Mass functions of a compact manifold

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Mass functions of a compact manifold

Andreas Hermann, Emmanuel Humbert

To cite this version:

Andreas Hermann, Emmanuel Humbert. Mass functions of a compact manifold. 2018. �hal-01818684�

ANDREAS HERMANN AND EMMANUEL HUMBERT

Abstract. LetM be a compact manifold of dimensionn. In this paper, we introduce theMass Functiona≥07→X₊^M(a) (resp. a≥07→X^M

−(a)) which is defined as the supremum (resp. infimum) of the masses of all metrics onM whose Yamabe constant is larger thanaand which are flat on a ball of radius 1 and centered at a pointp∈M. Here, the mass of a metric flat aroundp is the constant term in the expansion of the Green function of the conformal Laplacian atp. We show that these functions are well defined and have many properties which allow to obtain applications to the Yamabe invariant (i.e. the supremum of Yamabe constants over the set of all metrics onM).

Contents

1. Introduction 1

2. Notation 2

3. Upper and lower mass functions ofM 3

4. Basic properties ofX_±^M 4

5. X_±^M(a) and surgery 5

5.1. Proof of Theorem 5.1 6

5.2. Proof of Corollary 5.2 7

6. Application to the Yamabe invariant 7

Appendix A. A lemma on the Yamabe constant 15

References 17

1. Introduction

Let (M, g) be a closed Riemannian manifold of dimension n≥3 and denote by Lg:= ∆g+ n−2

4(n−1)sg: C^∞(M)→C^∞(M)

the conformal Laplace operator ofg, wheresg is the scalar curvature ofg and ∆g

is the Laplace-Beltrami operator with non-negative spectrum. Assume that the metricg is flat on an open neighborhood of a pointp∈M and that all eigenvalues ofLg are strictly positive. Then it is well-known that there exists a unique Green function Gg of Lg at p, i. e. in the sense of distributions we have LgGg =δp, the function Gg is smooth and strictly positive onM \ {p}and asx→pwe have

Gg(x) = 1

(n−2)ωn−1r(x)ⁿ⁻² +m(g, p) +o(1)

Key words and phrases. Yamabe operator; Yamabe invariant; surgery; positive mass theorem.

1

whereωn−1is the volume of the standard sphere of dimensionn−1, the function r denotes the Riemannian distance from p and m(g, p) ∈ R is a number called the mass of g at p. This quantity is related to the so-called ADM mass of an asymptotically flat Riemannian manifold. The study of the mass has led to many interesting results in geometric analysis and General Relativity. An example is an application to the so-called conformal Yamabe constant of (M, g) defined by

Y(M, g) := inf Z

M

sgdv^g,

where the inf is taken over the set of all Riemannian metrics on M which have unit volume and are conformal to g. Namely, in a famous article [8], Richard Schoen used positivity of the massm(g, p) to prove that Y(M, g)< Y(Sⁿ, gcan) if (M, g) satisfies the assumptions above and is not conformally diffeomorphic to the standard sphere (Sⁿ, gcan).

In this article we consider the dependence of the mass on the Yamabe constant Y(M, g). We define two functions a7→X₊^M(a) anda7→X₋^M(a) whose values are a sup and an inf of massesm(g, p) respectively taken over the set of all Riemannian metricsg withY(M, g)> awhich are flat on a ball of radius 1 centered atp∈M (see Definition 3.1). We prove that for small values of a the values X₊^M(a) and X₋^M(a) decrease and increase respectively under surgery of codimension at least 3 (see Theorem 5.1). Finally, we give an application to the smooth Yamabe invariant ofM defined by

σ(M) := supY(M, g)

where the sup is taken over the set of all Riemannian metrics onM. The question of whether for a given smooth manifold M one has σ(M) < σ(Sⁿ) is open in general. We prove that if X₊^M(σ(M)) > 0 then we have σ(M) < σ(Sⁿ). The precise statement is given in Theorem 6.1.

In the proofs of these theorems we use a surgery result obtained by the second author together with Ammann and Dahl [1] and a variational characterization of the massm(g, p) obtained by the two authors of the present article [5].

Acknowledgement: E. Humbert is supported by the project THESPEGE (APR IA), R´egion Centre-Val de Loire, France, 2018-2020.

2. Notation

LetM be a closed manifold of dimensionn≥3. The set of Riemannian metrics onM will be denoted byMM. Forg∈ MM, we denote by

Lg:= ∆g+ n−2

4(n−1)sg: C^∞(M)→C^∞(M)

the conformal Laplace operator ofg, wheresg is the scalar curvature ofg and ∆g

is the Laplace-Beltrami operator with non-negative spectrum. Moreover, we write N := _n−2²ⁿ and we denote by

Y(g) :=Y(M, g) = infnR

MuLgu dv^g kuk²_LN

u∈C^∞(M)\ {0}o

the (conformal) Yamabe constant ofgand byσ(M) := sup_g∈MMY(g) the (smooth) Yamabe invariant ofM. We will writeY(g) instead ofY(M, g) sinceM will always be clear from the context. We have Y(g) >0 if and only if all eigenvalues of Lg

are strictly positive. In the following, we will always assume that σ(M)>0.We define, for anya∈[0, σ(M)[,

ZM(a) :={g∈ MM |Y(g)> a}

Ω^a_M :=

(g, p)∈ZM(a)×M |B^g_p(1) is isometric to B .

whereB_p^g(1) stands for the ball with centerpand radius 1 with respect to the met- ricg and where Bis the standard Euclidean unit ball of dimension n. Note that these sets are not empty as soon asσ(M)>0 (see the first item of Proposition 4.1).

Letηbe a smooth function onM such thatη≡_(n−2)ω¹

n−1 onB^g_p(¹₂) and supp(η)⊂ B_p^g(1), whereωn−1denotes the volume ofSⁿ⁻¹with the standard metric. If (g, p)∈ Ω^a_M for some a≥0 then there exists a unique Green function Gg of Lg at pand we have for allx∈M\ {p}:

Gg(x) =η(x)r(x)²⁻ⁿ+m(g, p) +α(x),

where r(x) denotes the Riemannian distance of x and p with respect to g, α is a smooth function defined on all of M which is harmonic on B_p^g(¹₂) and satisfies α(p) = 0 andm(g, p)∈Ris a number called the mass ofg atp.

We recall thatm(g, p) has a variational characterization established in [5]. Namely, the functionFη: M →Rdefined by

Fη(x) :=

∆g(ηr²⁻ⁿ)(x), x6=p

0, x=p

is smooth onM. For everyu∈C^∞(M) we define J_p^g(u) :=

Z

M\{p}

ηr²⁻ⁿFηdv^g+ 2 Z

M

uFηdv^g+ Z

M

uLgu dv^g.

Then, it was proven in [5] that

−m(g, p) = inf{J_p^g(u)|u∈C^∞(M)} (1) and that the infimum is attained for the smooth function β defined by

β(x) :=m(g, p) +α(x).

We say that a closed manifold satisfies PMT (for Positive Mass Theorem) if for every metric g on M and for all points p ∈ M such that g is flat on an open neighborhood ofpandY(g)>0 we havem(g, p)≥0. It is conjectured that every closed manifold satisfies PMT. This conjecture has been proved in some special cases (see e. g. [7], [11], [9]). A complete proof has been announced by Lohkamp [6]

and Schoen-Yau [10].

3. Upper and lower mass functions ofM

Definition 3.1. The upper (resp. lower) mass function X₊^M : [0, σ(M)] → R∪ {±∞}(resp. X₋^M : [0, σ(M)]→R∪ {±∞}) are defined by : for alla∈[0, σ(M)]

X₊^M(a) := lim sup

ε→0

sup

(g,p)∈Ω^max(a_M ^−ε,0)

m(g, p)

(resp.

X₋^M(a) := lim inf

ε→0 inf

(g,p)∈Ω^max(a_M ^−ε,0)

m(g, p).)

Note that the maximum in the definitions above is only to ensure thatX₋^M(a) and X₊^M(a) are well defined whena= 0. Ifa >0, one can just replace Ω^{max(a−ε,0)}_M by Ω^a−ε_M in these defintions. The goal of this paper is to establish several properties of X_±^M(a).

4. Basic properties of X_±^M Proposition 4.1. It holds that

(1) X₊^M andX₋^M are well defined for all a∈[0, σ(M)]as soon asσ(M)>0 ; (2) For alla∈[0, σ(M)],X₊^M(a)≥X₋^M(a) ;

(3) X₊^M is a decreasing function of a and X₋^M is an increasing function of a and they are both left continuous;

(4) For alla∈[0, σ(M)],0≥X₋^M(a)>−∞; (5) For alla >0,X₊^M(a)<+∞;

(6) X₋^M(0) = 0if and only ifM possesses the property PMT ;

(7) LetM, N be compact manifolds of dimension n≥3 with positive Yamabe invariant. Then, for all a >0 we have

X₊^M∐N(a) = max(X₊^M(a), X₊^N(a)) and X₋^M∐N(a) = min(X₋^M(a), X₋^N(a)).

Proof. (1) It suffices to show that Ω^a(M) is not empty if 0 < a < σ(M). We fix a^′ ∈ (a, σ(M)). First, it is clear that there exists a metric g with Y(g) = a^′. Hence, letξ=P

idx²_i, where (x1,· · · , xⁿ) is a system of normal coordinates at some p∈M, be a flat metric aroundpand letgε:= (1−ηε)g+ηεξ, whereηε:M →[0,1]

is a cut-off function equal to 1 on B_p^g(ε), equal to 0 outside B_p^g(2ε) and such that

|dηε| ≤ ²_ε and |∇²ηε| ≤ _ε²2. By Lemma A.1 we have limε→0Y(gε) = Y(g). Now, the metrichε=_ε¹2gεis flat onB_p^hε(1). Ifεis small enough thenY(hε) =Y(gε)> a which implies thathε∈Ω^a(M).

(2) and (3) are clear from the definitions.

(4) Let u be any nonzero smooth function compactly supported in the Euclidean ball B. Let (g, p) ∈ Ω^a_M for some a. From the definition of Ω^a_M, we can identify (Bp(1), g) with Bso thatucan be considered as a test function in the variational characterization (1) which provides

−m(g, p)≤J_g^p(u).

The inequality X₋^M(a)>−∞ follows by noticing thatJ_g^p(u) does not depend on the choice ofa≥0 nor on the choice of (g, p)∈Ω^a_M.

Let us prove now thatX₋^M(a)≤0. It comes from the facts that if (g, p)∈Ω^a_M then (bg, p)∈Ω^a_M for anyb≥1, and also that for any suchb

m(bg, p) =b¹⁻ⁿ²m(g, p).

(5) Let ta >0 and let (g, p)∈Ω^a_M. We have to show thatm(g, p) is bounded by a constant which depends only ona but not on (g, p). Letu∈C^∞(M). In what follows, C > 0 denotes a positive constant which might depend on a but not on (g, p). By the variational characterization (1), chooseuso that

−m(g, p) + 1≥J_g^p(u)

From the definition ofJ_g^p, one has J_g^p(u)≥ −C+ 2

Z

M

uFηdv^g+ Z

M

uLgu dv^g. Using that fact thatY(g)≥aand using H¨older inequality, one gets J_g^p(u)≥ −C−2kFηkL^∞

Z

Bp^g(1)

|u|^Ndv^g

!_N¹

vol(B_p^g(1))^NN−1+a Z

M

|u|^N, dv^g _N²

.

Set now

Xg = Z

M

|u|^Ndv^g _N¹

,

we obtain that there exists someC^′, C^′′>0 independent of (g, p) such that J_g^p(u)≥C−C^′Xg+C^′′X_g². (2) This quantity is bounded from below independently of (g, p). This show thatm(g, p) is bounded from above by a constant independent of (g, p)∈Ω^a_M. This implies that for alla >0,X₊^M(a)<+∞.

(6) Clearly the property PMT for a manifold is equivalent to X₋^M(0) ≥0. Since X₋^M(0)≤0 by item (4), the result follows.

(7) Leta∈(0, σ(M∐N)] = (0,min(σ(M), σ(N))] andε >0. On the one hand, let (g, p)∈Ω^a−ε_M∐N wherep∈M. Thengdecomposes asg=gM∐gN wheregM ∈ MM

and gN ∈ MN. We have a−ε < Y(g) = min(Y(gM), Y(gN)). Sincep∈M, this implies that (gM, p)∈Ω^a−ε_M . Letρ >0. Ifεis small enough, it follows from the def- inition ofX₊^M(a) thatm(g, p) =m(gM, p)≤X₊^M(a)+ρ≤max(X₊^M(a), X₊^N(a))+ρ.

In the same way, ifp∈N,m(g, p)≤max(X₊^M(a), X₊^N(a)) +ρ. From these inequal- ities, we obtain

X₊^M∐N(a)≤max(X₊^M(a), X₊^N(a)) +ρ and sinceρis arbitrary

X₊^M∐N(a)≤max(X₊^M(a), X₊^N(a)).

On the other hand, let (gM, p)∈Ω^a−ε_M and letgN any metric onN withY(gN)≥ a−ε. Ifεis small enough, thenm(gM, p) =m(gM∐gN, p)≤X₊^M∐N(a) +ρ. Hence X₊^M(a)≤X₊^M∐N(a) +ρ. The same holds forN and the result follows.

The proof forX₋^M(a) is similar.

5. X_±^M(a)and surgery

In this section, we first establish the following theorem, whose proof is a conse- quence of the results in [5] and [1]

Theorem 5.1. Let M be a compact manifold of dimension n ≥ 3 and M^♯ be obtained fromM by a surgery of dimensionk≤n−3. Then, for alla∈[0, σ(M)], one has

X₊^M(a)≤X₊^M♯(min(a,Λn,k)) and X₋^M(a)≥X₋^M♯(min(a,Λn,k)).

whereΛn,0= +∞and whereΛn,k>0 depends only onn andk.

A consequence of Theorem 5.1 is

Corollary 5.2. Let M0 be any compact non spin (resp. spin) simply connected manifold of dimension n ≥ 5 such that σ(M0) > 0 and let a > 0. Then, for all compact (resp. compact spin) manifolds M of the same dimension one has

0≥X₋^M(min(a,Λn))≥X₋^M0(min(a,Λn)) and X₊^M(min(a,Λn))≤X₊^M0(min(a,Λn)) whereΛn= min1≤k≤n−3Λn,k >0.

This has the following obvious consequence:

Corollary 5.3. LetM0, M1be two compact non spin (resp. spin) simply connected manifolds of dimension n ≥ 5 such that σ(M0), σ(M1) > 0 and let a ∈ (0,Λn).

Then we have

X_±^M0(a) =X_±^M1(a).

Remark 5.4. By Corollary C in [4], if M is a compact simply connected non- spin manifold of dimension at least 5 then σ(M) >0. By [1], whenM is simply connected andσ(M)>0, it holds that

σ(M)≥min{Λn, σ(W1), ..., σ(Wk)}

whereW1, ..., Wk are generators of the oriented cobordism group in dimensionn.

Remark 5.5. This corollary allows to recover a result in [5]: ifM0not spin, simply connected of dimension n ≥5 satisfies PMT, then all the manifolds of the same dimension satisfy PMT. Indeed, assume thatM0 satisfies PMT thenX₋^M0(0) = 0 (see Proposition 4.1) and henceX₋^M(0) = 0 which implies PMT. Note that Lohkamp [6] and Schoen and Yau [10] recently announced a complete proof of the Positive Mass Theorem (i.e. all manifolds satisfy PMT).

Another consequence is the following:

Corollary 5.6. Assume that M is simply connected, that σ(M) > 0 and that a <Λn. Then,X₊^M(a)>0.

Remark 5.7. Again, if the proof of the Positive Mass Theorem by Lohkamp in [6] or by Schoen and Yau announced in [10] is confirmed then, for all M and all a < σ(M),X₊^M(a)>0.

5.1. Proof of Theorem 5.1. Letg ∈Ω^a_M. In [5], we constructed a sequence of metricsgkonM^♯such that limkm(gk, p) =m(g, p). In the construction, the metric gk can be made isometric togin B_p^g(1) (as soon asB_p^g(1) is topologically trivial).

Moreover, we used exactly the same metrics as in the main result of [1] where it was proved that

limk Y(gk)≥min(Λn,k, Y(g))

where Λn,0= +∞and Λn,k >0 depends only onnandk. This proves that for all ε >0 we havegk ∈Ω^min(a,Λ_M♯ ^n,k)−εas soon askis large enough. Theorem 5.1 easily follows.

5.2. Proof of Corollary 5.2. (1) Let M0 be a compact non-spin (resp. spin) simply connected manifold with σ(M0) >0 and M any compact (resp. compact spin) manifold of the same dimension. By Proposition 4.1,

X₊^M∐(−M)(a) =X₊^M(a),

where (−M) isM equipped with the opposite orientation. Theorem 5.1 then shows that

X₊^M(a) =X₊^M∐(−M)(a)≤X₊^M♯(−M)(min(a,Λn,k)) (3) where ♯ denotes the connected sum. Here, we used that the connected sum is a surgery of dimension 0.

(2) The manifolds M ♯(−M) and M0♯(−M0) are oriented (resp. spin) cobordant since they are both oriented (resp. spin) cobordant to Sⁿ. Since M0♯(−M0) is simply connected and not spin (resp. spin), it is obtained from M ♯(−M) by a finite sequence of surgeries of dimension k≤ n−3 (see the proofs of Theorem B and Theorem C in the article [4] by Gromov-Lawson). Theorem 5.1 then implies that

X₊^M♯(−M)(min(a,Λn,k))≤X₊^M0♯(−M0)(min(a,Λn,k)). (4) Inequality (3) remains true whenM is replaced byM0. As a consequence, we get from Proposition 4.1 that

X₊^M0♯(−M0)(a) = max(X₊^M0♯(−M0)(a), X₊^M0(a)) =X₊^{M0♯(−M0)∐M0}(a).

Using Theorem 5.1, we obtain

X₊^M0♯(−M0)(min(a,Λn,k))≤X₊^{M0♯(−M0)♯M0}(min(a,Λn,k))

Now, M0♯(−M0)♯M0 is oriented (resp. spin) cobordant to M0 and M0 is simply connected and not spin (resp. spin): by the same argument as above,M0is obtained fromM by a finite sequence of surgeries of dimensionk≤n−3. This proves that X₊^M0♯(−M0)(min(a,Λn,k))≤X₊^{M0♯(−M0)♯M0}(min(a,Λn,k))≤X₊^M0(min(a,Λn,k)).

Together with Inequalities (3) and (4), we obtain the desired inequality X₊^M(min(a,Λn,k))≤X₊^M0(min(a,Λn,k)).

(3) The argument forX₋^M(a) is similar.

6. Application to the Yamabe invariant Theorem 6.1.

1. For any compact manifoldM withσ(M)>0, one has X₊^M(0) = +∞.

2. For every n ≥ 3 there exists a constant dn > 0 such that for all compact manifolds M of dimensionnwith σ(M)>0 and for alla∈(0, σ(M)) we have

X₊^M(a)≤dn

(σ(Sⁿ)−a)^1/n

a .

Corollary 6.2. Let dn be the constant in part 1 of Theorem 6.1 and suppose that M is a compact manifold of dimensionn withσ(M)>0 such that

dn

σ(Sⁿ) <lim sup

ε→0

X₊^M(σ(M)−ε) ε^1/n . Then we haveσ(M)< σ(Sⁿ).

Note that the hypothesis of Corollary 6.2 is satisfied if X₊^M(σ(M)) > 0 since the functiona7→X₊^M(a) is continuous from the left. This fact leads to a natural question: is this possible thatX₊^M(σ(M))>0 ? The answer is given by

Proposition 6.3. It holds that

X₊^RP3(σ(RP³))>0 and X₊^Sn(σ(Sⁿ)) = 0.

Proof of Proposition 6.3. The fact that X₊^Sn(σ(Sⁿ)) = 0 is an immediate con- sequence of Corollary 6.2. Beside, it was proven by Bray and Neves [3] that σ(RP³) < σ(S³) is attained by the standard metric. Since the standard metric ofRP³ is locally conformally flat, one can choose a metricg in its conformal class such that B_p^g(1) is flat where p∈ RP³ is fixed. Then, sinceRP³ satisfies P M T, one has

X₊^RP3(σ(RP³))≥m(g, p)>0.

Proof of Corollary 6.2. Assume thatσ(M) =σ(Sⁿ). Using Theorem 6.1 we get

dn

σ(Sⁿ) <lim sup

ε→0

dnε^1/n

(σ(Sⁿ)−ε)ε^1/n = lim sup

ε→0

dn

σ(Sⁿ)−ε

which is a contradiction.

Proof of Theorem 6.1. For the first statement, let p ∈ M and gm ∈ Ω⁰_M be a sequence of metrics converging in C² to some metric g∞ such that Y(g∞) = 0.

To see the existence of such a sequence, it suffices to constructg∞. For this, just consider any metric (g, p) ∈ Ω^a_M for some a > 0. It is standard that one can modify g locally outside B_p^g(1) to get a metric h such that Y(h) <0. Then, set gt=tg+ (1−t)hfort∈[0,1]. Let

t∞:= max{t∈[0,1]|Y(gt)≤0}.

We can then setg∞:=gt∞ andgm:=g_t∞+¹

m. It was then proven in [5] and in [2]

that limmm(gm, p) = +∞which proves thatX₊^M(0) = +∞.

The second statement is much harder to prove. Let a ∈ (0, σ(M)) and let (gε)ε∈(0,a) be a sequence of Riemannian metrics onM such that for allεwe have gε∈Ω^a−ε_M and

Aε:=m(gε, p)≥X₊^M(a) +ε.

For everyεletβε be the smooth function onM such that−Aε=J_p^g(βε). We put Xε:=Z

M

|βε|^Ndv^gε_N¹ .

By H¨older’s inequality and by the definition ofY(gε) we have

−Aε=J_p^gε(βε) = Z

M\{p}

ηr²⁻ⁿFηdv^gε+ 2 Z

M

βFηdv^gε+ Z

M

βLgβ dv^gε

≥ Z

M\{p}

ηr²⁻ⁿFηdv^gε−2Z

M

|Fη|^NN−1dv^gεN

−1

N Xε+ (a−ε)X_ε². We put

Cn:=

Z

M\{p}

ηr²⁻ⁿFηdv^g, Dn := 2Z

M

|Fη|^NN−1dv^gN

−1 N .

The numbersCn,Dn are independent of (M, g) sinceg is flat on the supports ofη andFη. We get

−Aε≥Cn−DnXε+ (a−ε)X_ε².

We haveCn≥0 since forM =Sⁿ with the standard metricgcan we have 0 =−m(gcan, p)≤J_p^gcan(0) =Cn.

We also haveCn+Aε≥0 for allεsince

−Aε=m(gε, p)≤J_p^gε(0) =Cn.

The quadratic functionf(x) :=Aε+Cn−Dnx+ (a−ε)x² satisfiesf(Xε)≤0 and attains its minimum value atx0=−_2(a−ε)^Dn . We get

0≥f(x0) =Aε+Cn− D_n² 4(a−ε) and thus

X₊^M(a) +ε≤Aε≤ D_n²

4(a−ε)−Cn≤ D²_n 4(a−ε). Asε→0 we obtain

X₊^M(a)≤ D_n²

4a (5)

Moreover, sincef(Xε)≤0, the number Xεis less than or equal to the largest root of the equationf(x) = 0. Using thatAε+Cn≥0 we get

Xε≤Dn+p

D²_n−4(Aε+Cn)(a−ε)

2(a−ε) ≤ Dn

a−ε.

Recall that the function βε is harmonic on B_p^gε(¹₂). Thus for all x ∈ B^g_p^ε(¹₃) we have

βε(x) = 6ⁿ vol(B)

Z

Bx^gε(¹₆)

βε(y)dv^gε,

where B denotes the Euclidean unit ball, since B_x^gε(¹₆)⊂B_p^gε(¹₂). Using H¨older’s inequality we get for allx∈B_p^gε(¹₃):

|βε(x)| ≤ 6ⁿ vol(B)vol

B_x^gε1 6

^N

−1 N Z

B^gεx (¹₆)

|βε|^N_N¹

≤6^Nnvol(B)^−N1 Xε

≤6^Nnvol(B)^−N1 Dn

a−ε

=: Bn

a−ε. (6)

For everyεwe defineρε>0 such that ρⁿ⁻²_ε =δn

|Aε|

(Bn+|Aε|)² (7)

where δn > 0 can be chosen such that for all ε the number ρε is as small as we want since the function x7→ _(Bn^|x|_+|x|)2 is bounded. We chooseδn such thatρε<¹₆ for allε. Then for everyεwe choosehε∈C^∞(M) such that 0≤hε≤1,hε≡1 on B_p^gε(ρε), hε≡0 onM \B_p^gε(2ρε) and |dhε| ≤ _ρ²

ε. Moreover, for every εwe write the Green functionGεof Lgε as

Gε(x) =η(x)r(x)²⁻ⁿ+Aε+αε(x)

whereαε∈C^∞(M) is harmonic onB_p^gε(¹₂) and satisfiesαε(p) = 0.

Step 1. For ε close enough to0 we have Z

M

|d(hεαε)|²dv^gε ≤ (n−2)ωn−1

4 |Aε|.

Since αε is harmonic on the support of dhε, we can use Identity (3) in [5] and H¨older inequality to write

Z

M

|d(hεαε)|²dv^gε = Z

Cε

|dhε|²α²_εdv^gε ≤ Z

Cε

|dhε|ⁿdvgε

_n²Z

Cε

|αε|^Ndvgε

_N²

whereCε:=B^g_p^ε(2ρε)\B^g_p^ε(ρε) is the support ofdhε. Observe that the definition ofhε and the fact that the volume of the support ofdhε is bounded by Cρⁿ_ε with C independent ofεimply that there existsα0>0 which is independent ofεsuch that

Z

Cε

|dhε|ⁿdvgε

n²

≤α0.

Hence, for allεsmall enough, Z

M

|d(hεαε)|²dv^gε ≤α0

Z

Cε

|αε|^Ndvgε

_N²

. (8)

Now, withαε=βε−Aεand using the equations (6) and (7) we get forεclose to 0:

Z

Cε

|αε|^Ndv^gε_N¹

≤Z

Cε

|βε|^Ndv^gε_N¹

+|Aε|vol(Cε)^N1

≤ Bn

a−ε+|Aε|

vol(Cε)^N1

= Bn

a−ε+|Aε|

(2ⁿ−1)^N1ρ

n−2

ε2 vol(B)^N1

≤ Bn

a−ε+|Aε|

(2ⁿ−1)^N1δ

n−2

n2

p|Aε|

Bn+|Aε|vol(B)^N1

≤En

p|Aε|

whereEn >0 is independent ofε. By choosingδn in equation (7) smaller we may assume thatE_n²α0≤^(n−2)ω₄ ⁿ⁻¹. Therefore the assertion of Step 1 follows from the equation (8).

For everyεwe define

βε:=ζn|Aε|¹²ρεⁿ² (9) whereζn>0 will be fixed later. We have for allε:

βε=ζn|Aε|¹²ρ

n−2

ε2 ρε=ζnδ

1

n2 |Aε|

Bn+|Aε|ρε≤ζnδ

1

n2ρε. (10) We defineuε∈C^∞(M) by

uε(r) = βε

β_ε²+r^{2 n−2}₂

andψε∈C^∞(M) by ψε=

uε if r≤ρε

ℓε(Gε−hεαε) if ρε≤r≤2ρε

ℓεGε if r≥2ρε

whereℓε=uε(ρε)(_(n−2)ω¹

n−1ρ²⁻ⁿ_ε +Aε)⁻¹ so thatψε is continuous.

Step 2. Conclusion.

We set

Eε= Z

M

|dψε|²+ n−2

4(n−1)sgε|ψε|² dv^gε and

Dε= Z

M

|ψε|^Ndv^gε _N²

so that

Qε(ψε) = Eε

Dε

. (11)

We write

Eε=E1+E2

where

E1= Z

Bε

|dψε|²+ n−2

4(n−1)sgε|ψε|² dv^gε

and

E2= Z

M\Bε

|dψε|²+ n−2

4(n−1)sgε|ψε|² dv^gε

whereBε:=B_p^gε(ρε) which is isometric to the Euclidean ball of radiusρε. OnBε, it holds that

∆gεuε=n(n−2)u^N−1_ε

and we get from multiplying this equation byuεand integrating by parts that Z

Bε

|duε|²dv^gε− Z

∂Bε

uε

∂uε

∂r da^gε =n(n−2) Z

Bε

|uε|^Ndv^gε. (12) One also has

σ(Sⁿ) = R

Rⁿ|duε|²dx R

Rⁿ|uε|^Ndx_N² =n(n−2) Z

Rⁿ

|uε|^Ndx ²_n

which leads to

Z

Bε

|uε|^Ndx n²

≤ σ(Sⁿ) n(n−2) Plugging this estimate into equation (12), we obtain

E1≤σ(Sⁿ) Z

Bε

|ψε|^Ndv^gε _N²

+ Z

∂Bε

uε

∂uε

∂r da^gε. (13) Now, we evaluateE2. For this, we integrate by parts:

E2= Z

M\Bε

ψεLgεψεdvgε+E3

whereE3 is a boundary term which will be computed below. Since LgεGε= 0 on M \Bε, we obtain

E2=ℓ²_ε Z

B^′_ε\Bε

Lgε(−hεαε)(Gε−hεαε)dvgε+E3

where B^′ :=B_p^gε(2ρε). Note that since αε is harmonic andhε is constant onBε, one has onBε,

Lgε(−hεαε) = ∆gε(−hεαε) = 0.

Hence, by definition of the Green functionGε, one has:

Z

B^′_ε\Bε

L_gε(−hεα_ε)Gεdv^gε = Z

M

L_gε(−hεα_ε)Gεdv^gε=−αε(p) = 0.

We also have Z

B_ε^′\Bε

Lgε(hεαε)(hεαε)dv^gε = Z

M

Lgε(hεαε)(hεαε) = Z

M

|d(hεαε)|²dv^gε. Since, by Step 1 we have

Z

M

|d(hεαε)|²dv^gε ≤(n−2)ωn−1

4 |Aε|=:γε (14)

we obtain:

E2≤ℓ²_εγε+E3.

So let us evaluateE3: E3=−ℓ²_ε

Z

∂Bε

(Gε−hεαε)∂(Gε−hεαε)

∂r

=−ℓ²_ε 1 (n−2)ωn−1

ρ²⁻ⁿ_ε +Aε

− 1 ωn−1

ρ¹⁻ⁿ_ε

vol(∂Bε)

=ℓ²_ε 1 (n−2)ωn−1

ρ²⁻ⁿ_ε +Aε

.

Combining this with (13) E1+E2≤σ(Sⁿ)

Z

Bε

|ψε|^Ndv^gε N²

+ Z

∂Bε

uε

∂uε

∂r da^gε+ℓ²_ε 1 (n−2)ωn−1

ρ²⁻ⁿ_ε +Aε

+ℓ²_εγε. (15) It remains to compute

E4:=

Z

∂Bε

uε

∂uε

∂r da^gε. Using the fact that on∂Bεwe have

uε=ℓε

1 (n−2)ωn−1

ρ²⁻ⁿ_ε +Aε

and that

∂uε

∂r =−(n−2) ρε

β_ε²+ρ²_εuε

we obtain

E4=−(n−2)ωn−1

ρⁿ_ε

β_ε²+ρ²_εℓ²_ε 1 (n−2)ωn−1

ρ²⁻ⁿ_ε +Aε

2

which together with the definition (9) ofβε leads to Z

∂Bε

uε

∂uε

∂r da^gε+ℓ²_ε 1 (n−2)ωn−1

ρ²⁻ⁿ_ε +Aε

=ℓ²_ε 1 (n−2)ωn−1

ρ²⁻ⁿ_ε +Aε

1− ρⁿ_ε

β_ε²+ρ²_ε(ρ²⁻ⁿ_ε + (n−2)ωn−1Aε)

=ℓ²_ε 1 (n−2)ωn−1

ρ²⁻ⁿ_ε +Aε

β²_ε−ρⁿ_ε(n−2)ωn−1Aε

β²_ε+ρ²_ε

=ℓ²_ε 1 (n−2)ωn−1

ρ²⁻ⁿ_ε +Aε

(ζ_n²|Aε| −(n−2)ωn−1Aε)ρⁿ_ε ζ_n²|Aε|ρⁿ_ε +ρ²_ε

=ℓ²_ε 1 (n−2)ωn−1

+Aερⁿ⁻²_ε ζ_n²|Aε| −(n−2)ωn−1Aε

ζ_n²|Aε|ρⁿ⁻²ε + 1 By assumption we have

Aε≥X₊^M(a) +ε.

If X₊^M(a)≤0, then the assertion 1 of Theorem 6.1 holds trivially. Thus we may assume thatAε>0 for allε. We put

ζn :=

p(n−2)ωn−1

2 .

Ifρεis small enough, we obtain Z

∂Bε

uε

∂uε

∂rda^gε+ℓ²_ε 1 (n−2)ωn−1

ρ²⁻ⁿ_ε +Aε

≤ℓ²_ε 1 (n−2)ωn−1

−(n−2)ωn−1Aε

2

.

Inserting this into equation (15) and using the definition (14) ofγεwe get

E1+E2≤σ(Sⁿ) Z

Bε

|ψε|^Ndv^gε _N²

−ℓ²_ε 1 (n−2)ωn−1

−(n−2)ωn−1Aε

2 +(n−2)ωn−1Aε

4

=σ(Sⁿ) Z

Bε

|ψε|^Ndv^gε _N²

−ℓ²_εAε

4 . Moreover, ifρε is small enough, we obtain

ℓ²_ε= βε

β²_ε+ρ²_ε

n−2 1 (n−2)ωn−1

ρ²⁻ⁿ_ε +Aε

−2

=ζn|Aε|^1/2ρ^n/2ε

ζ_n²Aερⁿ_ε+ρ²_ε

n−2 1 (n−2)ωn−1

ρ²⁻ⁿ_ε +Aε

−2

=ζn|Aε|^1/2ρ^n/2ε

ζ_n²Aερⁿ⁻²ε + 1 n−2

ρ⁴⁻²ⁿ_ε 1 (n−2)ωn−1

+Aερⁿ⁻²_ε −2

ρ²ⁿ⁻⁴_ε

≥ζn|Aε|^1/2 2

n−2

ρ

n(n−2)

ε 2

(n−2)²ω_n−1²

4 .

From this and the definition (7) ofρε it follows that

E1+E2≤σ(Sⁿ) Z

Bε

|ψε|^Ndv^gε N²

−4⁻ⁿ (n−2)ωn−1ⁿ⁺²2 |Aε|ⁿ²ρ

n(n−2)

ε 2

=σ(Sⁿ) Z

Bε

|ψε|^Ndv^gε _N²

−4⁻ⁿ (n−2)ωn−1ⁿ⁺²₂

δnⁿ² |Aε|ⁿ (Bn+|Aε|)ⁿ

=:σ(Sⁿ) Z

Bε

|ψε|^Ndv^gε _N²

−E_n^′ |Aε|ⁿ (Bn+|Aε|)ⁿ. We have

Dε≥ Z

Bε

|ψε|^Ndv^gε _N²

and therefore

Qε(ψε) = E1+E2

Dε

≤σ(Sⁿ)−E_n^′ |Aε|ⁿ (Bn+|Aε|)ⁿ

Z

Bε

|ψε|^Ndv^gε −_N²

Moreover, using the substitutionr=βεswe get Z

Bε

|ψε|^Ndv^gε =ωn−1

Z ρε

0

βε

β_ε²+r² n

rⁿ⁻¹dr

=ωn−1

Z ρε/βε

0

1 1 +s²

n

sⁿ⁻¹ds

≤ωn−1

Z ∞

0

1 1 +s²

n

sⁿ⁻¹ds

=:Fn. It follows that

Qε(ψε)≤σ(Sⁿ)− E_n^′ Fn^2/N

|Aε|ⁿ (Bn+|Aε|)ⁿ and therefore withGn:= (E^′_nFn^−2/N)^1/n we get

a−ε < Y(gε)≤σ(Sⁿ)−(Gn)ⁿ |Aε|ⁿ (Bn+|Aε|)ⁿ. We take the limit ε→0 and we get

a≤σ(Sⁿ)−(Gn)ⁿ X₊^M(a)ⁿ

(Bn+X₊^M(a))ⁿ =σ(Sⁿ)−(Gn)ⁿ 1 (_X^BMⁿ

+(a)+ 1)ⁿ It follows that

Bn

X₊^M(a)+ 1≥ Gn

(σ(Sⁿ)−a)^1/n We putαn:=σ(Sⁿ)−^(G₂ⁿn⁾ⁿ and we distinguish two cases.

a) Ifa≥αn we get

Bn

X₊^M(a)≥ 1 2

Gn

(σ(Sⁿ)−a)^1/n and sincea < σ(Sⁿ) we get

X₊^M(a)≤2Bn

Gn

(σ(Sⁿ)−a)^1/n≤ 2Bn

Gn

σ(Sⁿ)

a (σ(Sⁿ)−a)^1/n b) Ifa≤α_n we have

σ(Sⁿ)−a≥σ(Sⁿ)−αn

and therefore using equation (5) X₊^M(a)≤ D²_n

4a ≤ D_n²(σ(Sⁿ)−a)^1/n

4(σ(Sⁿ)−αn)^1/na =D²_n(σ(Sⁿ)−a)^1/n 2Gna .

This shows the second statement.

Appendix A. A lemma on the Yamabe constant

LetM be a compact manifold of dimensionn≥3. For any Riemannian metric g onM we denote by

Lg:= ∆g+ n−2 4(n−1)sg

the Yamabe operator ofgand by

Qg: C^∞(M)→C^∞(M), Qg(u) :=

R

MuLgu dv^g (R

M|u|^N)^2/N