Constant Scalar Curvature Metrics on Connected Sums

Let (M,g) be a compact Riemannian manifold with dimension n>2. The Yamabe problem is to find a metric with constant scalar curvature in the conformal class of g, by minimizing the total scalar curvature. The proof was completed in 1984. Suppose (M',g') and (M'',g'') are compact Riemannian n-manifolds with constant scalar curvature. We form the connected sum M' # M'' of M' and M'' by removing small balls from M' and M'' and joining the S^{n-1} boundaries together. In this paper we use analysis to construct metrics with constant scalar curvature on M' # M''. Our description is quite explicit, in contrast to the general Yamabe case when one knows little about what the metric looks like. There are 9 cases, depending on the signs of the scalar curvature on M' and M'' (positive, negative, or zero). We show that the constant scalar curvature metrics either develop small"necks"separating M' and M'', or one of M', M'' is crushed small by the conformal factor. When both have positive scalar curvature, we construct three different metrics with scalar curvature 1 in the same conformal class.


Introduction
Let (M ′ , g ′ ) and (M ′′ , g ′′ ) be compact manifolds of dimension n 3. The connected sum M = M ′ #M ′′ is the result of removing a small ball B n from each manifold, and joining the resulting manifolds at their common boundary S n−1 . By smoothly joining g ′ and g ′′ we can also construct a 1-parameter family of metrics g t on M for t ∈ (0, δ), where t measures the radius of the excised balls B n .
In this paper we suppose g ′ and g ′′ have constant scalar curvature (not necessarily the same value), and study (some of) the metricsg t with constant scalar curvature in the conformal class of g t for small t. Our method is to write down explicit metrics whose scalar curvature is close to constant, and show using analysis that they can be adjusted by a small conformal change to give metrics with constant scalar curvature.
By the proof of the Yamabe problem, every conformal class on M contains a metric with constant scalar curvature. Our proofs are simpler than the solution of the Yamabe problem, as our problem is much easier, and they have the advantage of giving a good grasp of what the Yamabe metrics actually look like as the underlying conformal manifold decays into a connected sum. We can for instance say that one obvious sort of behaviour, that of developing long 'tubes' resembling S n−1 × R does not happen, but that small 'pinched necks' may develop instead, or else M ′ or M ′′ may be crushed very small by the conformal factor.
Other authors have also considered the scalar curvature of metrics on connected sums. Two important early papers on manifolds with positive scalar curvature are Gromov and Lawson [3] and Schoen and Yau [11], and both show [3,Th. A], [11,Cor. 3] that the connected sum of two manifolds with positive scalar curvature carries metrics with positive scalar curvature.
Writing after the solution of the Yamabe problem, Kobayashi [6] defines the Yamabe number of a manifold to be the supremum over conformal classes of the Yamabe invariant defined in §1.2, and proves an inequality [6,Th. 2(a)] between the Yamabe number of two manifolds and the Yamabe number of their connected sum. His formula can be related to our results below in the limit t → 0.
The first version of this work formed part of the author's D.Phil. thesis [5] in 1992, supervised by Simon Donaldson. In the interval between then and the publication of the present paper several papers on similar topics have appeared, and we mention in particular Mazzeo, Pollack and Uhlenbeck [8]. They study connected sums of compact or noncompact manifolds with constant positive scalar curvature, and the part of Theorem 3.9 below dealing with the metrics of §2.2 also follows from their main result.
The remainder of this section goes over some necessary backgound material. In §2 we define two families of metrics g t upon connected sums, make estimates of the scalar curvature of these metrics, and prove a uniform bound on a Sobolev constant for a particular embedding of Sobolev spaces.
The main existence results for metrics of constant scalar curvature conformal to the metrics g t above are proved in §3. We begin with a quite general existence proof using a sequence method and then apply it, first to the case of scalar curvature −1, and then to the case of scalar curvature 1. The former is simple, but the latter is more difficult, and the proof of a result on the eigenvalues of the Laplacian ∆ on the metrics g t in the positive case has been deferred until the appendix.
In §4 we deal with the cases left over from §3, which are the connected sums involving manifolds of zero scalar curvature. To do so requires some additions to the methods of §3, as the problem of rescaling a metric with scalar curvature close to ±1 to get exactly ±1 is different to the problem of rescaling a metric with scalar curvature close to zero to get scalar curvature a small but unknown constant. We shall see that each combination of positive, negative and zero scalar curvature manifolds has distinctive features.
Acknowledgements. The author would like to thank Simon Donaldson for many useful conversations, and Rafe Mazzeo and the referees for helpful comments.

Analytic preliminaries
We define Lebesgue and Sobolev spaces, principally to establish notation. An introduction to them may be found in Aubin [1, §2]. Let (M, g) be a Riemannian manifold. For q 1, the Lebesgue space L q (M ) is the set of locally integrable functions u on M for which the norm g u L q = u L q = M |u| q dV g 1/q is finite. Here dV g is the volume form of the metric g. Then L q (M ) is a Banach space (with the convention that two functions are equal if they differ only on a null set) and L q (M ) a Hilbert space. Let r, s, t 1 with 1/r = 1/s + 1/t. If φ ∈ L s (M ) and ψ ∈ L t (M ) then φψ ∈ L r (M ) and φψ L r φ L s ψ L t ; this is Hölder's inequality.
Let q 1 and let k be a nonnegative integer. The Sobolev space L q k (M ) is the subspace of u ∈ L q (M ) such that u is k times weakly differentiable and |∇ i u| ∈ L q (M ) for i k. The Sobolev norm on L q k (M ) is . This makes L q k (M ) into a Banach space, and L 2 k (M ) a Hilbert space. Since we will often have to consider different metrics on the same manifold, we will use the superscript notation g . L q and g . L q k to mean Lebesgue and Sobolev norms computed using the metric g.
We also write C k (M ) for the vector space of continuous, bounded functions on M with k continuous, bounded derivatives, and C ∞ (M ) = k 0 C k (M ). The Sobolev Embedding Theorem [1,Th. 2.20] says that if 1 r 1 q − k n then L q k (M ) is continuously embedded in L r (M ) by inclusion, and if − r n > 1 q − k n then L q k (M ) is continuously embedded in C r (M ) by inclusion.

The Yamabe problem
We now briefly discuss the Yamabe problem, concerning the existence of metrics of constant scalar curvature in a conformal class on a compact manifold. An introduction to the problem and its solution may be found in the survey paper [7] by Lee and Parker. Fix a dimension n 3. Throughout the paper we shall use the notation a = 4(n − 1) n − 2 , b = 4 n − 2 and p = 2n Let M be a compact manifold of dimension n. Define a functional Q upon the set of Riemannian metrics on M by where S is the scalar curvature of g. Then Q is known as the total scalar curvature, or Hilbert action. Let g,g be metrics on M . We say that g,g are conformal ifg = φg for some smooth conformal factor φ : M → (0, ∞). Usually we write φ = ψ p−2 for smooth ψ : M → (0, ∞). The conformal class [g] of g is the set of metricsg on M conformal to g. Applying the calculus of variations to the restriction of Q to a conformal class, one finds that a metric g is a stationary point of Q on [g] if and only if it has constant scalar curvature.
(2) Thusg = ψ p−2 g has constant scalar curvature ν if and only if ψ satisfies the Yamabe equation: One can show using Hölder's inequality that Q is bounded below on a conformal class. So we define λ [g] = inf Q(g) :g is conformal to g . The Yamabe problem [13]. Given a compact Riemannian n-manifold (M, g), find a metricg conformal to g which minimizes Q on [g], so that Q(g) = λ [g] . Theng is a stationary point of Q on [g], and so has constant scalar curvature.
A solutiong of the Yamabe problem is called a Yamabe metric. The problem was posed in 1960 by Yamabe [13], who gave a proof that such a metricg always exists. His idea was to choose a minimizing sequence for Q on [g], and show that a subsequence converges to a smooth minimizerg for Q. Unfortunately the proof contained an error, discovered by Trudinger [12]. The proof was eventually repaired by Trudinger, Aubin, Schoen and Yau.
If λ [g] 0 then constant scalar curvature metrics in [g] are unique up to homothety [1, p. 135], and are Yamabe metrics. Thus, if λ [g] < 0 then [g] contains a unique metricg with scalar curvature −1, and if λ [g] = 0 then [g] contains a unique metricg with scalar curvature 0 and volume 1. However, if λ [g] > 0 then constant scalar curvature metrics in [g] are not necessarily unique up to homothety, and may be higher stationary points of Q rather than Yamabe metrics. Thus, [g] may contain several different metrics with scalar curvature 1. We will see an example of this in §3. 3.
In §3 and §4, weak solutions in L 2 1 (M ) to the Yamabe equation (3) will be constructed for certain special compact manifolds M . For these solutions to give metrics of constant scalar curvature, it is necessary that they be not just L 2 1 solutions, but C ∞ solutions. We quote a result of Trudinger [12,Th. 3,p. 271] showing this is the case. Proposition 1.1 Let (M, g) be a compact Riemannian n-manifold with scalar curvature S, and u a weak L 2 1 solution of a∆u + Su =S|u| (n+2)/(n−2) , for S ∈ C ∞ (M ). Then u ∈ C 2 (M ), and where u is nonzero it is C ∞ .

Stereographic projections
Let (M, g) be a compact Riemannian manifold with positive scalar curvature. Then by [7, Lem. 6.1, p. 63] the operator a∆ + S in (3) admits a Green's function Γ m for each m ∈ M , which is unique and strictly positive. That is, The stereographic projection of S n is R n . In general, stereographic projections (M ,ĝ) are asymptotically flat, that is, their noncompact ends resemble R n in a way made precise in [7,Def. 6.3]. Following Lee and Parker [7, p. 64], we use the notation that φ = O ′ (|x| q ) means φ = O(|x| q ) and ∇φ = O(|x| q−1 ), and φ = O ′ (|x| q ) means φ = O(|x| q ), ∇φ = O(|x| q−1 ) and ∇ 2 φ = O(|x| q−2 ), for small or large x in R n , depending on the context. For simplicity we shall suppose that g is conformally flat near m in M . In this case the asymptotic expansion of the metricĝ is particularly simple, [7, Th. 6.5(a)]. Proposition 1.3 Let (M, g) be a compact Riemannian manifold with positive scalar curvature, and suppose m ∈ M has a neighbourhood that is conformally flat. Let (M ,ĝ) be the stereographic projection of M from m. Then there exist µ ∈ R and asymptotic coordinates {x i } onM with respect to whichĝ ij satisfieŝ This constant µ is proportional to an important invariant of asymptotically flat manifolds called the mass, which is defined and studied by Bartnik [2]. For physical reasons it was conjectured that the mass must be nonnegative for complete asymptotically flat manifolds of nonnegative scalar curvature, and zero only for flat space. This was proved for spin manifolds by Witten [14], whose proof was generalized to n dimensions by Bartnik [2, §6].
A proof of the general case when n 7 is given by Schoen [10, §4]. He also claims the result for all dimensions [9, p. 481], [10, p. 145], but the proof has not been published. This 'positive mass theorem' in dimensions 3,4 and 5 was important in the completion of the proof of the Yamabe problem by Schoen in 1984 [9]. We will only need the case when the metric g of M is conformally flat about m, which we state as follows: In the situation of Proposition 1.3, (perhaps with the additional assumption that M is spin or n 7), we have µ 0, with equality if and only if M is S n , and its metric is conformal to the round metric.

Glued metrics on connected sums
In this section, we shall define two families of metrics g t on the connected sum M of constant scalar curvature Riemannian manifolds (M ′ , g ′ ) and (M ′′ , g ′′ ).
The first family, in §2.1, is made by choosing M ′ of constant scalar curvature ν, cutting out a small ball, and gluing in a stereographic projection of M ′′ , homothetically shrunk very small. The second family, in §2.2, is made by choosing M ′ and M ′′ both of constant scalar curvature ν, and joining them by a small 'neck' of zero scalar curvature. The relation between these families when ν = 1 is discussed in §2.3.
We finish with two results, Propositions 2.1 and 2.2, about the families of metrics. The first gives explicit bounds for their scalar curvature, to determine how good an approximation to constant scalar curvature they are. The second shows that the Sobolev constant for a certain Sobolev embedding of function spaces can be given a bound independent of the width of the neck, for small values of this parameter.

Combining constant and positive scalar curvature
Let (M ′ , g ′ ) be a compact Riemannian n-manifold with constant scalar curvature ν. Applying a homothety to g ′ if necessary, we may assume that ν = 1, 0 or −1. For simplicity, assume that m ′ ∈ M ′ has a neighbourhood in which g ′ is conformally flat; this assumption will be dropped in §3. 4.
Let (M ′′ , g ′′ ) be a compact Riemannian n-manifold with scalar curvature 1. As with M ′ , suppose m ′′ ∈ M ′′ has a neighbourhood in which g ′′ is conformally flat. Let (M ,ĝ) be the stereographic projection of M ′′ from m ′′ , as in Definition 1.2, so thatM = M ′′ \ {m ′′ }, andĝ is asymptotically flat with zero scalar curvature, and is conformal to g ′′ . By Proposition 1.3, there is an immersion Ξ ′′ : R n \ B R (0) →M for some R > 0, whose image is the complement of a compact set inM , such that A family of metrics {g t : t ∈ (0, δ)} on M = M ′ #M ′′ will now be written down. For any t ∈ (0, δ), define M and the conformal class of g t by where t is the equivalence relation defined by The conformal class [g t ] of g t is then given by the restriction of the conformal classes of g ′ andĝ to the open sets of M ′ , M ′′ that make up M ; this definition makes sense because the conformal classes agree on the annulus of overlap where the two open sets are glued by t . Let A t be this annulus in M . Then A t is diffeomorphic via Φ ′ to the annulus {v ∈ R n : t 2 < |v| < t} in R n .
To define a metric g t within the conformal class just given, take g t = g ′ on the component of M \ A t coming from M ′ , and g t = t 12ĝ on the component coming fromM . It remains to choose a conformal factor on A t , which is identified with {v ∈ R n : The conditions for smoothness at the edges of the annulus A t are that ψ t (v) should join smoothly onto ψ ′ (v) at |v| = t, and onto ξ(t −6 v) at |v| = t 2 . Choose a C ∞ function σ : R → [0, 1], that is 0 for x 2 and 1 for x 1 and strictly decreasing in [1,2]. Let β 1 (v) = σ(log |v|/ log t) and β 2 (v) = 1 − β 1 (v) for all v ∈ R n with t 2 < |v| < t. Finally, define ψ t by This finishes the definition of g t for t ∈ (0, δ). The reasoning behind the definition -whyĝ is shrunk by a factor of t 6 , but the cut-off functions change between radii t and t 2 , for instance -will emerge in §2.4, where we show that for this definition the scalar curvature of g t is close to the constant function ν in the L n/2 norm.

Combining two metrics of constant scalar curvature ν
A family of metrics will now be defined on the connected sum of two Riemannian manifolds with constant scalar curvature ν. We shall do this by writing down a zero scalar curvature Riemannian manifold with two asymptotically flat ends, and gluing one end into each of the constant scalar curvature manifolds using the method of §2.1; the new manifold will form the 'neck' in between.
Let (M ′ , g ′ ) and (M ′′ , g ′′ ) be Riemannian n-manifolds with constant scalar curvature ν; applying homotheties if necessary we shall assume that ν = 1, 0 or −1. We shall use the gluing method of §2.1 to glue the two asymptotically flat ends of (N, g N ) into (M ′ , g ′ ) and (M ′′ , g ′′ ). Let M = M ′ #M ′′ . A family of metrics {g t : t ∈ (0, δ)} on M will be defined, such that g t resembles the union of (M ′ , g ′ ) and (M ′′ , g ′′ ) joined by a small 'neck' of radius t 6 , modelled upon (N, t 12 g N ). It will be done briefly, as the treatment generalizes §2.1.
Suppose that M ′ , M ′′ contain points m ′ , m ′′ with neighbourhoods in which g ′ , g ′′ are conformally flat. (In §3.4 this assumption will be dropped.) Thus For any t ∈ (0, δ), define M and the conformal class of g t by where t is the equivalence relation defined by The conformal class [g t ] of g t is then given by the restriction of the conformal classes of g ′ , g ′′ and g N to the open sets of M ′ , M ′′ and N that make up M ; this definition makes sense because the conformal classes agree on the annuli of overlap where the three open sets are glued by t . Let A t be this region of gluing in M . Then A t is diffeomorphic via Φ ′ and Φ ′′ to two copies of the annulus {v ∈ R n : t 2 < |v| < t}.
To define a metric g t within this conformal class, let g t = g ′ on the component of M \ A t coming from M ′ , g t = g ′′ on the component of M \ A t coming from M ′′ , and g t = t 12 g N on the component on M \ A t coming from N . So it remains to choose a conformal factor on A t itself. Using Φ ′ and Φ ′′ , this is the same as choosing a conformal factor for two copies of the subset {v ∈ R n : t 2 < |v| < t} of R n .
As in §2.1 define a partition of unity β 1 , β 2 on A t , and define ψ t by ψ t (v) = β 1 (v)ψ ′ (v) + β 2 (v)(1 + t 6(n−2) |v| −(n−2) ) on the component of A t coming from M ′ , and ψ t (v) = β 1 (v)ψ ′′ (v) + β 2 (v)(1 + t 6(n−2) |v| −(n−2) ) on the component coming from M ′′ . Here ψ t is thought of as a function on A t , which is identified by Φ ′ and Φ ′′ with two disjoint copies of {v ∈ R n : t 2 < |v| < t}. Now let g t be ψ p−2 t h in A t , where h is the push-forward to A t by Φ ′ , Φ ′′ of the standard metric on {v ∈ R n : t 2 < |v| < t}. This completes the definition of g t . However, the parametrization of the conformal classes by t differs in §2.1 and §2.2. Identifying M ′ and M ′′ with R n conformally near m ′ , m ′′ , and reasoning from the definition of stereographic projection, we find that in §2.
Because of this, the conformal class [g t ] constructed in §2.2 is the same as the conformal class [g t 2 ] constructed in §2.1. We adopted this mildly inconvenient convention because with it we will be able to prove results simultaneously for the metrics of §2.1 and §2.2, without changing the powers of t involved.

Estimating the scalar curvature of g t
We now show that the scalar curvature of g t approaches the constant value ν in the L n/2 norm as t → 0. Proposition 2.1 Let {g t : t ∈ (0, δ)} be one of the families of metrics defined on M = M ′ #M ′′ in §2.1 and §2.2. Let the scalar curvature of g t be ν − ǫ t . Then there exist Y, Z > 0 such that |ǫ t | Y and gt ǫ t L n/2 Zt 2 for t ∈ (0, δ).
Proof. The proof will be given for the metrics g t of §2.1 only, the modifications for §2.2 being left to the reader. We first derive an expression for ǫ t . Outside A t , the scalar curvature of g t is ν and 0 on the regions coming from M ′ and M ′′ . On A t , calculating with (2) gives since β 1 + β 2 = 1.
. The reason for choosing to scaleĝ by a factor of t 12 whilst making β 1 change between t 2 and t is to make this estimate work -the first power has to be as high as 12 to work in dimension 3. Substituting it into (5) gives that Using a lower bound for ψ ′ we find that on the subannulus t 2 < |v| < t, the estimate |ψ t (v)| C 0 > 0 holds for some C 0 and all t ∈ (0, δ). Using this to get rid of the ψ t terms on the r.h.s., and an upper bound for ψ ′ , it can be seen that on the subannulus t 2 < |v| < t. But and in a similar way |∆β 1 | = O |v| −2 . Substituting into (6), we find that for all t ∈ (0, δ), |ǫ t | Y on the subannulus t 2 < |v| < t, for some Y |ν|. Thus |ǫ t | Y on A t , and outside A t , ǫ t = 0 on the component coming from M ′ , and |ǫ t | = |ν| Y on the component coming from M ′′ . Therefore |ǫ t | Y , giving the first part of the proposition. To prove the second part, observe that by the estimates on ψ t above, the support of ǫ t has volume C 1 t n , for some C 1 > 0. So gt ǫ t L n/2 Zt 2 , where Z = Y C 2/n 1 .

A uniform bound for a Sobolev embedding
If (M, g) is a compact Riemannian n-manifold, then L 2 1 (M ) is continuously embedded in L 2n/(n−2) (M ) = L p (M ) by the Sobolev Embedding Theorem, [1,Th. 2.20]. This means that L 2 1 (M ) ⊂ L p (M ), and φ L p A φ L 2 1 for all φ ∈ L 2 1 (M ), and some A > 0 depending on g. We shall prove this holds for the metrics g t of §2.1 and §2.2, with A independent of t.
Proof. First consider the metrics of §2.1. The proof works by proving similar Sobolev embedding results on the component manifolds (M ′ , g ′ ) and (M , t 12ĝ ) that make up (M, g t ), and then 'gluing' them together. These are given in the following lemmas, the first following from the Sobolev Embedding Theorem [1, Th. 2.20].
Proof. The inequality φ L p D 2 ∇φ L 2 is invariant under homotheties, and so it is enough to prove the lemma when t = 1. Bartnik Combining the last two equations and taking square roots proves the lemma, with D 2 = (B(1 + C)) 1/2 .
The volume form of g t on A t is ψ p t dV h , so the contribution from A t to M φ p dV gt is At (ψ t φ) p dV h . Using (4) we may eliminate ψ t , divide into integrals on the component manifolds, and show that gt by Hölder's inequality. But as dβ 1 | At L n is a conformally invariant norm we Combining the last few equations and the obvious analogues forM , and remembering that dβ 1 + dβ 2 = 0, we obtain for all t ∈ (0, δ) and some D 3 , D 4 > 0 depending on D 1 , D 2 and bounds for the ratios of the various conformal factors on A t . From the last equation we see that Proposition 2.
where ω n−1 is the volume of the unit sphere S n−1 in R n . So when t is small, This completes the proposition for the metrics of §2.1. The proof for the metrics of §2.2 requires only simple modifications, and we leave it to the reader.

Existence results for scalar curvature ±1
Let M be the manifold of §2.1 or §2.2 with one of the metrics g t defined there, and denote its scalar curvature by S. As in §1.2, a conformal change tog t = ψ p−2 g t may be made, and the condition forg t to have constant scalar curvature ν is the Yamabe equation Now the metrics g t have scalar curvature close to ν, so let S = ν − ǫ; then by Proposition 2.1, gt ǫ L n/2 Zt 2 . Also we would like the conformal change to be close to 1, so put ψ = 1 + φ, where we aim to make φ small. Substituting both of these changes into (8) gives where b = 4/(n − 2) and f (t) = |1 + t| (n+2)/(n−2) − 1 − (n + 2)t/(n − 2). In this section, we shall suppose that ν = ±1, as the zero scalar curvature case requires different analytic treatment and will be considered in §4. Equation (9) has been written so that on the left is a linear operator a∆ − νb applied to φ, and on the right are the 'error terms'.
The method of §3.1 is to define by induction a sequence This depends upon being able to invert the operator a∆ − νb, and we consider the existence and size of the inverse in §3.2 and §3.3. Given this invertibility, we show that if ǫ is sufficiently small, {φ i } ∞ i=0 converges to φ ∈ L 2 1 (M ) which is a weak solution of (9). Finally we show that φ is smooth and ψ = 1 + φ is positive, so thatg t has constant scalar curvature ν.
In §3.2 and §3.3 we state the existence theorems for constant positive and negative scalar curvature respectively on connected sums, the main results of this section. Note that §3.3 produces three distinct metrics of scalar curvature 1 in the conformal class of each suitable connected sum of manifolds with scalar curvature 1, in contrast to the negative scalar curvature case, where any metric of scalar curvature −1 is unique in its conformal class.

An existence result for constant scalar curvature
Fix ν = ±1, and suppose (M, g) is a compact Riemannian n-manifold. Let A, B, X and Y be positive constants, to be chosen later. We write down four properties, which (M, g) may or may not satisfy: Property 2. Let the scalar curvature of g be ν − ǫ. Then |ǫ| Y .
Our first goal is to prove that if ǫ L n/2 is sufficiently small, then this sequence converges in L 2 1 (M ). Setting φ as the limit of the sequence, (10) implies that φ will satisfy (9), as we would like. This will be achieved via the next lemma.
From the form of this equation, it is clear that there exists W > 0 depending only on F 0 , F 1 , F 2 , F 3 , n such that when s is small, there exists an x with 0 < x W s and 2χ(x) = x. Suppose s is small enough, so that such an x exists. Now x that we have just proved. Dividing the equation x = 2χ(x) by x and subtracting some terms it follows that 1 > F 1 s + 2F 2 x + 2F 3 x 4/(n−2) > 0, and so {φ i } ∞ i=0 converges, by comparison with a geometric series. Let the limit of the sequence be φ. Then as φ i x W s for all i, by continuity φ also satisfies φ W s.
To apply the lemma we must show that T : L 2 1 (M ) → L 2 1 (M ) defined above satisfies the hypotheses. Let s = ǫ L n/2 . We will define F 0 , F 1 , F 2 , F 3 > 0 depending only on A, B, X and n, such that (11) holds.
Putting η = 0 in (10), and applying Properties 1 and 4, we see that where F 4 , F 5 are constants depending only on n, and F 4 = 0 if n 6. Let η 1 , η 2 ∈ L 2 1 (M ), and let T (η i ) = ξ i . Then taking the difference of (10) with itself for i = 1, 2 we get Applying Property 4 and making various estimates gives The calculation uses Hölder's inequality, (12), Properties 1 and 3, the expression η L r η L s (vol M ) (s−r)/rs when 1 r < s and η ∈ L r (M ) ⊂ L s (M ), and the fact that This inequality is (11) for the operator T : . . , F 3 and n, and these depend only on n, A, B and X, so W depends only on n, A, B and X. Since φ i = T (φ i−1 ) and T is continuous, taking the limit gives φ = T (φ), so (10) shows that φ satisfies (9) weakly. Thus we have proved the following lemma: Thus weak solutions φ of (9) do exist for small ǫ L n/2 , and for these ψ = 1 + φ is a weak solution of (3). But forg = ψ p−2 g to be a metric we need ψ to be smooth and positive. Proposition 1.1 shows that ψ ∈ C 2 (M ), and is C ∞ wherever it is nonzero, so it remains to show that ψ > 0.
Examples of manifolds (with negative scalar curvature) can be found for which (3) admits solutions that change sign, so there is something to be proved. This difficulty does not arise in the proof of the Yamabe problem, as there ψ is the limit of a minimizing sequence of positive functions, so ψ 0 automatically.
We deal with this problem in the following proposition, by proving that if ψ = 1 + φ is a solution to (3) that is negative somewhere, then φ L 2 1 must be at least a certain size. So if φ is small in L 2 1 (M ), then ψ = 1 + φ 0. We then show that ψ > 0 using a maximum principle.
We are now ready to define the constant c in Theorem 3.1. Let c be small enough that three conditions hold: firstly, ǫ L n/2 c implies ǫ L n/2 is sufficiently small to satisfy Lemma 3.3, so thata φ exists and satisfies φ L 2 1 is sufficiently small to satisfy Proposition 3.4, so that ψ = 1 + φ 0; and thirdly, that φ L 2 1 cW implies φ L 2 1 is sufficiently small that φ cannot be the constant −1. (As X/2 vol(M ) by Property 1, this depends only on X.) Then c depends only on n, A, B, X and Y , as the three conditions each separately do. Thus if ǫ L n/2 c, then there exists φ with φ L 2 1 W ǫ L n/2 , such that ψ = 1 + φ 0 and satisfies (3). By the third condition on c, ψ is not identically zero. By Proposition 1.1, ψ ∈ C 2 (M ), and is C ∞ wherever it is nonzero. It remains to show that ψ > 0 forg = ψ p−2 g to be nonsingular and have constant scalar curvature ν. This we achieve using the strong maximum principle [7, Th. 2.6]: Theorem 3.5 Suppose h is a nonnegative, smooth function on a connected manifold M , and u ∈ C 2 (M ) satisfies (∆ + h)u 0. If u attains its minimum m 0, then u is constant on M .
As M is compact and S and ψ are continuous, they are bounded on M , and there is a constant h 0 such that S − νψ p−2 h on M . Now M is connected, and ψ ∈ C 2 (M ) satisfies (3) and is nonnegative, so ψ satisfies a∆ψ + hψ 0. Thus by the strong maximum principle, if ψ attains the minimum value zero, then ψ is identically zero on M . But it has already been shown that this is not the case, so ψ cannot be zero anywhere and must be strictly positive. The proof of Theorem 3.1 is therefore complete.

Constant negative scalar curvature
We now construct metrics of scalar curvature −1 on connected sums using the results of §3.1. Fix ν = −1, and consider the metrics g t of §2.1 and §2.2. Properties 1-3 of §3.1 have already been dealt with, so it remains to show that Property 4 holds for the metrics g t , and that gt ǫ t L n/2 is small when t is small. As ν = −1 Property 4 is about the invertibility of a∆ + b, which is simple as the eigenvalues of ∆ are nonnegative.

Proof.
As ∆ is self-adjoint and all its eigenvalues are nonnegative, and as a, b > 0, by some well-known analysis a∆ + b has a right inverse, T say, from We may define φ ∈ L 2 (M ) by φ = T ξ, and a∆φ + bφ = ξ will hold weakly.
It must first be shown that φ ∈ L 2 1 (M ) and that it satisfies the inequality. Since φ, ξ ∈ L 2 (M ), M φξdV gt exists, and by subtraction M φ∆φdV gt exists as well. This is M |∇φ| 2 dV gt , and so φ ∈ L 2 1 (M ) by definition. Multiplying the expression above by φ and integrating gives a M |∇φ| 2 , and the r.h.s. is at most gt φ L p gt ξ L 2n/(n+2) by Hölder's inequality, since φ ∈ L p (M ) by the Sobolev embedding theorem. But gt φ L p A gt φ L 2 1 by Proposition 2.2. Putting all this together gives b gt φ 2 . So far we have worked with ξ ∈ L 2 (M ) rather than L 2n/(n+2) (M ). It has been shown that the operator T : is linear and continuous with respect to the L 2n/(n+2) norm on L 2 (M ), and bounded by B. But therefore, by elementary functional analysis, the operator T extends uniquely to a continuous operator on the closure of L 2 (M ) in L 2n/(n+2) (M ), that is, L 2n/(n+2) (M ) itself. Call this extended operator T . Then for ξ ∈ L 2n/(n+2) (M ), φ = T ξ is a well-defined element of L 2 1 (M ), satisfies gt φ L 2 1 B gt ξ L 2n/(n+2) , and a∆φ + bφ = ξ holds in the weak sense, by continuity. This concludes the proof.
All the previous work now comes together to prove the following two existence theorems for metrics of scalar curvature −1:  As in §2.2, define the family {g t : t ∈ (0, δ)} of metrics on M = M ′ #M ′′ . Then there exists C > 0 such that g t admits a smooth conformal rescaling tõ g t = (1 + φ) p−2 g t with scalar curvature −1 for small t, and gt φ L 2 1 Ct 2 .
The proofs of the theorems are nearly the same, so only the first will be given. To get the second proof, change vol(M ′ ) to vol(M ′ ) + vol(M ′′ ) in the definition of X.
Proof of Theorem 3.7. Applying Propositions 2.1 and 2.2 to the family {g t : t ∈ (0, δ)} gives a constant Y for Property 2 of §3.1, and constants A, ζ such that if t ζ then Property 3 holds for g t with constant A. By Lemma 3.6, there is a constant B such that Property 4 also holds for g t when t ζ.
It is clear that as t → 0, vol(M, g t ) → vol(M ′ ) > 0. So there is a constant X > 0 such that X/2 vol(M, g t ) X for small enough t. This gives Property 1. Thus there are constants n, A, B, X, Y such that Properties 1-4 of §3.1 hold for (M, g t ) when t is small. Theorem 3.1 therefore gives a constant c such that if gt ǫ t L n/2 c, we have the smooth conformal rescaling to a constant scalar curvature metric that we want.
But by Proposition 2.1, gt ǫ t L n/2 Zt 2 . So for small enough t, gt ǫ t L n/2 c, and there exists a smooth conformal rescaling to a metricg t = (1 + φ) p−2 g t which has scalar curvature −1. Moreover, φ L 2 1 W gt ǫ t L n/2 W Zt 2 , where W is the constant given by Theorem 3.1. Therefore putting C = W Z completes the proof.

Constant positive scalar curvature
Now we construct metrics of scalar curvature 1 on connected sums. The problems we encounter are in proving Property 4 of §3.1, which now deals with the invertibility of a∆ − b, and they arise because a∆ may have eigenvalues close to b. Our strategy is to show that if a∆ has no eigenvalues in a fixed neighbourhood of b on the component manifolds of the connected sum, then for small t, a∆ has no eigenvalues in a smaller neighbourhood of b on (M, g t ).
This is the content of the next theorem. We shall indicate here why the theorem holds, but the proof we leave until the appendix, because it forms a rather long and involved diversion from the main thread of the paper.
Here is a sketch of the proof. Suppose φ is an eigenvector of a∆ on (M, g t ) for small t. Restricting φ to the portions of M coming from M ′ and M ′′ and smoothing off gives functions on M ′ , M ′′ . We try to show that one of these is close to an eigenvector of a∆ on M ′ or M ′′ . This can be done except when φ is large on the neck compared to the rest of the manifold.
But as the neck is a small region when t is small, for φ to be large there and small elsewhere means that φ must change quickly around the neck, so that M |∇φ| 2 dV gt has to be large compared to M φ 2 dV gt . Thus the eigenvalue of φ must be large. Conversely, if the eigenvalue of φ is close to b, then φ cannot be large on the neck compared to the rest of M , and therefore either M ′ or M ′′ must also have an eigenvalue close to b.
Using this result, Property 4 of §3.1 can be proved for the metrics: But the restriction on the eigenvalues of a∆ means that and these together with (17) and Hölder's inequality imply that For t ζ, we apply Proposition 2.2 to 1 , and similarly, the integral on the l.h.s. above is gt φ 2 Therefore Dividing by γa gt φ L 2 1 /(a + b + γ) then gives gt φ L 2 1 B gt ξ L 2n/(n+2) for small t, where B = (a + b + γ)A/aγ. So the lemma holds for ξ ∈ L 2 (M ). This may be extended to ξ ∈ L 2n/(n+2) (M ) as in the proof of Lemma 3.6, and the argument is complete.
We now prove two existence theorems for metrics of scalar curvature 1:  These metrics are stationary points of the Hilbert action Q, but they need not be absolute minima, i.e. Yamabe metrics. The third is never a minimum. If g ′ and g ′′ are Yamabe metrics, the author expects that generically one of the first and second metrics will be a Yamabe metric, and in a codimension 1 set of cases when vol(M ′ ) ≈ vol(M ′′ ) both the first and second metrics will be distinct Yamabe metrics.

Extending to conformally curved metrics
In defining the metrics {g t : t ∈ (0, δ)} in §2.1 and §2.2, we assumed for simplicity that g ′ , g ′′ are conformally flat in neighbourhoods of m ′ , m ′′ . It turns out that if we drop this assumption, then provided the metrics g t on M are suitably defined the results of §3.2 and §3.3 still hold without change. The principal difference is that the expression (5) for the scalar curvature of g t becomes more complicated, with new error terms that have to be estimated and controlled.
Following the method of §2.1, we can choose an identification of a ball about m ′ in M ′ with B δ (0), such that the induced metric on B δ (0) is g ′ = h+q ′ , where h is the standard metric on R n and q ′ = O ′′ (|v| 2 ) in the sense of §1.3. To glue in the neck metric g N of §2.2, for instance, we would define g t = β 1 (h + q ′ ) + β 2 (1 + t 6(n−2) |v| −(n−2) ) p−2 h, where (β 1 , β 2 ) is the partition of unity defined in §2.1. Writing out the scalar curvature explicitly, we see that as q ′ = O ′′ (|v| 2 ), the terms involving q ′ can be absorbed into the existing error terms in §2.4, so that (6) holds for the new metrics g t . Therefore Proposition 2.1 holds for the new metrics g t as well. It can be seen by following the proof of Proposition 2.2 and the proof of Theorem 3.9 in the appendix that no other nontrivial modifications are required to prove these results for the more general families of metrics {g t : t ∈ (0, δ)} discussed above. Thus the new metrics satisfy all the necessary conditions, and the results of §3.2 and §3.3 apply to them without change.

Connected sums with zero scalar curvature
In this section the methods of §2 and §3 will be adapted to study zero scalar curvature manifolds. We have three cases to consider, when the scalar curvatures of g ′ and g ′′ are 0 and 1, or 0 and 0, or −1 and 0. Each case introduces specific difficulties, and each needs some additional methods to prove the existence of constant scalar curvature metrics.
The first two cases fit into a common analytic framework, and will be handled together. The differences with the previous method are that g t must be defined more carefully than before to control the errors sufficiently, and in the analysis the operator a∆ − νb now has one or two small eigenvalues. Thus when the sequence {φ i } ∞ i=0 is defined inductively using the inverse of this operator, the components in the directions of the corresponding eigenvectors have to be attended to, to prevent the sequence diverging. The third case is discussed in §4.5. We shall outline what the constant scalar curvature metrics look like and how to prove existence results, but will not go into much detail.
Choose k with (n − 2)(n + 2)/2(n + 1) < k < (n − 2)(n + 2)/2n, which will remain fixed throughout this section. Choose δ ∈ (0, 1) such that δ 2k/(n+2) r and δ −2/n R. We will write down a family of metrics {g t : t ∈ (0, δ)} on M = M ′ #M ′′ , in a similar way to §2.1. For any t ∈ (0, δ), define M and the conformal class of g t by where t is the equivalence relation defined by whenever v ∈ R n and t (n−2)/n < |v| < t 2k/(n+2) . As in §2.1, the conformal class [g t ] of g t is the restriction of the conformal classes of g ′ andĝ to the open sets of M ′ ,M making up M , and is well-defined because the conformal classes agree on the annulus of overlap A t , where the two open sets are glued by t . Define g t within this conformal class by g t = g ′ on the component of M \ A t coming from M ′ , and g t = t 2ĝ on the component coming fromM . It remains to choose a conformal factor on A t . This is done as in §2.1, except that the annulus {v ∈ R n : t (n−2)/n < |v| < t 2k/(n+2) } in R n replaces {v ∈ R n : t 2 < |v| < t} in R n in the definition of the partition of unity. This completes the definition of g t for t ∈ (0, δ). Lemma 4.1 Let the scalar curvature of g t be −ǫ t . Then ǫ t is zero outside A t . There exists Y > 0 such that for all t ∈ (0, δ), ǫ t satisfies |ǫ t | Y , and the volume of A t with respect to g t satisfies vol(A t ) = O(t 2nk/(n+2) ). Therefore gt ǫ t L 2n/(n+2) = O(t k ) and gt ǫ t L n/2 = O(t 4k/(n+2) ).
Proof. Outside A t , the metric g t is equal to g ′ or homothetic toĝ, and so has zero scalar curvature, verifying the first claim of the lemma. The proof that |ǫ t | Y is the same as that for the corresponding statement in Proposition 2.1, setting ν = 0. The estimate on the volume of A t also follows by the method used in Proposition 2.1, and the last two estimates are immediate. We introduced k above to make the estimate on gt ǫ L 2n/(n+2) easy to write down.
Proof. Calculating with (2) gives Let F be the quadratic form on R n given by the second derivatives of ψ ′ ; then ψ ′ = 1 + F + O ′ (|v| 3 ). As the scalar curvature of g ′ is zero, the trace of F is zero. Now dV gt = ψ p t dV h . Multiplying through by this equation and making various estimates gives that Integrate this over A t . Now β 1 (v) = β(|v|) where β(|v|) = σ(log |v|/ log t), from §2.1. So (∇β 1 (v))·(∇F ) = 2|v| −1 F dβ dx and (∇β 1 (v))·(−µt n−2 ∇(|v| 2−n )) = (n − 2)µt n−2 |v| 1−n dβ dx and ∆β 1 Using a Fubini theorem, we may write the integral on the right hand side as a double integral over S n−1 and |v|. The volume forms are related by dV h = |v| n−1 dΩd|v|, where dΩ is the standard volume form on S n−1 with radius 1. But as the trace of F w.r.t. h is zero, S n−1 F dΩ = 0 and the terms on the right hand side of (20) involving F vanish. So viewing (20) as a double integral and integrating over S n−1 gives where ω n−1 is the volume of S n−1 .
The integral on the right is an exact integral, by parts. By definition, β changes from 0 to 1 and dβ dx from 0 to 0 over the interval, so it evaluates to At ǫ t (v) dV gt = (n − 2)ω n−1 µt n−2 + error terms, which is nearly the conclusion of the lemma; it remains only to show that the 'error terms' are of order t n−2+α . This is a simple calculation and will be left to the reader, the necessary ingredients being that as |v| lies between t (n−2)/n and t 2k/(n+2) , O(|v|) may be replaced by O t 2k/(n+2) and tO |v| −1 may be replaced by O t 2/n , dβ dx = O |v| −1 and d 2 β The error term that is usually the biggest is O |v| n+1 , and in order to ensure that this error term is smaller than the leading term calculated above, that is, to ensure α > 0, k must satisfy k > (n + 2)(n − 2)/2(n + 1), which was one of the conditions in the definition of k above.
The lemma shows that the average scalar curvature of (M, g t ) is close to −(n − 2)ω n−1 µt n−2 vol(M ′ ) −1 , so we will choose this value for the scalar curvature ofg t in §4.4.

Combining two metrics of zero scalar curvature
Let M ′ , M ′′ , M, g ′ , g ′′ , m ′ and m ′′ be as usual, with g ′ , g ′′ of zero scalar curvature and conformally flat in neighbourhoods of m ′ , m ′′ . Rescaling g ′ , g ′′ by homotheties still gives metrics of zero scalar curvature. Thus gluing these rescaled metrics using the method of §2.2 gives a 2-parameter family of metrics in the same conformal class [g t ] that all have small scalar curvature. Which of these metrics do we expect to be close to a metric of constant scalar curvature?
The necessary condition is that vol(M ′ ) = vol(M ′′ ); we will explain why at the end of §4.3. Suppose, by applying a homothety to g ′ or g ′′ if necessary, that vol(M ′ ) = vol(M ′′ ). A family of metrics {g t : t ∈ (0, δ)} will be defined on M following §4.1, such that when t is small, g t resembles the union of M ′ and M ′′ with their metrics g ′ and g ′′ , joined by a small 'neck' of approximate radius t, which is modelled upon the manifold N of §2.2, with metric t 2 g N .
Choose k with (n − 2)(n + 2)/2(n + 1) < k < (n − 2)(n + 2)/2n, and apply the gluing method of §4.1 twice, once to glue one asymptotically flat end of (N, t 2 g N ) into M ′ at m ′ , and once to glue the other asymptotically flat end into  4.3 Let the scalar curvature of g t be −ǫ t . Then ǫ t is zero outside A t . There exists Y > 0 such that for all t ∈ (0, δ), ǫ t satisfies |ǫ t | Y , and the volume of A t with respect to g t satisfies vol(A t ) = O(t 2nk/(n+2) ). Therefore gt ǫ t L 2n/(n+2) = O(t k ) and gt ǫ t L n/2 = O(t 4k/(n+2) ).
Proof. This is identical to Lemma 4.1, and its proof is the same, except that g t may also be homothetic to g N in the first sentence.
Proof. This is merely Lemma 4.2 applied twice, firstly to the gluing of N into M ′ and secondly to the gluing of N into M ′′ . We have also used the observation that for both asymptotically flat ends of N , the constant µ of §4.1 takes the value 1. To see this, compare the definition of µ in Proposition 1.3 with the definition of (N, g N ) in §2.2. Proof. The proof follows that of Proposition 2.2, applied to the metrics of §4.1 and §4.2 rather than §2.1 and §2.2, except for some simple changes to take into account the different powers of t used to define the new metrics.

Inequalities on the connected sum
As in §3.3, to calculate with the inverse of a∆ − νb we need to know about the spectrum of a∆ on (M, g t ). The next three results give the necessary information; the proofs are again deferred to the appendix.   Here λ t = O(t n−2 ), and The proposition is proved by a series method, starting with a function that is 1 on the part of M coming from M ′ and −1 on the part coming from M ′′ , and then adding small corrections to get to an eigenvector of a∆. Note that β t takes the approximate values ±1 on the two halves because vol(M ′ ) = vol(M ′′ ) by assumption; if the volumes were different, then the approximate values would have to be adjusted so that M β t dV gt = 0.
We may now state the analogue of Theorem 4.6 for the metrics of §4.2, which will be proved in the appendix.   Proof. Proposition 4.7 shows that β t = 1 + O(t 2(n−2)/n ) on A ′ t and β t = −1 + O(t 2(n−2)/n ) on A ′′ t , as these are annuli in which t (n−2)/n < |v| < t 2k/(n+2) . Applying these and Lemmas 4.3 and 4.4 to the integral of β t ǫ t over M gives and as vol(A t ) = O(t 2nk/(n+2) ), the second term is O(t 2nk/(n+2)+2(n−2)/n ). But by the definitions of k and α, it is easily shown that n − 2 + α < 2nk/(n + 2) + 2(n − 2)/n, and so the first error term is larger and subsumes the second, as required.
We note that this lemma is the reason for requiring that vol(M ′ ) = vol(M ′′ ). For if the two are not equal, then Lemma 4.4 still shows that A ′ t ǫ t dV gt and A ′′ t ǫ t dV gt are equal to highest order, but β t takes values approximately proportional to vol(M ′ ) −1 on A ′ t , and to vol(M ′′ ) −1 on A ′′ t . Thus in this case M β t ǫ t dV gt is O(t n−2 ) rather than O(t n−2+α ). But we will need M β t ǫ t dV gt = o(t n−2 ) for the proof in §4.4.

Existence of constant scalar curvature metrics
Now we give the existence results for constant scalar curvature metrics on the connected sums of §4.1 and §4.2. Then g t has scalar curvature −D 0 t n−2 − η. As in §3, the condition forg t = (1 + ρ + τ ) p−2 g t to have scalar curvature −D 0 t n−2 is Define a vector space P of functions on (M, g t ) by P = 1 in §4.1, and P = 1, β t in §4.2, where β t is as in Proposition 4.7. We shall construct ρ, τ satisfying (21), with ρ ∈ P and τ ∈ P ⊥ with respect to the L 2 1 inner product. Define inductively sequences {ρ i } ∞ i=0 of elements of P and {τ i } ∞ i=0 of elements of P ⊥ ⊂ L 2 1 (M ) by ρ 0 = τ 0 = 0, and having defined the sequences up to i − 1, let ρ i and τ i be the unique elements of P and P ⊥ satisfying If we can show that these sequences converge to ρ ∈ P and τ ∈ P ⊥ that are small when t is small, then the arguments of §3 complete the theorem. The difficulty lies in inverting the operator a∆ + bD 0 t n−2 : by Theorems 4.6 and 4.8, the operator is invertible on P ⊥ with inverse bounded by γ −1 , as all the eigenvectors of a∆ in P ⊥ have eigenvalues at least γ. But on P , the inverse is of order t 2−n , which is large; so ρ i may be large even if the right hand side of (22) is small.
The solution is to ensure that the P components of η are smaller even than t n−2 , so that after applying the inverse of a∆ + bD 0 t n−2 to them, they are still small. Let π denote orthogonal projection onto P ; both the L 2 and the L 2 1 inner product give the same answer, and in fact the projection makes sense even in L 1 (M ). Then from (22) we make the estimates for some constants D 1 , D 2 , D 3 independent of t. The norms on the right hand side of (23) would normally be L 2n/(n+2) norms, but as P is a finite-dimensional space all norms are equivalent, and we may use the L 1 norm.
Our strategy is to show that if ρ i−1 L 2 1 D 4 t α and τ i−1 L 2 1 D 5 t k for large enough constants D 4 , D 5 , then ρ i L 2 1 D 4 t α and τ i L 2 1 D 5 t k also hold for small t, so by induction the sequences are bounded; convergence for small t easily follows by a similar argument to that used in Lemma 3.2.
From Lemmas 4.2, 4.4 and 4.9 we deduce that π(η) L 1 = O(t n−2+α ), so the first term on the right of (23) contributes O(t α ) to ρ i L 2 1 , consistent with ρ i L 2 1 D 4 t α if D 4 is chosen large enough. The third term π(ητ i−1 ) L 1 is bounded by A η L 2n/(n+2) τ i−1 L 2 1 , and η L 2n/(n+2) = O(t k ) by Lemmas 4.1 and 4.3; the third term therefore contributes O(t 2k+2−n ) to ρ i L 2 1 , and by the definition of α, this error term is strictly smaller than O(t α ). The fourth error term is also easily shown to be smaller than O(t α ).
Thus the only problem term in (23) is the second term, and the only reason it is a problem is that the P ⊥ component of η, multiplied by ρ i−1 , may have an appreciable component in P . We get round this as follows. Suppose ξ ∈ P ⊥ and ρ ∈ P , and consider the P -component of ξρ. In §4.1 this component is zero, and there is no problem; in §4.2 there may be a component in the direction of β t , and it is measured by M ξβ 2 t dV gt . But by the description of β t in Proposition 4.7, β 2 t is close to 1, and ξ is orthogonal to the constants, and so in general the P component of ξρ will be small compared to the sizes of ξ and ρ. Taking this into account, it is easy to get a good bound on π(ηρ i−1 ) L 1 .
The rest of the proof will be left to the reader. What remains to be done is to prove inductively that bounds ρ i L 2 1 D 4 t α , τ i L 2 1 D 5 t k hold for small enough t, and then to prove the convergence of the sequences, and these may both be done using the methods of Lemma 3.2, working from (23) and (24). Setting φ = ρ + τ , where ρ, τ are the limits of the sequences, the reader may then rejoin the proof of Theorem 3.1 after Lemma 3.3.
As the metrics constructed have negative scalar curvature, they are unique in their conformal classes, and are Yamabe metrics. The theorem thus tells us that the Yamabe metric on the connected sum, with small neck, of two zero scalar curvature manifolds, balances the volumes of the two component manifolds so that they are equal, a fact which seems rather appealing.

Combining zero and negative scalar curvature
There is just one case left, that of gluing a metric of zero scalar curvature into a metric of scalar curvature −1. This case can be handled using the results of §3, using the following simple extension of the method: we define a family of metrics {g t : t ∈ (0, δ)} on the connected sum M , with −1 − ǫ t the scalar curvature of g t , and then prove that |ǫ t | Y , ǫ t L n/2 Zt ι as in Proposition 2.1, and that φ L p At −κ φ L 2 1 for φ ∈ L 2 1 (M ) as in Proposition 2.2, where norms are taken with respect to g t .
Here ι, κ > 0. The idea is that if ι is large compared to κ, then we may follow the proofs of §3 adding in powers of t, and at the crucial stages when we need some expression to be sufficiently small, the power of t will turn out to be positive, and so we need only take t small enough. To do this in practice, we modify the proof slightly to cut down the number of applications of Proposition 2.2, and thus obtain a more favourable necessary ratio of ι to κ. Now κ is essentially determined by what the Yamabe metric on the connected sum actually looks like -if t is the radius of the 'neck', any family of good approximations to the Yamabe metrics will have κ = (n − 2)/n (this value will be justified below). So the problem is to make ι large enough, in other words, to start with a family of metrics g t that are a good approximation to scalar curvature −1. We shall not go through the construction and proof again for this case, but will describe how to define metrics g t that have close enough to constant scalar curvature for the method outlined above to work.
Consider the connected sum M of (M ′ , g ′ ) with scalar curvature −1 and (M ′′ , g ′′ ) with scalar curvature 0. To get a good enough approximation to scalar curvature −1, we have to rescale g ′′ so that its scalar curvature approximates −1 rather than 0. Let ξ be the Green's function of a∆ at m ′′ on M ′′ satisfying a∆ξ = δ m ′′ − vol(M ′′ ) −1 in the sense of distributions. Since ξ is only defined up to the addition of a constant, choose ξ to have minimum value 0. Then ξ is a C ∞ function onM = M ′′ \ {m ′′ } with a pole at m ′′ , of the form (n − 2)ω −1 n−1 |v| 2−n + O ′ (|v| 1−n ), in the usual coordinates. Letĝ t = t (n−2) 2 /2n + t (n−2)(n+2)/2n vol(M ′′ )ξ p−2 g ′′ onM . Calculating its scalar curvatureŜ t using (2) giveŝ so that −1 Ŝ t < 0, andŜ t is close to −1 away from m ′′ for small t. Outside a small neighbourhood of m ′′ ,ĝ t is close to t 2(n−2)/n g ′′ , so that the diameter of M ′′ is multiplied by t (n−2)/n . But in a small neighbourhood of m ′′ ,ĝ t resembles a 'neck' metric of radius proportional to t, as in §2.2. Soĝ t looks like M ′′ rescaled by t (n−2)/n , and with a 'neck' of radius proportional to t, opening out to an asymptotically flat end.
We construct g t by gluingĝ t into g ′ using the natural 'neck'. Thus a rough description of g t is that it is g ′ on the M ′ part and t 2(n−2)/n g ′′ on the M ′′ part, and the two parts are joined by a 'neck' with radius proportional to t. With a family {g t : t ∈ (0, δ)} of metrics defined in this way, the modified method outlined above may be applied to show that there exist small conformal deformations of g t to scalar curvature −1, for t sufficiently small.
One final point: we can now see the reason for the failure of Proposition 2.2, which necessitated this whole detour. Consider a smooth function φ on (M, g t ) that is 0 on the M ′ part, 1 on the M ′′ part, and changes only on the 'neck'. A simple calculation shows that for such a function, φ L p ∼ t (2−n)/n φ L 2 1 , and so the value for κ given above is the least possible.

Doing without conformal flatness
In §3.4, we explained that the results of §3 still hold if the assumption that g ′ , g ′′ are conformally flat in neighbourhoods of m ′ , m ′′ is dropped. However, the results of §4 do require modifications to generalize in this way. The problem is in extending Lemmas 4.2 and 4.4 to the curved case: we need a quite precise evaluation of the total scalar curvature of g t , and have to be careful that the error terms do not swamp the term that we can evaluate.
To deal with the case of §4.1 first, it can be shown that the proof of Lemma 4.2 still holds when g ′ has conformal curvature near m ′ , because defining g t as in §3.4, the extra error terms introduced in the scalar curvature can be absorbed into the error terms already in (19), and so the proof of Lemma 4.2 holds from that point. But if we allow g ′′ to be conformally curved near m ′′ , then Proposition 1.3 doesn't hold, and the scalar curvature of g t is dominated (except for n = 3, 4, 5) by error terms that seem to have no nice expression.
Therefore the situation is this. Theorem 4.10 applies without change when the metrics g t of §4.1 are defined using g ′′ conformally flat near m ′′ , but g ′ not necessarily conformally flat near m ′ . To include the case g ′′ not conformally flat, the result will hold if we weaken it so as not to prescribe the constant value that the scalar curvature takes, but merely give an estimate of its magnitude. Also, I believe that the result applies as stated when n = 3, 4 or 5, because then the mass term is large enough to dominate the errors.
The case of §4.2 is easier: Theorem 4.10 applies without change to the metrics g t of §4.2 defined using g ′ , g ′′ not supposed conformally flat about m ′ and m ′′ . This is because the metrics g t are made by gluing in the neck metric t 2 g N , which is conformally flat, so it reduces to the case of §4.1 when g ′′ is locally conformally flat, which we have already seen works.
This leaves the case of §4.5. The generalization of Proposition 2.2 suggested there will extend without change to the conformally curved case, so the problem is to define the metrics g t in such a way that the extra error terms introduced in the expression for the scalar curvature still give good enough approximations to constant scalar curvature for the existence result to apply. I think this can be done quite readily just by working on how to produce good approximations g t , say by adding a well-chosen conformal factor to the existing definition.
A Appendix. The spectrum of a∆ on M ′ #M ′′ In this appendix we prove Theorems 3.9, 4.6 and 4.8, and Proposition 4.7. They are results on the eigenvalues and eigenvectors of the operator a∆ on M with the metrics g t defined in §3 and §4. They appear here and not in the main text because the proofs are long calculations. Theorem 3.9 takes up §A.1 and §A.2. Its proof divides naturally into considering eigenvalues of a∆ smaller than b and eigenvalues larger than b. The eigenvectors with eigenvalues smaller than b form a finite dimensional space E. In §A.1 we define a vector space E t that is a good approximation to E when t is small, and using this we show that all eigenvalues of vectors in E are at most b − γ. In §A.2 we consider eigenvectors with eigenvalues larger than b, which must therefore be orthogonal to E, and by considering their inner product with E t we can show that their eigenvalues must be at least b + γ.
In §A.3, we prove similar results for use in the zero scalar curvature material of §4. Most of the work needed to prove them has already been done in §A.1 and §A.2, and the main problem is the construction of an eigenvector β t of a∆ with a small eigenvalue λ t . This is done by a sequence method, the basic idea being to start with an approximation to β t and repeatedly invert a∆ upon it; as λ t is the smallest positive eigenvalue, the β t -component of the resulting sequence grows much faster than any other and so comes to dominate.

A.1 Eigenvalues of a∆ smaller than b
We now prove Theorem 3.9, which is reproduced here. Proof. It is well known that the spectrum of the Laplacian on a compact Riemannian manifold is discrete and nonnegative, and that the eigenspaces are finite-dimensional. Therefore on M ′ and M ′′ there are only finitely many eigenvalues of a∆ smaller than b, and to each is associated a finite-dimensional space of eigenfunctions. Let E ′ be the finite-dimensional vector space of smooth functions on M ′ generated by eigenfunctions of a∆ on M ′ associated to eigenvalues less than b; we think of E ′ as a subspace of L 2 1 (M ′ ). For §2.2, define E ′′ on M ′′ in the same way. As a∆ has no eigenvalues in the interval (b − 2γ, b + 2γ), we see that: and also two analogous inequalities for M ′′ in §2.2. The perpendicular subspace (E ′ ) ⊥ of (26) may be taken with respect to the inner product of L 2 1 (M ′ ) or with respect to that of L 2 (M ′ ) -both give the same space, as E ′ is a sum of eigenspaces of a∆. Now if we have two statements like (25) and (26) but applying to M rather than M ′ , then we can prove the result. This is the content of the next lemma.
defining (E t ) ⊥ with the L 2 1 inner product. Then Theorem 3.9 holds.
Proof. Suppose E t exists. We must show that l ∈ (b − γ, b + γ) cannot be an eigenvalue of a∆ on (M, g t ). Suppose φ is an eigenfunction of a∆ with this eigenvalue l. Let φ 1 and φ 2 be the components of φ in E t and (E t ) ⊥ respectively. Then, as a∆φ − lφ = 0, using (27) and (28) in the last line. But as γ + l − b, γ + b − l > 0, this shows that φ 1 = φ 2 = φ = 0, which is a contradiction.
To complete the proof of the theorem, we therefore need to produce some spaces E t of functions on M satisfying (27) and (28). In §2.1, E t should be modelled on E ′ , and in §2.2, on E ′ ⊕ E ′′ .
As a half-way stage between E ′ , E ′′ and E t , spacesẼ ′ ,Ẽ ′′ of functions on M ′ , M ′′ will be made that are close to E ′ , E ′′ , but which vanish on small balls around m ′ , m ′′ . Let σ ′ be a C ∞ function on M ′ that is 1 on the complement of a small ball about m ′ , 0 on a smaller ball about m ′ , and otherwise taking values By choosing the ball outside which σ ′ is 1 to be small, we can ensure thatẼ ′ is close to E ′ in L 2 1 (M ′ ) in the following sense: the two have the same dimension, and anyṽ ∈Ẽ ′ may be written asṽ Suppose that σ ′ has been chosen so that these hold. Then two statements similar to (25) and (26) hold forẼ ′ , as we shall see in the next lemma.
Lemma A.2 The subspaceẼ ′ satisfies the following two conditions: where the inner product used to construct (Ẽ ′ ) ⊥ is that of L 2 1 (M ′ ).
Proof. First we prove (30). Let φ ∈Ẽ ′ ; then Because v 1 and v 2 are orthogonal in both L 2 and L 2 1 , Here between the third and fourth lines we have used (29), between the fifth and sixth lines we have used (25), and between the last two we have used the L 2 -orthogonality of v 1 and v 2 and the trivial inequality (b − 2γ)[1 + γ/2(a + b + 2γ)] + aγ/2(a + b + 2γ) b − 3γ/2. This proves (30).
To prove (31), observe that by (29), orthogonal projection fromẼ ′ to E ′ is injective, and as they have the same (finite) dimension, it must also be surjective.
that is, the E ′ -component ofṽ 1 is v 1 , the same as that ofṽ 2 . Butṽ 1 andṽ 2 are orthogonal in L 2 1 (M ′ ), so taking their inner product gives that v 1 . The result is that which is the analogue of (29) for (Ẽ ′ ) ⊥ instead ofẼ ′ . This is the ingredient needed to prove (31) by the method used above for (30), and the remainder of the proof will be left to the reader.
For the case of §2.2, a subspaceẼ ′′ of functions on M ′′ is created in the same way, and Lemma A.2 applies to this space too. We now define the spaces E t . In §2.1, let E t be the space of functions that are equal to some function inẼ ′ on the subset of M identified with M ′ \ Φ ′ [B t (0)], and are zero outside this subset. In §2.2, let E t be the direct sum of this space of functions, and the corresponding space made fromẼ ′′ .
For small t the functions in E t are C ∞ , and on their support g t is equal to g ′ (or g ′′ in case §2.2). Thus (30) applies to functions in E t .
A fortiori, it satisfies the inequality (27) of Lemma A.1.
Proof. In §2.1 this follows immediately from (30), as the g t and g ′ agree upon the support of the functions of E t . In §2.2 φ is the sum of elements ofẼ ′ and E ′′ ; both sides of (32) split into two terms, each involving one function. So (32) is the sum of two inequalities, which follow immediately from (30) as before, and from the counterpart of (30) applying toẼ ′′ .
The previous lemma showed that the space of functions E t upon M satisfies inequality (27) of Lemma A.1. In the next proposition, proved in §A.2, we show that the inequality (28) is satisfied too.
Proof. For simplicity we shall prove the proposition for the metrics of §2.1 only, and the modifications for §2.2 will be left to the reader. We will start from (31) of Lemma A.2. The constants in (33) and (31) are different -the first has b + γ, the second b + 3γ/2. Choose constants b 0 > b 1 > · · · > b 5 with b 0 = b + 3γ/2 and b 5 = b + γ. These will be used to contain five error terms.
Shortly we shall choose r 1 , r 2 , r 3 with 0 < r 1 < r 2 < r 3 . Define three compact Riemannian submanifolds with boundary R t ⊂ S t ⊂ T t in (M, g t ) to be the subsets of M coming from subsets R, S and T of M ′′ respectively, where When t is small, R t , S t and T t lie in the region of M in which the function β 2 , used in §2.1 to define g t , is 1. Then R t , S t , T t are homothetic to R, S, T respectively, by a homothety multiplying their metrics by t 12 .
The idea is this. A diffeomorphism Ψ ′ t from M ′ \ {m ′ } onto M \ R t will be constructed, which will be the identity outside T t . Using Ψ ′ t any function in L 2 1 (M ) defines a function in L 2 1 (M ′ ). Applying (31) of Lemma A.2 therefore induces an inequality upon functions in L 2 1 (M ). We will be able to show that for functions that are not too large in S t , this inequality implies (33) as we require. Then only the case of functions that are large in S t remains.
Suppose, for the moment, that r 1 , r 2 , r 3 are fixed with r 1 < r 2 < r 3 . For R t to be well defined, r 1 must satisfy r 1 > δ −4 . For T t to be well defined, t must be sufficiently small that t 6 r 3 < δ. We also suppose that t is small enough that the functions of E t vanish on T t . Let Let φ ∈ L 2 1 (M ), and define φ ′ = (Ψ ′ t ) * (φ). Then φ ′ ∈ L 2 1 (M ′ ), as we shall see. An easy calculation shows that where F t is a function on M that is 1 on that part of M coming from M ′ \ Φ ′ B t 6 r3 (0) , is zero on that part of M not coming from M ′ \ Φ ′ B t 6 r1 (0) , and in the intermediate annulus is given by Similarly, we may easily show that where G t is a function on M that is 1 on that part of M coming from M ′ \ Φ ′ B t 6 r3 (0) , is zero on that part of M not coming from M ′ \ Φ ′ B t 6 r1 (0) , and in the intermediate annulus is given by Here, the first term in the max(. . . ) is the multiplier for the radial component of ∇φ, and the second term is the multiplier for the nonradial components. As F t , G t are bounded, we see from (34) and (35) that φ ′ ∈ L 2 1 (M ′ ), as was stated above.
Suppose now that φ ∈ (E t ) ⊥ ⊂ L 2 1 (M ). For small enough t this implies that φ ′ ∈ (Ẽ ′ ) ⊥ , and so (31) applies by Lemma A.2. Combining this with (34) and (35) gives that Now by the definition of ψ t , ψ ′ (v)ψ t (v) −1 approaches 1 as t → 0. In fact it may be shown that C 0 t 6(n−2) |v| −(n−2) when t 6 |v| t 6−2/(n−2) , for some constant C 0 . For t small enough this certainly holds in the region t 6 r 1 |v| t 6 r 3 , and in this region we have Choose r 1 greater than δ −4 , and large enough that b 1 (1 + C 0 r −(n−2) 1 . Then for small t, the ψ ′ ψ −1 t terms in F t and G t can be absorbed by putting b 2 in place of b 0 . Next, r 2 is defined uniquely in terms of r 3 to satisfy r 1 < r 2 < r 3 and b 3 (r 2 − r 1 ) n−1 r n 3 r 1−n 2 (r 3 − r 1 ) −n = b 4 . Then b 3 (|v| − t 6 r 1 ) n−1 r n 3 |v| 1−n (r 3 − r 1 ) −n b 4 when t 6 r 2 |v| t 6 r 3 . This is to bound F t below on the region |v| t 6 r 2 .
The last two definitions are circular, as r 2 is defined in terms of r 3 , and vice versa because S t depends on r 2 . However, manipulating the definition of r 2 reveals that however large r 3 is, r 2 must satisfy r 2 r 1 (1−b 1/(n−1) 3 b −1/(n−1) 2 ) −1 , and so vol(S) is bounded in terms of r 1 , whereas vol(T ) can grow arbitrarily large. Therefore (37) does hold for r 3 sufficiently large.
The above estimates show that G t b 1 /b 3 and F t b 1 b 4 /b 0 b 3 on M \ S t for small t. Substituting these into (36) gives that when t is sufficiently small, Suppose that St φ 2 dV gt (b 4 /b 5 − 1)· M\St φ 2 dV gt . Then b 4 M\St φ 2 dV gt b 5 M φ 2 dV gt , and from (38) we see that (33) holds for φ, which is what we have to prove. Therefore it remains only to deal with the case that St φ 2 dV gt > (b 4 /b 5 − 1) · M\St φ 2 dV gt .
Suppose that this inequality holds. The basic idea of the remainder of the proof is that when t is small, the volume of S t is also small, and this forces φ to be large on S t compared to its average value elsewhere. Therefore φ must change substantially in the neighbourhood T t of S t , and this forces ∇φ to be large in T t .
Restrict t further, to be small enough that t 6 r 3 t 2 . Then T t is contained in the region of gluing in which β 2 = 1. So the pair (S t , T t ) is homothetic to a pair (S, T ) of compact manifolds with C ∞ boundaries and with S contained in the interior of T ; the metrics on (S t , T t ) are the metrics on (S, T ) multiplied by t 12 . For these S, T the following lemma holds.
Lemma A.5 Let S, T be compact, connected Riemannian n-manifolds with smooth boundaries, such that S ⊂ T but S = T . Then there exists C 1 > 0 such that for all φ ∈ L 2 1 (T ), Proof. We begin by quoting a theorem on the existence of solutions of the equation ∆u = f on a manifold with smooth boundary.
Theorem A.6 Suppose that T is a compact manifold with smooth boundary, and that f ∈ L 2 (T ) and satisfies T f dV g = 0. Then there exists ξ ∈ L 2 2 (T ), unique up to the addition of a constant, such that ∆ξ = f , and in addition n·∇ξ vanishes at the boundary, where n is the unit outward normal to the boundary.
Proof. This is a partial statement of [4, Ex. 2, p. 65]. Hörmander's example is only stated for C ∞ functions f and ξ, but the proof works for f ∈ H (0) (T ) and ξ ∈ H (2) (T ) in his notation, which are L 2 (T ) and L 2 2 (T ) in ours.
Put f = vol(S) −1 in S and f = (vol(S)−vol(T )) −1 in T \S. Then T f dV g = 0, so by the theorem, there exists a function ξ ∈ L 2 2 (M ) satisfying ∆ξ = f , and that ∇ξ vanishes normal to the boundary. Because of this vanishing, the boundary term has dropped out of the following integration by parts equation: Substituting in for ∆ξ and using Hölder's inequality gives Now S is connected, so the constants are the only eigenvectors of ∆ on S with eigenvalue 0 and derivative vanishing normal to the boundary. By the discreteness of the spectrum of ∆ on S with these boundary conditions, there is a positive constant K S less than or equal to all the positive eigenvalues. It easily follows that for φ ∈ L 2 1 (S), Also, a simple application of Hölder's inequality yields 1 vol(T ) − vol(S) T \S φ dV g T \S φ 2 dV g vol(T ) − vol(S) Adding together (40), (41) and (42) gives (39), as we want, with constant C 1 = T |∇ξ| 2 dV g 1/2 + (K S · vol(S)) −1/2 .
The point of this calculation is that because (S t , T t ) are homothetic to (S, T ) by the constant factor t 12 , Lemma A.5 implies that for all φ ∈ L 2 1 (T t ), of constants interpolating between b + 2γ and b + γ, we choose constants interpolating between 2γ and γ, and some simple changes must be made to the proof as the powers of t used in defining the metrics of §4.1 and §4.2 are different to those used in §2.1 and §2.2.
identifying subsets of M ′ , M ′′ with subsets of M .
Proof. Let y be the unique element of C ∞ (M ) satisfying M y dV gt = 0 and a∆y = e. To make β t = e+w and a∆β t = λ t β t , we must find w and λ t such that a∆w = λ t (e + w) − a∆e. Define inductively a sequence of real numbers {λ i } ∞ i=0 and a sequence {w i } ∞ i=0 of elements of C ∞ (M ) beginning with λ 0 = w 0 = 0. Let and let w i be the unique element of C ∞ (M ) satisfying M w i dV gt = 0 and Note that as M e dV gt = M w i−1 dV gt = 0, the right hand side has integral zero over M , and so w i exists. Thus the sequences {λ i } ∞ i=0 and {w i } ∞ i=0 are well-defined, provided only that the integral on the right hand side of (45) is nonzero; we will prove later that the integral is bounded below by a positive constant.
If both sequences converge to λ t and w respectively, say, then (46) implies that a∆w = λ t (e + w) − a∆e, so that β t = e + w is an eigenvector of a∆ associated to λ t . The rôle of (45) is as follows: multiply (46) by y and integrate over M . Integrating by parts gives |λ i | w i L 2 e L 2 + w i−1 L 2 + a ∇w i L 2 ∇e L 2 |λ i |a 1/2 γ −1/2 ∇w i L 2 e L 2 + w i−1 L 2 + a ∇w i L 2 ∇e L 2 , applying (47) between the second and third lines, and Hölder's inequality. Dividing by a 1/2 γ −1/2 ∇w i L 2 and using (47) on the left hand side gives where D 0 = (aγ) 1/2 ∇e L 2 . Define D 1 = (2 vol(M ′ )) −1 M ye dV gt . Then y = D 1 e+z, where M z dV gt = M ze dV gt = 0. So z ∈ (Ẽ t ) ⊥ , and by Lemma A.8 a ∇z 2 As M ze dV gt = 0 and e = a∆y = aD 1 ∆e + a∆z we have M z D 1 ∆e + ∆z dV gt = 0, so Multiplying by (aγ) 1/2 ∇z −1 L 2 and substituting (49) into the l.h.s. then gives γ z L 2 D 0 D 1 . Since y = D 1 e + z and M ew i−1 dV gt = 0, from (45) we find Now (48) and (50) are what we need to prove that the sequences {λ i } ∞ i=0 and {w i } ∞ i=0 are well-defined and convergent, provided D 0 is sufficiently small and D 1 sufficiently large. It can be shown that if 2D 2 0 γ 2 vol(M ′ ) and D 1 is large enough, then the two sequences converge to λ t and w respectively satisfying a∆w = λ t (e + w) − a∆e, where w L 2 2D 0 /γ and The proof uses the same sort of reasoning as Lemma 3.2, and will be left to the reader.