Surgery and the Yamabe invariant

Jimmy Petean Max-Planck Institut für Mathematik, Bonn, Germany and Gabjin Yun^{1} Department of Mathematics Myong-Ji University, Korea

Abstract

We study the Yamabe invariant of manifolds obtained as connected sums along submanifolds of codimension greater than 2. In particular: for a compact connected manifold M with no metric of positive scalar curvature, we prove that the Yamabe invariant of any manifold obtained by performing surgery on spheres of codimension greater than 2 on M is not smaller than the invariant of M.

1 Introduction

Given a smooth compact manifold M^{n} of dimension n the Yamabe invariant of M is defined in the following way: first pick a conformal class C of Riemannian metrics on M and let

Y (M, C) = inf_{g∈C} \frac{∫_{M} s_{g} dvol_{g}}{(V ol_{g}(M))^{\frac{n−2}{n}}} .

Here s_{g} is the scalar curvature of g and V ol_{g}(M) is the volume of M with respect to the metric g. Then

Y (M) = sup_{C} Y (M, C)

1Supported in part by KOSEF 960-7010-201-3.

is the Yamabe invariant of M. This invariant was introduced by O. Kobayashi in [5] and it is also frequently called the sigma constant of M. Note that Y (M) is an invariant of the smooth structure of M.

Yamabe first considered what we called Y (M, C) in [10]. He gave a proof that Y (M, C) is always achieved by a metric in C. His proof contained a mistake pointed out by Trudinger [9], and fixed in succesive steps by Trudinger, Aubin [3] and Schoen [8]; so proving that in every conformal class there is a metric with constant scalar curvature. A basic fact about the Yamabe invariant is that it is positive if and only if the manifold admits a metric of positive scalar curvature. The study of the invariant is then naturally divided into two cases: when Y (M) > 0 and when Y (M) ≤ 0.

We will be concerned in this paper with manifolds for which Y (M) ≤ 0. In this case, every conformal class admits a unique metric of volume 1 and constant scalar curvature. That constant is precisely the Yamabe constant of the conformal class. Hence the Yamabe invariant in this case is the supremum of the scalar curvature on the space of metrics of constant scalar curvature and unit volume. Also in this case,

Y (M) = − (inf_{M} ∫ _{M}|s_{g}|^{n/2}dvol_{g})^{2/n} ,

where M is the space of Riemannian metrics on M (c.f. [2] [6]). So the study of the Yamabe invariant in this case is equivalent to the study of the infimum of this natural Riemannian functional.

Computations of the invariant are not easy to carry out. Nevertheless, there has been some success in low dimensions. In dimension 4, LeBrun computed the invariants for all compact Kähler surfaces with non-positive Yamabe invariant (see [6]). The computations of the invariant for 3-dimensional manifolds (always with non-positive invariant) are a by-product of Anderson’s program for the hyperbolization conjecture (see [1]). The invariants are readily computable in dimension 2 from the Gauss-Bonnet formula.

We will prove the following:

Theorem 1 Let M_{1}, M_{2} be compact smooth n-dimensional manifolds. Suppose that a k-dimensional manifold W embedds into both M_{1} and M_{2} with trivial normal bundle. Assume k < n − 2. Let M_{1,2}^{W} be the manifold obtained by gluing M_{1} and M_{2} along W. Then

a) If Y (M_{i}) ≤ 0, then Y (M_{1,2}^{W}) ≥ −[(−Y (M_{1}))^{n/2} + (−Y (M_{2}))^{n/2}]^{2/n}.

b) If Y (M_{1}) ≤ 0 and Y (M_{2}) > 0 then Y (M_{1,2}^{W}) ≥ Y (M_{1}).

Here as usual M_{1,2}^{W} is constructed by deletting the image of W on both M_{1} and M_{2} and then identifying x ∈ M_{1} − W with h_{2}jh_{1}^{−1}(x), where h_{i} is a trivialization of the normal bundle of W in M_{i} (i.e. a diffeomorphism between the normal bundle and a tubular neighbourhood of W) and j is the inversion in R^{n−k} − {0} (j(v) = v/∥v∥^{2}). The resulting M_{1,2}^{W} will depend on (the homotopy class of) the trivializations h_{i} although we do not make this explicit in our notation.

In particular, we have:

Corollary 1 If M̂ is obtained from M by performing surgery of codimension at least 3 and Y (M) ≤ 0 then Y (M̂) ≥ Y (M).

Note that O. Kobayashi [5] has proved the 0-dimensional case of this result (i.e. for connected sums at points).

In the next section we will carry out the main computations. We will show that given a metric g on M, we can deform g on the end of M − W (W is an embedded submanifold of codimension at least 3) without appreciably changing the volume or the minimum of the scalar curvature (more precisely, for any positive ϵ we will show that we can construct a deformation with scalar curvature bounded below by s_{g} − ϵ and volume bounded above by V ol_{g}(M) + ϵ). Moreover we will be able to “choose” how the metric looks like at the end of M − W. This will be done following the constructions of Gromov-Lawson [4] and Schoen-Yau [7]. In the last section we will use this construction to prove the results we mentioned.

2 The main construction

Let (M, g) be a compact smooth Riemannian manifold of dimension n. Let W be a k-dimensional embedded submanifold (k ≤ n−3) with trivial normal bundle N. Also let S be the unit sphere bundle of N. Of course, M − W has an end diffeomorphic to S × (0, 1). In this section we will construct, for any positive (small) constant ϵ_{0}, a metric ĝ on M − W which verifies:

a) ĝ = g away from the end.

b) the scalar curvature, sĝ, of ĝ is greater than s_{g} − ϵ_{0} everywhere.

c) V olĝ(M − W) ≤ V ol_{g}(M) + ϵ_{0}.

d)at the end ĝ = h + dE^{2} + dt^{2}, where h is any metric on W picked previously, dE^{2} is the Euclidean metric on the sphere S^{n−k−1}(r) and dt^{2} is the Euclidean metric on the line. The positive number r can be taken as small as desired.

In the next section we will use this construction to prove the results mentioned in the introduction.

We will separate our task into two steps:

Step 1: construct ĝ satisfying a), b) and c) so that in the end it looks like the product of the Euclidean metric on the line and the metric g_{δ} induced by g on the δ-sphere bundle of N (here we will be able to pick δ as small as we want).

Step 2: for δ small enough find a homotopy ds_{t}, 0 ≤ t ≤ 1, between the metric g_{δ} and any metric like the one described in d) so that the “total” metric ds_{t} + dt^{2} on S × [0, 1] has positive scalar curvature everywhere and has volume as small as we want. Note that we will use this construction for metrics of non-positive scalar curvature, hence this will be enough to verify condition d).

Let us begin with Step 1. The construction can be done following the one by Gromov and Lawson [4] in their study of metrics with positive scalar curvature (see also [7]). We will only sketch briefly those parts where the construction in [4] applies directly to our case.

First choose a positive number ϵ so that the exponential map gives a diffeomorphism between the ϵ ball in the normal bundle and an open neighbourhood U_{ϵ} of W in M. Of course, we can pick ϵ as small as we want. Let N_{ϵ} denote the ϵ ball in the normal bundle, i.e. N_{ϵ} = {(x, y) ∈ N : ∥y∥ < ϵ}. Pull back the metric g from M to N_{ϵ} via the exponential map. We will still call this metric on N_{ϵ} by g.

For any 0 < δ < ϵ we will denote by S^{δ}N the (n-k-1)-sphere bundle of points in N of norm δ and we will call g_{δ} the restriction of g to S^{δ}N. Also, for any point (x, y) ∈ N, let r(x, y) = ∥y∥ denote the distance (measured by g) from (x, y) to W.

Now pick any r_{1} so that 0 < r_{1} < ϵ. In the plane with coordinates (t, r) consider a smooth curve γ which moves vertically from (0, ϵ) to (0, r_{1}), then is the graph of a strictly decreasing function h joinning the points (0, r_{1}) and (t_{f}, r_{f}), and then stays on the horizontal line r = r_{f}.

Let M^{γ} = {((x, y), t) ∈ N_{ϵ} × R : (r(x, y), t) ∈ γ}. Let g^{γ} denote the restriction to M^{γ} of the product metric g + dt^{2} (dt^{2} denotes the Euclidean metric on the line). To complete Step 1 we need to show that we can pick γ so that the metric g^{γ} has the required restrictions on scalar curvature and volume.

Let us study the volume of g^{γ}. More precisely, we will describe what conditions must verify γ in order that the volume is bounded by some chosen small positive constant u.

The horizontal piece of γ can be taken arbitrarily small. So we only need to consider the volume of the part of γ corresponding to the graph of h.

We will then restrict our attention to the piece of γ given by the graph of h. But to avoid introducing new notation we will keep calling γ the smaller curve and g^{γ} the restriction of the metric to this piece. To estimate the volume of g^{γ} we will compare its volume element to the one of the original metric g. Of course, dvol_{g^{γ}} = dvol_{g^{γ}}(e_{1}, ..., e_{n})dvol_{g}; where e_{1}, ..., e_{n} is a g-orthonormal basis (at certain point).

Let h^{−1} : (r_{f}, r_{1}) → (0, t_{f}) be the inverse function of h. Consider the map

F : N_{r_{1}} − \overline{N_{r_{f}}} → M^{γ}

given by F (x, y) = (x, y, h^{−1}(r(x, y))). F is clearly a diffeomorphism (onto the part of γ corresponding to the graph of h) and we will use it to bring the metric g^{γ} to N_{r_{1}} − \overline{N_{r_{f}}}.

The function r : N_{r_{1}} − \overline{N_{r_{f}}} → R is a smooth submersion. So, at any point q = (x, y) ∈ N_{r_{1}} − \overline{N_{r_{f}}}, the kernel of r_{∗q} has dimension n − 1. Let e_{2}, ..., e_{n} be a g-orthonormal basis of Kernel(r_{∗q}). And let e_{1} be a unitary vector orthogonal to that space.

We have to compute (dvol_{g^{γ}})_{F (q)}(F_{∗q}e_{1}, ..., F_{∗q}e_{n}). Now for i = 2, .., n we have F_{∗q}e_{i} = (e_{i}, 0) as elements of T_{q}M ⨁ T_{F (q)}R. Moreover,

(dvol_{g^{γ}})_{F (q)}(F_{∗q}e_{1}, ..., F_{∗q}e_{n}) = \sqrt{det(g^{γ}(F_{∗}e_{i}, F_{∗}e_{j})_{ij})}.

It follows from the previous considerations that the (n-1)×(n-1) matrix obtained by deletting the first row and column of g^{γ}(F_{∗}e_{i}, F_{∗}e_{j})_{ij} is the identity. But also,

g^{γ}(F_{∗}e_{i}, F_{∗}e_{1}) = δ_{i1} + dt^{2}((h^{−1}r)_{∗}e_{i}, (h^{−1}r)_{∗}e_{1}).

Hence the matrix is diagonal, having 1 in all but the first diagonal entry and 1 + dt^{2}((h^{−1}r)_{∗}e_{i}, (h^{−1}r)_{∗}e_{1}) in the first.

We need to compute, or estimate, (h^{−1}r)_{∗}e_{1}.

Claim: Pick any small positive constant u. If r_{1} is small enough then at any q in N_{r_{1}} −W we have |r_{∗q}e_{1} −1| ≤ u. Here we are identifying the tangent space of the real line with the real numbers, and we are picking the e_{1} “ with the right sign”.

To prove the claim, let (x_{1}, ..., x_{k}) be coordinates for W on an open set U. Pick an orthonormal frame for N over U and let (y_{1}, ..., y_{n−k}) be the induced coordinates on the fibers. Hence (r(x, y))^{2} = y_{1}^{2} + ... + y_{n−k}^{2}.

In U × D^{n−k}(r_{1}) it is easy to find a g-orthonormal basis for the kernel of r_{∗}. At a point (x, y) let \frac{∂}{∂r} be the unitary vector tangent to the geodesic line from 0 to y. Let v_{2}, ...v_{n−k} be an orthonormal basis of the tangent space to the sphere of radius r(x, y). Let w_{n−k+1}, ..., w_{n} be a g-orthonormal basis of the tangent space to U at x. Note that at q ∈ N_{r_{1}},

g(v_{i}, w_{j}) ∈ o(r_{1}), g(∂/∂r, w_{j}) ∈ o(r_{1})

and since we are in geodesic coordinates g(\frac{∂}{∂r}, v_{i}) = 0. Applying the Gram-Schmidt process to the basis v_{2}, ..., v_{n−k}, w_{n−k+1}, ..., w_{n−1}, \frac{∂}{∂r} we get an orthonormal basis. The first n − 1 vectors will be an o.n. basis for the kernel of r_{∗}. The last vector, which is the vector e_{1} in the claim, will differ from \frac{∂}{∂r} by o(r_{1}). But it is clear that r_{∗}(\frac{∂}{∂r}) = \frac{∂}{∂t} and the claim follows.

Hence, with the notation we were using before the claim,

g^{γ}(F_{∗q}e_{1}, F_{∗q}e_{1}) = 1 + (1 + o(u))^{2} (\frac{∂h^{−1}}{∂t} (r(q)))^{2}

.

Finally,

dvol_{g^{γ}}(e_{1}, ..., e_{n}) = \sqrt{1 + (1 + o(u))^{2} (\frac{∂h^{−1}}{∂t} (r(q)))^{2}} ≤

≤ 1 − (1 + o(u))\frac{∂h^{−1}}{∂t} (r(q))

and we have

V ol_{g^{γ}}(N_{r_{1}} − N_{r_{f}}) ≤ V ol_{g}(N_{r_{1}}) − (1 + o(u)) ∫_{N_{r_{1}}−N_{r_{f}}} \frac{∂h^{−1}}{∂t} (r(q)) dvol_{g}. (1)

To estimate the last integral we will need the following result, whose proof is elementary.

Lemma 1 Given δ > 0 small enough, there exists a constant K, depending on g and δ, such that for all 0 < r_{a} < r_{b} < δ,

V ol_{g}(N_{r_{b}} − N_{r_{a}}) ≤ (r_{b} − r_{a})K.

Now we can see what are the conditions we need for γ. Suppose we want to make the volume of the portion of M − W with the newly defined metric less than ϵ_{0}. Picking r_{1} small enough we can make the first term in (1) less than ϵ_{0}/2. It follows from the previous lemma that the second term in (1) is bounded by 2(1 + o(u))K t_{f}. As we let r_{1} tend to 0, the first 3 factors in the previous product are bounded independently of r_{1} (with a little effort we could prove that the product actually tends to 0, but we will not need this).

So, to keep the volume under control, we just need to construct γ so that as r_{1} tends to 0, t_{f} also tends to 0.

We will now show that we can construct such a curve γ which satisties also our requirements about the scalar curvature. More precisely, given any small positive r_{1} we will construct γ so that t_{f} ≤ 7r_{1} and s_{g^{γ}} ≥ s_{g} −ϵ_{0} (where ϵ_{0} is a positive number we fixed previously).

Let θ be the angle between the r-axis and the tangent to γ (at some point). Let κ be the principal curvature of γ. We have the following formula for the scalar curvature of g^{γ} (it is computed in [4]),

s_{g^{γ}} = s_{g} + O(1)sin^{2}θ + \frac{(n − k − 1)(n − k − 2)}{r^{2}}sin^{2}θ − (n − k − 1)\frac{κ}{r} sin θ.

The function O(1) is bounded independently of r, θ and κ. Let A be an upper bound for it.

To construct γ we will be describing how we make θ vary from 0 to π/2. Let s be the arclength parameter. Let κ(s) be the curvature of γ at (length) s. The change of θ between s_{1} and s_{2} is given by the integral of κ(s) (between s_{1} and s_{2}, of course).

We will describe the function κ(s). First pick θ_{0} so that

(A + \frac{4n}{r_{1}^{2}}) sin θ_{0} < ϵ_{0}.  (2)

Bend γ changing θ from 0 to θ_{0} by a curve of length r_{1}/2. We can do this keeping κ(s) bounded by 2/r_{1}, and so it is easy to see from (2) that we will have s_{g^{γ}} ≥ s_{g} − ϵ_{0}. We will continue on the straight line of angle θ_{0} till we get to a point with r = r_{2} < (sin θ_{0})/A. We will now make a new bend of length r_{2}/2. In order to keep the desired inequality for the scalar curvature it is enough to keep κ(s) bounded by (sin θ_{0})/2r_{2}; hence we will increase θ by about (sin θ_{0})/4. Then we will continue again in a straight line (of angle θ_{1}) for an arbitrarily small period of time. We will repeat this process until we got to a horizontal line (θ = π/2). Note that the amount of θ we can increase at every bend is about (sin θ_{n−1})/4 (where θ_{n−1} is the angle previous to the new bending). It is then clear that we need to repeat the process only a finite number of times. Now pick i so that sin θ_{i} is about 1/2. Then we will clearly finish the process at the step i + 10. Hence the length of γ in this last part of the process is bounded by 5r_{1}. On the other hand all the previous bends will verify that r_{k−1} − r_{k} ≥ (1/4)r_{k−1}. This implies that the length of the first bends (and straight lines) are bounded by twice the decrease in r. Hence the total length of γ up to the i-th step is bounded by 2r_{1}. We can then construct the whole γ of length bounded by 7r_{1}. Therefore t_{f} ≤ 7r_{1}. This completes Step 1 of the construction.

Step 2: We have already deform g so that in the end we have the product metric g_{δ} + dt^{2}. We now want to find a homotopy between g_{δ} and some canonical metrics. We will have a family g^{t}, a ≤ t ≤ b of metrics of positive scalar curvature on a compact manifold X and we need to know conditions under which the metric g^{t} + dt^{2}, on X × [a, b], has positive scalar curvature and small volume.

The following two lemmas are the tools we need to study this problem. The first one will give us a bound on the total volume of a homotopy and is completely elementary,

Lemma 2 Let g^{t}, a ≤ t ≤ b, be a family of metrics on the compact manifold X. Let dt^{2} be the Euclidean metric on the interval (a, b) and consider the metric G = g^{t} + dt^{2} on X × (a, b). We have,

V ol_{G}(X × (a, b)) ≤ sup_{a≤t≤b}{V ol_{g^{t}}(X)}(b − a).

We will be concerned with the case in which all the metrics g^{t} have strictly positive scalar curvature. One can then see ([4], [7]) that doing the homotopy slowly in time (stretching the interval) the scalar curvature of the total metric is also positive. We will now write down explicitly how much we need to stretch the interval in order to get positive scalar curvature.

First we need to introduce some notation. Suppose we have a homotopy, H = {g^{t}}, 0 ≤ t ≤ 1, of positive scalar curvature metrics on a compact manifold X. Let

s(H) = min_{(x,t)}{s_{g^{t}}(x)}.

Now consider a system of coordinates x̄ = (x_{1}, ..., x_{n}) in some open neighbourhood in X. Let G_{a} be the metric g^{t} + a^{2}dt. Let g_{ik}(x, t) = g^{t}(∂/∂x_{i}, ∂/∂x_{k}) and let g^{ik} denote as usual the coefficients of the inverse of the matrix (g_{ik}). By a straightforward computation we get the following formula,

s_{G_{a}}(x, t) = s_{g^{t}}(x) + \frac{1}{4a^{2}} (g^{ik} \frac{∂g_{jk}}{∂t} g^{jp} \frac{∂g_{ip}}{∂t} − g^{ik} \frac{∂g_{ik}}{∂t} g^{jp} \frac{∂g_{jp}}{∂t} ) (x, t)

−\frac{1}{2a^{2}} (g^{ik} \frac{∂^{2}g_{ik}}{∂t^{2}} + \frac{∂}{∂t}(g^{ik} \frac{∂g_{ik}}{∂t})) (x, t),

where s_{G_{a}} and s_{g^{t}} denote the scalar curvatures of the metrics G_{a}and g^{t}, respectively.

We will call B(x, t) the expression multiplied by 1/a^{2} in the previous formula and B(H) the maximum of B(x, t) on X × [0, 1]. Note that B(x, t) does not depend on the system of coordinates. We have

Lemma 3 Let H = g^{t}, 0 ≤ t ≤ 1, be a family of metrics of strictly positive scalar curvature on the compact manifold X. The metric Ĝ_{a} = g^{t/a} + dt^{2} on X × [0, a] has positive scalar curvature if a > \sqrt{B(H)/s(H)}.

Proof: Consider the map between X × [0, 1] and X × [0, a] given by multiplication by a in the t-coordinate. Pulling back the metric Ĝ_{a} via this diffeomorphism we get the metric G_{a} = g^{t} + a^{2}dt^{2} on X × [0, 1]. The lemma now follows directly from our formula for the scalar curvature of G_{a}.

□

Now let us come back to our problem. We want to study the metric g close to W. Find a coordinate system of the form (x_{1}, ⋯, x_{k}, y_{k+1}, ⋯, y_{n}). Where the x_{i} are coordinates for W and the y_{i} are coordinates on the normal bundle induced by an orthonormal frame (make the identification via the exponential map). As before r(x, y) will denote the distance between the point (x, y) and W. So r^{2} = Σy_{i}^{2}.

The metric g can be expressed in these coordinates in the form,

g = Σ(g_{ij}(x, 0) + o(r)) dx_{i}dx_{j} + Σo(r) dx_{i}dy_{α}+
+Σ(δ_{αβ} + o(r^{2})) dy_{α}dy_{β}.

We will first homotope g_{δ} to the product metric g_{δ}^{N} = g|_{W} + dE^{2}(δ) on W × S^{n−k−1} (dE^{2}(δ) is the standard Euclidean metric on the δ-sphere). It is easy to see from our local formula that the metrics g_{δ} and g_{δ}^{N} will differ only by low order terms. Consider the simplest homotopy g_{δ}^{t} = tg_{δ}^{N} + (1 − t)g_{δ}. In local coordinates g_{δ}^{t} will look like g_{δ}^{N} plus some terms of order o(r) multiplied by t. If we call H^{1} this family of metrics we have then that B(H^{1}) is bounded independently of δ (for δ smaller than some fixed small δ_{0}). It is also clear that s(H^{1}) is bounded below by some positive multiple of 1/δ^{2}. We also have that the volume of the metrics g_{δ}^{t} go to 0 as we take δ small. It follows from the previous lemmas that given any ϵ_{0} > 0 for any δ small enough the metric g_{δ}^{t} +dt^{2} on W ×S^{n−k−1} ×[0, 1] will have positive scalar curvature everywhere and volume less than ϵ_{0}.

Hence we can assume now that at the end we have the product metric, g_{δ}^{N}, of the Euclidean metrics (on the δ-sphere and on the line) and the restriction of g to W.

Finally, we need to show that we can change the metric g|_{W} to any other metric h on W. Consider again the homotopy g^{t} = t(h + dE^{2}) + (1 − t)g_{δ}^{N}. Of course, the homotopy only works on the W -part, while we always have the Euclidean metric on the δ-sphere. Let us call H^{2} this homotopy. It is again easy to see that B(H^{2}) is bounded independently of δ (it is actually independent of δ), while s(H^{2}) is bounded below by some positive multiple of 1/δ^{2}. The volumes of the metrics g^{t} tend to 0. These observations and the previous lemmas show that if δ is small enough the metric g^{t} + dt^{2} on W × S^{n−k−1} × [0, 1] has positive scalar curvature and volume less than ϵ_{0}.

We have therefore finished Step 2 and so our construction.

3 Proof of Theorem 1

In this section we will use the previous construction to prove the results mentioned in the introduction. We will use the following well known result (see [5]),

Lemma 4 Let C be a conformal class of Riemannian metrics on the compact n-dimensional manifold M. Assume n ≥ 3 and Y (M, C) ≤ 0. Let g be any metric in C. Then,

Y (M, C) ≥ min(s_{g})(V ol_{g}(M))^{2/n}.

Proof of Theorem 1: We will use the same notations as in the introduction. To begin, fix any positive ϵ. Pick conformal classes C_{i} in M_{i} so that Y (M_{i}, C_{i}) ≥ Y (M_{i}) − ϵ (for i = 1, 2). Let g_{i} be the metric of volume λ_{i} in C_{i} realizing Y (M_{i}, C_{i}). Renormalize the metrics so that λ_{1} + λ_{2} = 1.

Now construct metrics ĝ_{i} in M_{i} −W, as in the previous section (using the ϵ we have just picked), so that they coincide in the ends. Then we can glue them together to obtain a metric ĝ on M_{1,2}^{W}. The volume of ĝ satisfies

V olĝ(M_{1,2}^{W}) ≤ 1 + 2ϵ

Note also that the scalar curvature of g_{i} is constant equal to

s_{g_{i}} = Y (M_{i}, C_{i})(λ_{i})^{−2/n}.

Hence,

sĝ ≥ min \{\frac{Y (M_{1}, C_{1})}{(λ_{1})^{2/n}} − ϵ, \frac{Y (M_{2}, C_{2})}{(λ_{2})^{2/n}} − ϵ\}

everywhere. So, from the previous lemma, we get:

Y (M_{1,2}^{W}, Cĝ) ≥ (1 + 2ϵ)^{2/n}min \{\frac{Y (M_{1}, C_{1})}{(λ_{1})^{2/n}} − ϵ, \frac{Y (M_{2}, C_{2})}{(λ^{2})_{2/n}} − ϵ\}

To simplify the notation, we will write a_{i} = Y (M_{i}, C_{i}). Assume first that both a_{1} and a_{2} are strictly negative. We can rescale the metrics so that λ_{1} is any number in (0,1), and λ_{2} = 1 − λ_{1}. The minimum appearing in the previous formula will be the greatest when the two numbers are equal. This happens when

λ_{i} = \frac{|a_{i}|^{n/2}}{|a_{1}|^{n/2} + |a_{2}|^{n/2}}

Then we get

min \{\frac{Y (M_{1}, C_{1})}{(λ_{1})^{2/n}} − ϵ, \frac{Y (M_{2}, C_{2})}{(λ_{2})^{2/n}} − ϵ\} = −(|a_{1}|^{n/2} + |a_{2}|^{n/2})^{2/n} − ϵ

and so,

Y (M_{1,2}^{W}, Cĝ) ≥ −(1 + 2ϵ)^{2/n} [((−Y (M_{1}) + ϵ)^{n/2} + (−Y (M_{2}) + ϵ)^{n/2})^{2/n} + ϵ] .

Since Y (M_{1,2}^{W}) = sup_{C}{(Y (M_{1,2}^{W}, C)} and ϵ was arbitrary, we get

Y (M_{1,2}^{W}) ≥ −[(−Y (M_{1}))^{n/2} + (−Y (M_{2}))^{n/2}]^{2/n},

finishing the proof of Theorem 1 in this case. If a_{i} = 0 the result follows easily making λ_{i} tend to 0. Part b) of Theorem 1 follows in the same way since the fact that Y (M_{2}) > 0 implies that M_{2} admits a scalar flat metric. We have therefore finished the proof of Theorem 1.

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Acknowledgements: The first author would like to thank the hospitality of the directors and staff of the Max-Planck Institut, where he stayed during the preparation of this work. He is also very grateful to Claude LeBrun for very helpful observations on previous versions of this paper and to Vyacheslav Krushkal for many explanations about surgery techniques. The second author would like to express his gratitude to M. T. Anderson for advice.

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