation for S 2 T M , the subbundle of symmetric tensors in T M ˝T M .
Then our convention for the Riemann tensor R ab c d is such that where r a is the Levi-Civita connection of a metric g ab , v any tangent vector field, and ! is any one-form.Using the metric to raise and lower indices, we have, for example, This may be decomposed as where the completely trace-free part W abcd is the Weyl tensor and P ab is the Schouten tensor.
It follows that where R bc D R ab a c is the Ricci tensor and its metric trace R D g ab R ab is the scalar curvature.We will use J to denote the metric trace of Schouten, i.e., J ´gab P ab .Lastly, we have C abc ´2r OEb P ca ; where C abc and B ab are Cotton and Bach tensors, respectively.It should be noted that the Bianchi identities imply .n3/C abc D r d W dabc .

The tractor bundle and connection.
To treat and work with objects that are conformally invariant, it is natural work, at least partly, in the setting of conformal manifolds.Here, by a conformal manifold .M; c/, we mean a smooth manifold of dimension n 3 equipped with an equivalence class c of Riemannian metrics, where g ab , y g ab 2 c means that y g ab D 2 g ab for some smooth positive function .On a general conformal manifold .M; c/, there is no distinguished connection on TM .But there is an invariant and canonical connection on a closely related bundle, namely the conformal tractor connection on the standard tractor bundle; see [2,7].
Here we review the basic conformal tractor calculus on Riemannian and conformal manifolds.See [2,11,19] for more details.Unless stated otherwise, calculations will be done with the use of generic g 2 c.
On an n-manifold M , the top exterior power of the tangent bundle ƒ n TM is a line bundle.Thus its square K ´.ƒ n TM / ˝2 is canonically oriented, and so one can take oriented roots of it: given w 2 R, we set EOEw ´K w 2n and refer to this as the bundle of conformal densities.For any vector bundle V, we write VOEw to mean V OEw ´V ˝EOEw.For example, E .ab/OEw denotes the symmetric second tensor power of the cotangent bundle tensored with EOEw, i.e., S 2 T M ˝EOEw on M .On a fixed Riemannian manifold, K is canonically trivialised by the square of the volume form, and so K and its roots are not usually needed explicitly.However, if we wish to change the metric conformally, or work on a conformal structure, then these objects become important.
Since each metric in a conformal class determines a trivialisation of K, it follows easily that, on a conformal structure, there is a canonical section g ab 2 .E .ab//OE2.This has the property that, for each positive section 2 .E C OE1/ (called a scale), g ab ´ 2 g ab is a metric in c.Moreover, the Levi-Civita connection of g ab preserves and therefore g ab .Thus it makes sense to use the conformal metric to raise and lower indices, even when we are choosing a particular metric g ab 2 c and its Levi-Civita connection for calculations.It turns out that this simplifies many computations, and so, in this section, we will do that without further mention.(In the subsequent sections, we will work with a fixed metric and use that to trivialise density bundles -so indices will be raised and lowered using the metric.) Considering Taylor series for sections of EOE1, one recovers the jet exact sequence at 2-jets, we see at once that T has a composition series (or filtration structure) What this notation means is that EOE 1 is a subbundle of T , and the quotient of T by EOE 1 (which is J 1 EOE1) has E a OE1 as a subbundle, whereas there is a canonical projection XW T !EOE1.In abstract indices, we write X A for this map and call it the canonical tractor.
Given a choice of metric g 2 c, the formula (where is the Laplacian r a r a ) gives a second-order differential operator on EOE1 which is a linear map J 2 EOE1 !EOE1 ˚Ea OE1 ˚EOE 1 that clearly factors through T and so determines an isomorphism In subsequent discussions, we will use (2.5) to split the tractor bundles without further comment.Thus, given g 2 c, an element V A of E A may be represented by a triple .; a ; /, or equivalently by The last display defines the algebraic splitting operators Y W EOE1 !T and ZW T M OE1 !T (determined by the choice g ab 2 c) which may be viewed as sections We call the sections X A , Y A and Z A a tractor projectors.
By construction, the tractor bundle is conformally invariant, i.e., determined by .M; c/ and independent of any choice of g 2 c.However, the splitting (2.6) is not.Considering the transformation of the operator (2.4) determining the splitting, we see that if y g D 2 f , the components of an invariant section of T should transform according to (2.7) where ‡ a D 1 r a , and conversely, this transformation of triples is the hallmark of an invariant tractor section.Equivalent to the last display is the rule for how the algebraic splitting operators transform Given a metric g 2 c, and the corresponding splittings, as above, the tractor connection is given by the formula where on the right-hand side the rs are the Levi-Civita connection of g.Using the transformation of components, as in (2.7), and also the conformal transformation of the Schouten tensor, (2.10) reveals that the triple on the right-hand side transforms as a one-form taking values in T , i.e., again by (2.7) except twisted by E a .Thus the right-hand side of (2.9) is the splitting into slots of a conformally invariant connection r T on (a section of) the bundle T .
There is a nice conceptual origin for the connection (2.9).Using (2.10) and the transformation of the Levi-Civita connection, it is straightforward to verify that the equation (2.11) r .ar b/ 0 C P .ab/0 D 0 on conformal densities 2 .EOE1/ is conformally invariant.As this is an overdetermined PDE, solutions in general do not exist.Overdetermined linear PDEs are typically studied by prolongation, and it is quickly verified that the tractor parallel transport given by (2.9) is exactly the closed system that arises from prolonging (2.11) (see [2,11]).From this observation and formula (2.9), it follows that non-trivial solutions of (2.11) are non-vanishing on an open dense set (for M connected) on which the metric 2 g ab is Einstein -an observation that has a number of applications; see [11,18] and references therein.The tractor bundle is also equipped with a conformally invariant signature .nC 1; 1/ metric h AB 2 .E .AB/ /, defined as quadratic form by the mapping Tractor inner product and the polarisation identity.This is important not only by dint of its conformal invariance, but it is easily checked that this tractor metric h is preserved by r T a , i.e., r T a h AB D 0. Thus it makes sense to use h AB (and its inverse) to raise and lower tractor indices, and we do this henceforth without further comment.In particular, X A D h AB X B is the canonical tractor (and hence our use of the same kernel symbol).For computations, Table 1 is useful.We see that h may be decomposed into a sum of projections Finally, for this section, we note that, of course, the curvature of the tractor connection Ä abCD is determined by and can be written in terms of tractor projectors as The bundle A ´ƒ2 T is often termed the adjoint tractor bundle, as it is a vector bundle modelled on the Lie algebra of the conformal group SO.n C 1; 1/.So, as expected, the tractor curvature is a two-form taking values in this bundle.
2.3.The differential splitting operator.In dimensions n 4, the tractor curvature is the image of a conformally invariant operator z acting on the Weyl curvature.The operator is as follows.
Lemma 2.1.In any dimension n 4, there is a conformally invariant differential map given by (2.13) where .ıU/ kij D r m U mkij .
The conformal invariance is easily verified directly using (2.8) and the conformal transformation of the Levi-Civita connection.It can also be deduced from the conformal invariance of the tractor curvature Ä abCD .In fact, the map (2.12) is a standard "BGG-splitting operator", as in the theory [5,8], and the conformal deformation sequence can be understood as arising from a twisting by ƒ 2 T of the de Rham complex [6,20].
We are, in particular, interested in the case of dimension n D 4. Then it is evident, from formula (2.13), that if U is SD (or ASD), then so is z.U / as a tractor-twisted two-form, as the Hodge-?commutes with the Levi-Civita connection.We obtain more if also U 2 ker D .Proposition 2.2.Suppose .M 4 ; g/ is ASD, and Proof.First note that, by standard BGG theory [5,8,15], this is true in the conformally flat setting.Now consider the curved case.
Certainly, ız is conformally invariant as it is a twisting of the usual divergence of twoforms (in dimension four) with the conformally invariant tractor connection.Thus, starting from the top (meaning from the left in the filtration on the adjoint tractor bundle that is induced from (2.3)), the first non-zero slot of ız must be conformally invariant and constructed from U , its covariant derivatives and possibly curvature contracted with U .From order considerations, the last of these can only happen in the bottom slot.The very top slot is rank 2 and involves no derivatives of U .Thus it is zero, as U is trace-free.At the next level, we have one derivative of U , but it is well known that, in dimension 4, there is no such conformal invariant.
Thus the image of ız lies in the bottom slot which contains a rank 2 tensor.Considering the knowledge of conformally flat case, it follows that this conformally invariant object must be a multiple of D U , plus possibly another conformally invariant rank 2 tensor constructed by contracting curvature into U .It is easily checked that it is not possible to construct a rank 2 conformal invariant by contracting curvature into U , as U is SD, while the Weyl curvature is, by assumption, ASD.Thus the only possibility is that the bottom slot is a non-zero multiple of D U .
We will give another proof of this result (by direct calculation) in the next section; see Corollary 3.3.Proof.z satisfies a twisted version of the Weitzenböck formula as in [23, Lemma 2.1].If .M 4 ; g/ is ASD, the only non-zero curvature term is given by 1  3 R.

Weitzenböck formula(s)
In this section, we use Proposition 2.2 and Corollary 2.3 to prove a Weitzenböck formula for z D z.U / 2 ƒ 2 C .A/ when U 2 ker D .We begin with some more general calculations.
Proposition 3.1.Let U 2 .W C / and let z D z.U / be given by (2.13).Then the covariant derivative of z is given by For the proof of this and the following proposition, we will use the following formulas (see [19, (6)]): Lemma 3.2.The following formulas hold: Proof.First, by the Leibniz rule, Proof of Proposition 3.1.By (2.13), We can now give another proof of Proposition 2.2.
Corollary 3.3.Suppose .M 4 ; g/ is ASD, and U is SD.Then U 2 ker D if and only if Proof.Since z is self-dual, it is harmonic if and only if ız D 0. By (3.1), Since U is a curvature-type tensor for which all contractions vanish, it follows that With the same assumptions as Proposition 3.1, the rough Laplacian of z, i.e., z ijAB D g k`r k r `zijAB , is given by Proof.By (3.1), Then, applying the formulas from Lemma 3.2 and using the Bianchi identity to write we get Since the very last term is skew-symmetric in k; `, we can rewrite this as The proposition will follow, once we prove the following lemma.
Proof.By definition of the divergence and the fact that Using the decomposition of the curvature tensor as in (2.1), it follows that .ı m j P k` g km P j ı j P km C g k`P m j /U `mi k : Using the fact that all contractions of U vanish, the above simplifies to (note that all terms involving the Schouten tensor can be seen to cancel after re-indexing).Finally, since U 2 .W C /, (3.3) follows.
This ends the proof of 3.4.
Theorem 3.6.If .M 4 ; g/ is ASD and U 2 ker D , then C .A/ is harmonic.Therefore, (3.4) and (3.5) follow from Proposition 3.4 and Corollary 2.3.Remark 3.7.Although we will not provide a proof, it is not difficult to show that (3.5) holds for any section U 2 .W C /, i.e., the condition U 2 ker D is not necessary.However, this is not the case for (3.4).

Proof of Theorem 1.5
The total Q-curvature is a conformal invariant, and we can rewrite (1.4) as a condition on the total Q-curvature and the Yamabe invariant as follows.By the Chern-Gauss-Bonnet formula, ( Since g is ASD, ( Z jW g j 2 dv g : By the Hirzebruch signature formula, Combining this with (4.2), we see that Therefore, assumption (1.4) is equivalent to Proof of Theorem 1.5.Since our assumptions are conformally invariant, it suffices to prove the result when g replaced by any metric in the same conformal class.The metric we will use is a solution of a variational problem related to the regularised determinant of an elliptic operator.The precise formulation is contained in Theorem A.2 in the appendix, while the following consequence will suffice for our purposes.
Then there is a smooth, unit volume conformal metric g 2 OEg 0 with J g D tr P g > 0 satisfying where V P g D P 1 4 Jg is the trace-free Schouten tensor.
The proof of this result is somewhat involved, and will also be given in the appendix.In the following, we will show how Theorem 1.5 follows from Theorem 4.1 and the Weitzenböck formulas of the preceding section.For the rest of the proof, the metric g is assumed to satisfy the conclusions of Theorem 4.1, and we will usually suppress the subscript g.
Assume U 2 ker D .We will record two integral identities that follow from Theorem 3.6 and inequality (4.3).First, by (3.5), Recalling that If we multiply both sides by J and integrate over M 4 , then Integrating by parts on the left and using (4.3), we find Combining (4.5) and (4.6), (4.7) 0 We now use (3.4) to get Integrating this over M 4 gives If we integrate by parts in the second term and use the contracted second Bianchi identity Substituting this result back into (4.8)gives Next, integrate by parts in the second-to-last term in (4.9) to get Substituting this back into (4.9)gives Finally, we rewrite the curvature terms above in terms of V P and J as follows: Multiplying by two and rearranging terms, we find We want to combine (4.10) with (4.7).To do so, we first use (4.10) to write Substituting this into (4.7),we have where Jg km : If we define the tensor then the term involving T in (4.11) can be estimated (via Cauchy-Schwarz) by By the arithmetic-geometric mean inequality, (4.12) By the definition of V , Consequently, (4.12) implies Substituting this into (4.11),we conclude ¹4J jıU j 2 C J jrU j 2 º dv: Since J > 0, it follows that rU D 0. However, it is then immediate from (4.7) that U Á 0.
A. Appendix: The proof of Theorem 4.1 In this appendix, we give the proof of Theorem 4.1.The material in this section is an extension of the existence work of Chang-Yang [10] for critical points of the regularised determinant of conformally covariant operators.We begin with a brief overview of their work, omitting the motivation from spectral geometry and limiting ourselves to the underlying variational problem.
Let .M 4 ; g/ be a closed, four-dimensional Riemannian manifold, and W 2;2 .M / the Sobolev space of functions whose weak derivatives up to order two are in L 2 .Consider the following functionals on W 2;2 .M /: where P g denotes the Paneitz operator, Q g the scalar curvature, and ª denotes the normalised integral (i.e., divided by the volume of g).Let 1 ; 2 ; 3 be constants and let F W W 2;2 .M / !R be given by We also define the associated conformal invariant As explained in [10], critical points of F determine a conformal metric satisfying a fourth order curvature condition.More precisely, if we define the U -curvature of g by then w is a smooth critical point of F if and only if the conformal metric g F D e 2w g satisfies U.g F / Á for some constant .A general existence result for critical points of F was proved in [10].
Regularity of extremals was proved by the second author in joint work with Chang-Yang [9]; later Uhlenbeck-Viaclovsky proved a more general regularity result for arbitrary critical points of F (see [39]).
To prove Theorem 4. Vol.z g/ 1=2 : Given a constant 4 , we define ˆW W 2;2 .M / !R by A trivial modification of the existence result of Chang-Yang and the regularity results of Chang-Gursky-Yang and Uhlenbeck-Viaclovsky gives the following.
Proof.Note that the functional ˆis scale-invariant: ˆOEw C c D ˆOEw for any constant c.Therefore, we may normalise a maximising sequence ¹w k º for ˆso that Z e 4w k dv g D 1: Assuming Consequently, if 4 Ä 0, then 4 IVOEw is bounded above.By (A.1), it is also bounded below (by the Yamabe invariant).Therefore, the addition of this term has no effect on the estimates in the existence proof of Chang-Yang.
We are now ready to prove Theorem 4.1.
Proof of Theorem 4.1.We first remark that if .M 4 ; g 0 / is conformally equivalent to the round sphere (suitably normalised), then g D g c satisfies the conclusions of the theorem.Therefore, we may assume .M 4 ; g 0 / is not conformally the round sphere.Taking

Theorem 4 . 1 .
Let .M 4 ; g 0 / be a closed Riemannian four-manifold with 2.T M 4 /.Let DW .S 2 .T M 4 // !.R.M 4 // denote the linearisation of W C at g, i.e., for h 2 .S 2 .T M 4 //, The choice of a conformal class of metrics OEg determines the bundle of algebraic Weyl tensors, W D W .M 4 ; OEg/, and the subbundle W C W S 2 0 .ƒ 2 C /, where S 2 0 .ƒ 2 C / is the bundle of symmetric, trace-free endomorphisms of ƒ 2 C .Note that .S 2 .T M 4 //, .ıg h/ j D g i k r k h ij is the divergence.The Atiyah-Singer index theorem can be used to calculate the index of (1.2): This complex is elliptic; see [29, Section 2].The associated cohomology groups are given by H 0 ASD .M 4 ; g/ D ker K g ; H 1 ASD .M 4 ; g/ D ¹h 2 .S 2 0 .T M 4 // W ı g h D 0; D g h D 0º; H 2 ASD .M 4 ; g/ D ker D g ; where, for h 2 4 / C 29 .M 4 / : and its sequence (2.2) are canonical objects on any smooth manifold.But, with a conformal structure c, we have the orthogonal decomposition of E ab OE1 into trace-free and trace parts E ab OE1 D E .ab/0 OE1 ˚gab EOE 1: we can canonically quotient J 2 EOE1 by the image of E .ab/0 OE1 under Ã (in (2.2)).The resulting quotient bundle is denoted T (or E A in abstract indices) and called the conformal cotractor bundle.Observing that the jet exact sequence at 1-jets (of EOE1), 1, we need to introduce another functional R g C , it follows from the Schwartz inequality thatIVOEw D Z .R g C 6jrw k j 2 /e 2w k dv g Z R g e 2w k dv g C ÂZ e 4w k dv g ZQ g 0 dv g 0 < 8 2 :By[22,Theorem B], (A.2) holds as long as .M 4 ; g 0 / is not conformally equivalent to the round sphere.Therefore, by Theorem A.2, there is a smooth conformal metric g D e 2w g 0 (which we can normalise to have unit volume) satisfying R g Y .M 4 ; OEg 0 /R g D :For the rest of the proof, we will omit the subscript g.If E D Ric 1 4Rg denotes the trace-free Ricci tensor of g, then we may use the definition of the Q-curvature to rewrite (A.3) as For now, let us assume the claim and see how the theorem follows.From (A.6) and (A.5),Let > 0 denote the eigenfunction associated to the first eigenvalue 1 .L/ of the conformal Laplacian It follows from the strong maximum principle that R= > 0 on M ; hence R > 0. This completes the proof of the theorem, once we prove Claim A.3.Proof of Claim A.3.If we integrate (A.3) over M and use the fact that g has unit volume, we obtain By definition of the Yamabe invariant (again using the fact that g has unit volume) and the fact that Y .M 4 ; OEg 0 / > 0,Y .M 4 ; OEg 0 / Z R g dv g Y .M 4 ; OEg 0 / 2 : Z Q g dv g Y .M 4 ; OEg 0 / 2 :Since the total Q-curvature is a conformal invariant, using assumption (ii) of the theorem, we see thatÄ 6 Z Q g dv g Y .M 4 ; OEg 0 / 2 D 6 Z Q g 0 dv g 0 Y .M 4 ; OEg 0 / 2 (A.8) L ´. 6 C R/ D 1 .L/:Since Y .M 4 ; OEg/ > 0, it follows that 1 .L/ > 0. An easy calculation using (A.7) and (A.g dv g Y .M 4 ; OEg 0 / Z R g dv g : 4 ; OEg 0 / 2 Y .M 4 ; OEg 0 / 2 D 1 2 Y .M 4 ; OEg 0 / 2 ; which proves (A.6).