José De Sousa
email: jdesousa@univ-paris1.fr

Xavier Fairise
email: xavier.fairise@univ-lemans.fr.

Stefano Comino

Pierre Fleckinger

Carl Gaigné

Laurent Linnemer

Jean-François Nivet

Patrick Schmitz

Jean-Marc Tallon

Jean-Philippe Tropéano

Luis Vasconcellos

On the value of partial commitment for cooperative investment in buyer-supplier relationship 1

Keywords: Incomplete contracts, relationship-specific investments, cooperation. JEL Classification : K12, L22, C7

Does formal contracting foster cooperation in a buyer-supplier relationship?

In line with the literature, we find that a renegotiable contract with relationshipspecific joint investments does not make it possible to reach the first-best. However, we show that a renegotiable contract may induce more cooperation than an informal arrangement can. This result may help to understand how cooperation emerges in Japanese procurement practices, which typically involve relationshipspecific joint investments and renegotiable contracts.

Introduction

A growing range of production activities is now subcontracted. Firms split their production process and outsource activities, from product design to assembly. This split expands vertical inter-firm relationships and trade in customized parts. In such vertical relationships, the "hold-up problem" appears to be crucial [START_REF] Klein | Vertical integration, appropriable rents and the competitive contracting process[END_REF]. The canonical example goes as follows. A buyer commissions a supplier to produce a tailor-made input. Either the supplier or both the buyer and the supplier make ex ante non-contractible and relationshipspecific investments to transact. At the (re)negotiation stage, investors are locked into the relationship and risk not receiving ex post the overall return of their autonomous investments. Two potential economic inefficiencies result from the hold-up problem. The investors may under-invest in specific assets [START_REF] Grossman | The costs and benefits of ownership: a theory of vertical and lateral integration[END_REF] and/or waste resources to protect her investment returns.

Following the classification introduced in Che and Hausch (1999), autonomous specific investments can be either 'selfish', when they generate a direct benefit only for the investor, 'purely cooperative', when the direct benefit only goes to the partner of the investor, or 'hybrid', when it offers direct benefits to both parties. A first stream of the literature concentrates on selfish investments and shows that simple renegotiable fixed-price contracts can solve the hold-up problem [START_REF] Chung | Incomplete contracts, specific investments and risk sharing[END_REF][START_REF] Aghion | Renegotiation design with unverifiable information[END_REF][START_REF] Edlin | Specific investment under negotiated transfer pricing: an efficiency result[END_REF][START_REF] Edlin | Holdups, standard breach remedies, and optimal investment[END_REF][START_REF] Nöldeke | Option contracts and renegotiation: a solution to the hold-up problem[END_REF]. 4 Che and Hausch (1999) concentrate on purely cooperative and hybrid investments. They show that with purely cooperative investments contracting has value only if the parties can commit not to renegotiate, while renegotiable contracts offer no advantage over a simple ex post negotiation. They also show that renegotiable contracts can be valuable when investments are hybrid. In all the literature we have referred to investments are autonomous, i.e., each party decides independently from the other and one party's investment has effect even if the other decides not to invest.

In this paper we study the value of partial commitment to solve the hold-up problem for joint investments. Joint investments require an explicit cooperation between the parties: one party investment generates benefits only if the other party also invests, so that investments are complementary. In that setting, we show that with joint investments renegotiable contracts are always potentially 1 valuable, but never able to restore first best incentives for both parties.

This result echoes the hybrid case of Che and Hausch where a simple contract is also potentially valuable without restoring first best incentives. However, the setting is different. They coined as purely cooperative or partly cooperative (i.e., hybrid) investments that are autonomous.5 Thus, a restriction of the Che and Hausch's investments lies in their lack of joint work. In our setting, firms cooperate in the sense that their investments are complementary.

Considering joint investments may help to understand how cooperation emerges in Japanese procurement practices. Our framework roughly fits some stylized facts about the Japanese buyer-supplier relationships. First, parties trade in customized parts, which require autonomous and joint relationship-specific investments (see among others Asanuma, 1985aAsanuma, ,b, 1989;;[START_REF] Aoki | Information, Incentives, and Bargaining in the Japanese Economy[END_REF][START_REF] Nishiguchi | Strategic Industrial Sourcing: The Japanese Advantage[END_REF][START_REF] Qiu | Keiretsu and relationship-specific investment: implications for market-opening trade policy[END_REF][START_REF] Spencer | Keiretsu and relationship-specific investment: A barrier to trade?[END_REF]. As an example of joint investments, Japanese car manufacturers cooperate with suppliers to design parts of the final product.6 They coordinate tasks, share information and meet each other. This cooperation is typically linked to joint investments, which are the source of productivity improvements over time. Second, Japanese practices tend to differ from American ones in key areas such as quality control and price determination. However, characterizing Japanese arrangements as informal and cooperative and Western arrangements as formal and antagonistic is a coarse generalization. In fact, the difference is blurred, as Western firms have adopted many Japanese practices (see e.g [START_REF] Cusumano | Supplier relations and management: a survey of Japanese, Japanese-transplant, and US auto plants[END_REF], while Japanese firms contract with their partners (Asanuma, 1985a,b, Nishiguchi, 1989). Based on a survey of automobile manufacturers, [START_REF] Cusumano | Supplier relations and management: a survey of Japanese, Japanese-transplant, and US auto plants[END_REF] provide evidence of the contractual nature of the Japanese buyer-supplier relationships. They find that, for each new model of car, parties sign a new contract. "The most common contract (62 percent of the sample) is 4 years, corresponding to the average model life-cycle" [START_REF] Cusumano | Supplier relations and management: a survey of Japanese, Japanese-transplant, and US auto plants[END_REF]. Given that the average total duration of a Japanese buyer-supplier relationship is about ten years, this means that parties sign up to three different contracts by relationship. Third, these contractual arrangements provide room for renegotiation. Parties write basic renegotiable contracts establishing basic rules covering a range of items including price determination, payment, delivery, property rights, the supply of materials and quality issues (Nishiguchi, 1989). Finally, these contractual arrangements also promote cooperation. The typical contract sets a target price for each input produced. Buyers then cooperate and help suppliers to reach their targets [START_REF] Cusumano | Supplier relations and management: a survey of Japanese, Japanese-transplant, and US auto plants[END_REF]; see also Nishiguchi, 1989). Therefore, overall Japanese firms seem to rely on partial commitment to their relationships and ex post bargaining to promote cooperation and joint investments.

Our result also echoes the literature on the value of partial commitment devices in other strategic and decision-theoretic environments. For instance, Colombo andMerzoni (2006, 2008) stress the value of partial commitment to a relationship in terms of the duration of contracts, while [START_REF] Amador | Commitment vs. flexibility[END_REF] show the value of commitment devices that partially limit the freedom of choice with time-inconsistent preferences.

Our contribution is also related to [START_REF] Mclaren | Supplier relations and the market context: a theory of handshakes[END_REF], who distinguishes joint from autonomous investments. However, the latter are assumed to be not relationship specific. This implies parties can pay an additional fee to adapt the tailor-made input to an alternative buyer. Under this assumption, a nonrenegotiable fixed-price contract gives optimal incentives for autonomous investments, but not for joint work. Ex post bargaining is thus required to foster cooperation, as the supplier can always get the buyer to share the costs ex post. We assume instead that autonomous investments are relationship-specific and we find that a fixed-price contract is valuable to promote cooperation.7 However, this result is not necessarily in conflict with [START_REF] Mclaren | Supplier relations and the market context: a theory of handshakes[END_REF]. In fact, in both settings, the possibility of bargaining ex post fosters cooperation: although the present setting underlines the value of contracting as partial commitment, contract incompleteness leaves room for renegotiation. 8The rest of the paper is organized as follows. In the next section, we present the model, a simple two-stage game between a buyer and a supplier. In section 3, we establish two benchmark outcomes to compare our results: the first-best and the ex post bargaining (without an initial contract). In section 4, we show how a formal fixed-price contract arrangement may foster cooperation. Finally, in section 5, we conclude.

Model

We consider a basic two-stage procurement model between a buyer (b) and a supplier (s). The buyer procures an input from the supplier. There are two simple ways of procuring the input: a formal or an informal arrangement.

In the formal arrangement, parties design a renegotiable fixed-price contract in the first stage and specify ex ante a fixed monetary transfer (t ∈ ℜ) of the buyer to the supplier for a fixed quantity of input (q ∈ ℜ + ). This initial allocation is enforceable by the court and ensures for the parties a status quo payoff. Contract terms are enforced in the second stage, unless they are renegotiated, in which case parties share ex post the surplus from renegotiation according to their bargaining strength.

In the informal arrangement, parties agree verbally, in the first stage, on the quantity of input without signing an initial contract. In the second stage, they bargain the terms of trade and determine the payment. Whatever the arrangement, not contractible autonomous and joint investments are made simultaneously in the first stage. They are relationship-specific, which rules out outside options and the possibility of adapting the input for an alternative buyer (see above).

Payoff functions and the nature of investments

Let v(q, j b , j s ) denote the buyer's gross value of procuring the good q ∈ ℜ + and c(q, a, j b , j s ) the supplier's gross monetary cost of producing q. Valuations are determined by relationship-specific investments. Let a ∈ ℜ + be the level (and cost) of autonomous investments made by the supplier. We first present the assumptions made throughout this study before commenting in more details on a, j b and j s .

Assumption 1 v and c are continuously differentiable in all arguments.

Assumption 2 v(q, j b , j s ) ≥ 0 is increasing in all arguments and strictly concave. For all q > 0 and (j b , j s ) ∈ ℜ 2 + , it satisfies:

lim q→0 v 1 (q, j b , j s ) = ∞, lim j b →0 v 2 (q, j b , j s ) = ∞, lim js→0 v 3 (q, j b , j s ) = ∞; lim q→∞ v 1 (q, j b , j s ) = 0, lim j b →∞ v 2 (q, j b , j s ) = 0, lim js→∞ v 3 (q, j b , j s ) = 0.
Assumption 3 c(q, a, j b , j s ) ≥ 0 is increasing in q, decreasing in investments and strictly convex. For all q > 0 and (a, j b , j s ) ∈ ℜ 3 + , it satisfies:

lim q→0 c 1 (q, a, j b , j s ) = 0 lim a→0 c 2 (q, a, j b , j s ) = -∞, lim j b →0 c 3 (q, a, j b , j s ) = -∞ lim js→0 c 4 (q, a, j b , j s ) = -∞. lim q→∞ c 1 (q, a, j b , j s ) = ∞ lim a→∞ c 2 (q, a, j b , j s ) = 0, lim j b →∞ c 3 (q, a, j b , j s ) = 0 lim js→∞ c 4 (q, a, j b , j s ) = 0.
Concavity and convexity of assumptions 2 and 3 imply decreasing returns for both parties.

Assumption 4 ∀(j b , j s ) ∈ ℜ 2 + v(0, j b , j s ) = 0, and ∀(a, j b , j s ) ∈ ℜ 3 + c(0, a, j b , j s ) = 0.

Assumption (4) says that when q = 0 both valuations do no depend on the level of investments. Since there is no outside market for investments, this assumption suggests that investments are relationship-specific [START_REF] Chung | Incomplete contracts, specific investments and risk sharing[END_REF](Chung, : 1034)).

Assumption 5 The cross derivatives of v(q, j b , j s ) and c(q, a, j b , j s ) satisfy v iℓ (q, j b , j s ) > 0 and c iℓ (q, a, j b , j s ) < 0 for all i = ℓ.

Assumption (5) says that investments are complementary.

Assumption 6

v 2 (q, j b , 0) = 0, and c 3 (q, a, 0, j s ) = 0.

Assumption (6) stipulates that the marginal return of joint investments is null when only one party is contributing. This assumption departs from other related structures of investments in the literature. Thus, our structure would be equivalent to the [START_REF] Edlin | Holdups, standard breach remedies, and optimal investment[END_REF]'s one-sided investment structure if we dropped j b and j s and only retained the autonomous investment a, which is a 'selfish' investment directly benefiting only to the supplier through its valuation c. Following the terminology coined by Che and Hausch (1999), a can also be purely cooperative or partly cooperative (i.e., hybrid). Recall that a is partly cooperative when it affects both valuations c and v and purely cooperative when it affects only the buyer's valuation v. However, Che and Hausch (1999)'s terminology may be somewhat misleading since "cooperative" has nothing to do with cooperative behavior. There is no joint work. Cooperation is achieved independently from the other party and a has an effect even if the other decides not to invest. In our setting, a is a selfish investment and cooperation is achieved through additional joint investments j b and j s affecting both valuations. Following assumptions ( 5) and ( 6), joint investments are complementary and require an explicit cooperation between the parties: one party investment generates benefits only if the other party also invests. Using this structure of investments, we study the value of partial commitment to solve the hold-up problem for joint investments.

Benchmark outcomes

We establish two useful benchmarks, the first-best and the no-contracting outcome, with which later results about contracting may be compared.

The first-best outcome

The first-best corresponds to the solution of the integrated firm program, which internalizes the effects of investment. The maximization program of the integrated firm is separable. In a first step, we determine the optimal quantity (q * ) given the investment levels. Then, we determine the investment levels given the optimal quantity.

Let Π denote the maximum gross joint surplus, such that:

Π(a, j b , j s ) = max q≥0 [v(q, j b , j s ) -c(q, a, j b , j s )] .
According to the optimality condition v 1 = c 1 , we obtain

q * = q * (a, j b , j s ) (1)
the quantity equalizing the marginal benefit to the marginal cost, therefore Π(a, j b , j s ) = v(q * , j b , j s ) -c(q * , a, j b , j s ).

The net joint surplus of investments S(a, j b , j s ) is given by:

S(a, j b , j s ) = Π(a, j b , j s ) -a -j b -j s ,
with Π(a, j b , j s ) strictly concave since v(.) is concave and c(.) convex. The efficient investments are such that (a * , j * b , j * s ) ∈ arg max a,j b ,js Π(a, j b , j s ) -a -j b -j s . Given the assumptions on v and c, (a * , j * b , j * s ) are unique and satisfy a system of first-order conditions (FOCs):

Π 1 (a * , j * b , j * s ) -1 = 0, (2)
Π 2 (a * , j * b , j * s ) -1 = 0, (3)
Π 3 (a * , j * b , j * s ) -1 = 0, (4)

The (informal) no-contracting outcome

We now consider the no-contracting game. Let us recall the sequence of events. Ex ante, parties agree verbally, without a prior contract, that the supplier will produce the input. Ex post, parties share the surplus according to their exogenous bargaining positions. 9 The optimal quantity q * ∈ ℜ + and the monetary transfer t ∈ ℜ are determined in the second stage, while investments are realized in the first stage. At the second stage, the negotiation outcome on q and t is solution of a Nash bargaining process, with µ ∈ [0, 1] the supplier's bargaining strength:

max t,q [v(q, j b , j s ) -t] 1-µ [t -c(q, a, j b , j s )] µ .
Therefore q * = q * (a, j b , j s ), is implicitly determined by v 1 (q * , j b , j s ) = c 1 (q * , a, j b , j s ), and t(a, j b , j s ) = (1 -µ)c(q * , a, j b , j s ) + µv(q * , j b , j s ).

At the first stage, the buyer and the supplier maximize their surplus:

• for the buyer:

U b = v(q * , j b , j s ) -(1 -µ)c(q * ,
a, j b , j s ) -µv(q * , j b , j s ) -j b ;

• for the supplier:

U s = (1 -µ)c(q * ,
a, j b , j s ) + µv(q * , j b , j s ) -c(q * , a, j b , j s ) -a -j s .

Given the optimal produced quantity q * , determined by equation ( 1) in both the first-best and the no-contracting outcome, we rewrite the above surplus using Π(a, j b , j s ). It follows that parties choose the investment levels of the nocontracting outcome ( a,  b ,  s ) satisfying

(  b ) ∈ arg max j b (1 -µ)Π(a, j b , j s ) -j b ,
( a,  s ) ∈ arg max a,js µΠ(a, j b , j s ) -a -j s , and the following system of FOCs:

∂U b ∂j b = (1 -µ)Π 2 ( a,  b , j s ) -1 = 0, (5)
∂U s ∂a = µΠ 1 ( a,  b ,  s ) -1 = 0, (6)
∂U s ∂j s = µΠ 3 ( a,  b ,  s ) -1 = 0. (7)
Since µ ∈ [0, 1], efficiency cannot be achieved. This may be explained as follows. Investments are made ex ante, while the surplus is shared ex post according to the bargaining positions. The payment t is determined independently of the investments made; therefore, externalities cannot be internalized.

What are the consequences of such an inefficiency? Given the concavity of Π(a, j b , j s ), the parties will invest less than the socially optimal level. Proposition 1 Under assumptions (1) to ( 6), the absence of contracting prior investing in specific assets induces under-investments, such that: a < a * ,  b < j * b and  s < j * s .

Proof See appendix (A).

Contracting and cooperation

We have seen that parties do not reach efficiency by simply bargaining ex post the terms of trade without a prior contract. Now suppose that parties sign a simple renegotiable fixed-price contract that specifies a fixed monetary transfer (t ∈ ℜ) of the buyer to the supplier for a fixed quantity of goods (q ∈ ℜ + ). Two questions arise. First, does the signing of this simple renegotiable contract make it possible to achieve efficiency? Second, failing that, does contracting offer a better outcome than the no-contracting game? If not, the contract has no value, and the optimal contract is the 'no contract.'

The contracting outcome

With regard to the first question, we find in a simple way that contracting does not make it possible to reach the first-best. This is not very surprising and can be shown formally.

Let first define the (gross) renegotiation surplus (RS), available ex post as:

RS = Π(a, j b , j s ) -[v(q, j b , j s ) -c(q, a, j b , j s )].
At the second stage, we assume a Nash bargaining process on q and t, solution of max t,q v(q, j b , j s ) -t -v(q, j b , j s ) + t 1-µ × t -c(q, a, j b , j s ) -t + c(q, a, j b , j s ) µ .

We obtain q * = q * (a, j b , j s ) implicitly determined by v 1 (q * , j b , j s ) = c 1 (q * , a, j b , j s ), and

t(a, j b , j s ) = (1 -µ) [c(q * , a, j b , j s ) -c(q, a, j b , j s )] + µ [v(q * , j b , j s ) -v(q, j b , j s )] + t.
The first stage objectives to be maximized are:

• for the buyer:

U b = v(q, j b , j s ) -t A + (1 -µ)RS B -j b .
(A) is the buyer's payoff given by the initial contract. It represents the buyer's status quo position. (B) is the payoff from the renegotiation process, depending on the buyer's bargaining strength (1 -µ).

• for the supplier:

U s = t -c(q, a, j b , j s ) C + µRS D -a -j s .
(C) is the supplier's cost given by the initial contract. It represents the supplier's status quo position. (D) is the payoff from the renegotiation process, depending on the supplier's bargaining strength µ.

Parties make the investment levels of the contracting outcome ( a,  b ,  s ), satisfying

(  b ) ∈ arg max j b v(q, j b , j s ) -t + (1 -µ)RS -j b ,
( a,  s ) ∈ arg max a,js t -c(q, a, j b , j s ) + µRS -a -j s .

and the following system of first-order conditions:

∂U b ∂j b = µv 2 (q,  b ,  s ) + (1 -µ)c 3 (q, a,  b ,  s ) + (1 -µ)Π 2 ( a,  b ,  s ) -1 = 0, (8) ∂U s ∂a = -(1 -µ)c 2 (q, a,  b ,  s ) + µΠ 1 ( a,  b ,  s ) -1 = 0, (9)
∂U s ∂j s = -µv 3 (q,  b ,  s ) -(1 -µ)c 4 (q, a,  b ,  s ) + µΠ 3 ( a,  b ,  s ) -1 = 0. (10)
Since µ ∈ [0, 1], it is not possible to implement the first-best outcome. This implies that contracting with renegotiation does not make it possible to achieve efficiency.

Contracting or no-contracting?

We fail to achieve efficiency with contracting. However, we wonder whether writing a contract is valuable, that is, if contracting offers a better outcome than no-contracting. A simple comparison of the no-contracting FOCs (5 -7) with the contracting FOCs (8 -10) shows that there is no obvious result regarding the improving effect of contracting.

The bargaining position (µ) plays an important role in deriving more precise results about the comparison between contracting and no-contracting outcomes. Before proceeding to the formal comparison, we consider some critical values of the parameter µ and two useful lemmas. Then we work out the comparison.

Let first define the set A:

A = {k ∈ [0, 1]/kv 3 (q, j b , j s ) + (1 -k)c 4 (q, a, j b , j s ) ≤ 0, ∀q, a, j b , j s } .

Let µ = sup A. This number exists; if k = 0, the above inequality, used to define sup A, reduces to c 4 (q, a, j b , j s ) ≤ 0, which is satisfied ∀q, a, j b , j s . The following useful lemma can now be stated.

Lemma 1 If µ < µ, then: µv 3 (q, j b , j s ) + (1 -µ)c 4 (q, a, j b , j s ) ≤ 0, ∀q, a, j b , j s .

Proof See appendix (B).

We now define one other critical value of µ. Consider the set C: C = {k ∈ [0, 1]/kv 2 (q, j b , j s ) + (1 -k)c 3 (q, a, j b , j s ) ≥ 0, ∀q, a, j b , j s } .

Let define µ = inf C. This number again exists; if k = 1, the inequality, used to define C, becomes v 3 (q, j b , j s ) ≥ 0, which is satisfied ∀q, j b , j s . A second useful lemma can now be stated.

Lemma 2 If µ > µ, then: µv 2 (q, j b , j s ) + (1 -µ)c 3 (q, a, j b , j s ) ≥ 0, ∀q, a, j b , j s .

Proof The proof is the same as the one of Lemma 1.

Using Lemmas 1 and 2, we now determine the value of the simple fixed-price contract in comparison with the no-contracting outcome:

Proposition 2 Suppose that assumptions (1) to (6) hold. If µ < µ < µ, then the no-contracting outcome generates a general under-investment in comparison with the contracting outcome. It follows that a < a,  b <  b and  s <  s .

Proof See appendix (C).

The results of propositions ( 1) and ( 2) state that no-contracting leads to under-investments compared to both the first-best and the contracting solutions. However, these propositions do not discriminate between contracting and nocontracting outcomes in terms of welfare. Contracting would be welfare improving, if we could prove that (suboptimal) contracting investments are lower than the first-best ones, given that over-investment is also a suboptimal solution. The comparison of the first-best and the contracting outcomes depends on the value of q fixed in the contract. We have the following proposition.

Proposition 3 Suppose that assumptions (1) to (6) hold. If q is such that q < q * ( a,  b ,  s ), then the contracting outcome generates a general under-investment in comparison with the first-best. It follows that a < a * ,  b <  * b and  s <  * s .

Proof See appendix (D).

The contracting investment levels ( a,  b ,  s ) depend on the value of q (see equations 8-10). It turns out that if q is such that q < q * ( a,  b ,  s ), then investment levels in the contracting case are lower than in the first-best outcome.

The above proposition can be usefully supplemented by a corollary stating the existence of a critical value for q.

Corollary 1 For a given µ, suppose there exists a value q such that the resulting contracting investment levels satisfy : q = q * ( a, j s , j b ). The contracting investment levels are then independent of µ.

Proof See appendix (E).

The contracting investment levels are function of µ and q. Thus, for each value of µ, it should exist a value of q compatible with q = q * ( a, j s , j b ). The above corollary states that the value of q ensuring that q = q * ( a, j s , j b ) is in fact independent of µ.

Numerical investigations

The two preceding propositions suggest that parameters µ and q play a crucial role in the comparison of the different outcomes (in terms of investments, produced quantities and total surplus). Unfortunately, it is not possible first to provide precise analytical results enabling the discrimination between contracting and no-contracting outcomes, and second, to perform surplus comparisons. We conduct instead numerical investigations to provide insight into the influence of parameters µ and q on the model outcomes.

The buyer's value and the supplier's cost functions are defined as follows:

v(q,  b ,  s ) = B 0 q b 0 ( β 2 b  β 2 s ) θ b 1 , c(q, a,  b ,  s ) = = A 0 q a 0 α 0 a -1-σ σ + α 1  -1-σ σ b + α 2  -1-σ σ s -ν σ 1-σ -a 1 , with B 0 > 0, b 0 , b 1 > 0, b 0 + b 1 < 1, β 1 , β 2 > 0, β 1 + β 2 = 1, 0 < θ < 1, A 0 > 0, a 0 > 1,a 1 > 0, a 0 -a 1 > 1, σ > 0, 0 < ν < 1, α 0 , α 1 , α 2 > 0 and α 0 + α 1 + α 2 = 1.
Numerical investigations are made using the following benchmark calibration: Let's first study the impact of a variation of the supplier's bargaining strength µ on the investment levels, the produced quantities and the total surplus, under three cases: first-best, no-contracting and contracting. Several values of the fixed quantity q are considered for the contracting case.11

Figure (1) provides a representation in terms of produced quantities and total surplus. Regarding produced quantities (q), the numerical investigations suggest that q is sensitive in the contracting case to the value of the fixed quantity q. More precisely, q increases as q grows. Results also show that the contracting produced quantities are always above the no-contracting ones and may be above the first best ones for high values of q.12 We identify a critical value of q, satisfying q = q * ( a, j b , j s ), which is roughly 0.24. If q is less than this value, the contracting produced quantity is increasing in µ. The converse applies if q is greater than the critical value. Finally, if q is equal to the critical value, the contacting produced quantities (and investment levels) are higher than the nocontracting values, lower than the first-best, and independent of the supplier's bargaining strength (consistently with corollary 1).

Concerning total surplus, our simulations in Figure ( 1) show that the nocontracting case surplus is significantly smaller than the first-best. However, Contracting (q = 0.12) Contracting (q = 0.24) Contracting (q = 0.36) Contracting (q = 0.48)

First best

No contracting

Contracting (q = 0.12) Contracting (q = 0.24) Contracting (q = 0.36) Contracting (q = 0.48) contracting allows significant improvement. The total surplus increases as q grows. Moreover, our numerical investigations suggest there exists values of µ and q that maximize the contracting surplus. Solving a second-best problem, 13 we determine the values of µ * and q * maximizing the contracting total surplus. Results reported in Table 3 suggest that contracting may significantly improve the allocation. If µ and q are appropriately set, the contracting surplus is very close to the first-best (it is less by 1.44%).

Table 3: Contracting results (q * = 0.4571 and µ * = 0.6144) q a  b  s S 0.3206 0.5436 1.1694 0.5259 2.0042 A principal function of the fixed-price contract is to fix an initial quantity of input (q ∈ ℜ + ). Then, for a given µ, how can we determine the value of q maximizing the contracting total surplus? To begin with, suppose that investments are such that v 3 (q, j b , j s ) = c 3 (q, a, j b , j s ) = 0. In that case, it can easily be checked that a contract setting q = q(a * , j * b , j * s ) implements the first-best, which is consistent with [START_REF] Edlin | Specific investment under negotiated transfer pricing: an efficiency result[END_REF]. However, this is no longer true with joint investments and only a second-best can be achieved. Figure (2) depicts the second-best quantities and surplus. 14 If the second best value of q is chosen, the surplus loss with respect to the first best values roughly varies from 1.4% to 4.2%. In contrast, in the no-contracting case, the minimum loss surplus is about 30%. Thus, implementing the second-best significantly improves efficiency.

We also provide results when q is set at specific values of interest: the firstbest (q * = 0.325) and the critical value (q = 0.24). The total surplus of the contracting outcome is again significantly higher than in the no-contracting case. In particular, choosing q = q * induces a surplus loss varying from 2.3% to 4.3%. In other words, choosing the first best value q = q * , in a setting with joint investments, significantly improves efficiency (compared to no-contracting). 15 In our framework, optimality is not attainable for two reasons. First, parties are linked by a relationship specific-investment. The good is not traded on a competitive market and partners have no outside options. So, inefficient investment 13 We determine the values of µ and q maximizing the total surplus subject to the set of constraints constituted by equations ( 8)-(10).

14 For a given value of µ, we determine the values of q maximizing the total surplus subject to the set of constraints ( 8)-( 10).

15 Note that without joint investment optimality is restored when q = q * . Contracting (q = 0.24) Contracting (q = q * = 0.3252)

First best

Second best

First best

Contracting (q = 0.24) Contracting (q = q * = 0.3252) occurs due to a traditional hold-up problem. Each party under-invests because it receives only a fraction of the marginal return of its investment. The private marginal return of investment is thus lower than the social marginal return.

Second, there are joint investments (and thus investment externalities). The buyer's value v and the seller's cost c are both influenced by j b and j s . Thus, the investment of one party has a positive effect on the marginal value of the other party (supermodularity assumption). In the no-contracting case, supermodularity leads parties to under-invest and optimality can not be achieved. Are parties made better off by contracting and renegotiating? The contracting game is also altered due to joint investments. However, our numerical investigations suggest that renegotiation may improve the total surplus. Contracting (equations ( 8) -( 10) ) induces a modification of the marginal return of investment that comprises two terms. First, under contracting, an investment increases the status quo position of the investor. But, second, if this investment is joint, this also increases the status quo payoff of the trading partner, which weakens the investor's position. The net gain obtained by the investor is a weighted mean (depending of the bargaining power) of these two effects. An increase in the investor's bargaining power tends to reduce its net gain through a reduction of its status quo position and an increase in the trading partner's payoff. So, a higher bargaining power leads to a smaller gain from renegotiation. In other words, a party has little interest in renegotiating if it has a strong bargaining power.

What are the effects of the fixed quantity q on the renegotiation stage? We discuss now how changes in q provide incentives to improve the total surplus. Recall that if q = 0, the no-contacting solution is attained, while increasing q improves the status quo position of the investors.16 However, in case of joint investments, the trading partner payoff is also improved, which leads to a reduction of the bargaining position of the investor. It follows that increasing q has an ambiguous effect on the bargaining position of the investor (which is not the case with an autonomous investment).

To sum up, q and µ may have ambiguous effects that can not be evaluated analytically. Our numerical investigations however show that writing a contract may be valuable and that the values of q and µ matter. As a whole, starting from an under-investment situation (no-contracting or q = 0), increasing q allows to improve the total surplus because it increases the investors' bargaining position.

Conclusion

In this paper, we analyze two simple ways in which the input could be procured:

(1) a renegotiable fixed-price contract and (2) an ex post bargaining of the terms of trade without a prior contract. We found that arrangements fail both to achieve efficiency and to provide an incentive for optimal joint investments. A direct implication of this result is that a process of vertical integration, with a unified direction, provides optimal incentives to cooperate.

We also aimed to compare the contracting and the no-contracting solutions. We found that contracting induces larger autonomous and joint investments compared to not contracting. Moreover, our formal analysis suggests that the supplier's bargaining strength and the fixed quantity of input play a crucial role in the comparison of the different outcomes (in terms of investments, produced quantities and total surplus). Formally, contracting is welfare-improving for values of the fixed quantity lower than a given threshold. Numerically, we have shown that the contracting surplus is very close to the first-best one for appropriate values of the supplier's bargaining strength and the fixed quantity of input. Therefore, ex post bargaining is not always needed to promote cooperation. This is in line with Japanese procurement practices, where firms rely both on partial commitment to their relationships and ex post bargaining to promote cooperation Assumption 5 implies c 2 and c 4 are decreasing in q and v 2 is increasing in q. It is easily deduced that if q satisfies q < q * ( a,  b ,  s ), one has: Π 1 (a * ,  * b ,  * s ) -Π 1 ( a,  b ,  s ) = (1µ) [c 2 (q * ( a,  b ,  s ), a,  b ,  s )c 2 (q, a,  b ,  s )] < 0, Π 2 (a * ,  * b ,  * s ) -Π 2 ( a,  b ,  s ) = µ [v 2 (q,  b ,  s )v 2 (q * ( a,  b ,  s ),  b ,  s )] + µc 3 (q * ( a,  b ,  s ),  b ,  s ) + (1µ)c 3 (q, a,  b ,  s ) < 0, Π 3 (a * ,  * b ,  * s ) -Π 3 ( a,  b ,  s ) = (1µ) [c 4 (q * ( a,  b ,  s ), a,  b ,  s )c 4 (q, a,  b ,  s )] -(1µ)v 3 (q * ( a,  b ,  s ),  b ,  s )µv 3 (q,  b ,  s ) < 0,

The rest of the proof is similar to the one of the under-investment result in the no-contracting outcome (see proposition 1).

E Proof of corollary 1

Consider a given value of µ and suppose there exists q such that q = q * ( a, j b , j s ). The contracting investment levels depend on µ and q. Any change in µ may change the value of q compatible with the previous equality.

It is easy to show that the optimality conditions providing the contracting investment level take the following form : µv 2 (q, j b , j s ) + (1µ)c 3 (q, a, j b , j s ) +(1µ)v 2 (q * ( a, j b , j s ), j b , j s ) -(1µ)c 3 (q * ( a, j b , j s ), j b , j s ) -1 = 0 -(1µ)c 2 (q, a, j b , j s )µc 2 (q * ( a, j b , j s ), a, j b , j s ) -1 = 0 -µv 3 (q, j b , j s ) -(1µ)c 4 (q, a, j b , j s ) +µv 3 (q * ( a, j b , j s ), j b , j s )µc 4 (q * ( a, j b , j s ), j b , j s ) -1 = 0 Knowing that q = q * ( a, j b , j s ), the above system can be rewritten as follows :

v 2 (q * ( a, j b , j s ), j b , j s ) -1 = 0 -c 2 (q * ( a, j b , j s ), a, j b , j s ) -1 = 0 -c 4 (q * ( a, j b , j s ), a, j b , j s ) -1 = 0

The solution of the above system, if it exists, is independent of µ. We conclude that if a is chosen so that q = q * ( a, j b , j s ), the contracting investment level is independent of µ.

Figure

  Figure 1: Produced quantities and total surplus

Table 1 :

 1 Benchmark calibrations Buyer's value B 0 b 0 b 1 β 1 β 2 Under this parametrization, assumptions (1-6) are satisfied. 10 First best results reported in Table (2) are graphically represented in Figures (1) and (2).

	θ

See Schmitz (2001) for an excellent review of the hold-up problem and the incomplete contract literature.

The term "cross investments" might be a more suitable name in that case. The term "cooperative investments" was coined byChe and Hausch (1999), although their main idea (i.e., that contracts are useless with purely cooperative investments when renegotiation cannot be rule out) appeared earlier in the literature. It was actually derived first in the 1988 discussion paper of[START_REF] Maskin | Implementation and renegotiation[END_REF], but these authors did not introduce a specific name for such investments.MacLeod and Malcomson (1983) also studied such investments but they also did not come up with a name.

For instance, with the introduction of the airbag systems car manufacturers initiated cooperations with plastic subcontractors to redesign the dashboard and bear the additional weight of the airbag.

We demonstrate this result for both a renegotiable and non-renegotiable fixed-price contract. The latter implies that the default point value is zero. Since specific investments are sunk, the break-up of the contract would generate a loss of value and give optimal incentives to invest. As a result, it can be demonstrated that if parties credibly commit not to renegotiate their initial fixed-price contract, optimal cooperation can be reached, by implementing a game of messages which discloses the relevant information to a third party. See supplementary material on http://jose.desousa.univ.free.fr/research/sup.htm.

We are grateful to a referee for suggesting this interpretation.

In an incomplete contract framework, it does not seem reasonable to assume that bargaining positions may be endogenously determined ex ante and enforced.

Assumptions (5) and (6) about the supplier's cost function are satisfied if νa 1 -1-σ σ < 0.

The contracting case with q = 0 is similar to the no-contracting case.

This is the case when q is about 1.5 times the value of the quantity produced in the fist-best case (see table 2).

Recall that v 1i > 0 and c 1i < 0 for all i > 1.

A Proof of proposition 1

It is worth noting that assumption (5) implies that Π il > 0 for all i = l.

Consider now the following problem:

with λ ∈ [0, 1].

The first-order conditions are:

The maximization problem (11) has a unique solution, that is, a(λ), j b (λ) and j s (λ).

Note that a(1) = a * , j b (1) = j * b , j s (1) = j * s and a(0) = a, j b (0) =  b , j s (0) =  s . Define V (λ) = Π(a(λ), j b (λ), j s (λ))λ + 1λ µ a(λ)

We necessarily have W (λ) ≤ 0 and W (λ 0 ) = 0. Thus, the function W (λ) attains a maximum at λ = λ 0 . At this point, the first and second order optimality conditions are necessarily satisfied, we thus have W ′ (λ 0 ) = 0 and W ′′ (λ 0 ) < 0.

The first and second derivatives of W (λ) are:

The first-order condition gives:

The above expression holds for any λ 0 . The first derivative of V (λ) is then:

We deduce the expression of the second derivative of V (λ):

The second order condition W ′′ (λ 0 ) < 0 also holds for any λ 0 . We thus have W ′′ (λ) < 0 for all λ. Consequently:

Suppose for example that a ′ (λ) > 0 and differentiate the FOCs with respect to λ to obtain:

The cross derivatives Π ij being negative, we get:

It immediately follows that:

We necessarily have j ′ b (λ) > 0 or j ′ s (λ) > 0. Suppose for example that j ′ b (λ) > 0. The same argument shows that j ′ s (λ) > 0.

B Proof of lemma 1

Consider any values of q, a, j b , j s . Define g(µ) = µv 3 (q, j b , j s ) + (1µ)c 4 (q, a, j b , j s ).

We get g ′ (µ) = v 3 (q, j b , j s )c 4 (q, a, j b , j s ) > 0.

It follows that g(µ) < g(μ) ≤ 0, ∀µ ∈ [0, μ[.

C Proof of proposition 2

Let ( a,  b ,  s ) ∈ ℜ 3 + be the investment levels of the contracting outcome, solutions of the following first-order conditions:

+ are the investment levels of the no-contacting solution:

(

Using Lemmas 1 and 2, it can be easily shown that for all µ ∈ [µ, µ]

The rest of the proof is similar to the one of the under-investment result in the no-contracting outcome (see proposition 1).

D Proof of proposition 3

Let ( a,  b ,  s ) ∈ ℜ 3 + be the investment levels of the contracting outcome, solutions of the following first-order conditions:

The above conditions can be rewritten as follows:

(a * ,  * b ,  * s ) ∈ ℜ 3 + are the investment levels of the first-best solution: www.tepp.eu


Bruno Colbois
email: bruno.colbois@unine.ch

AND Ahmad El Soufi
email: elsoufi@univ-tours.fr

Alessandro Savo
email: savo@dmmm.uniroma1.it

EIGENVALUES OF THE LAPLACIAN ON A COMPACT MANIFOLD WITH DENSITY

Keywords: 2010 Mathematics Subject Classification. 58J50, 35P15, 47A75 Manifold with density, Weighted Laplacian, Witten Laplacian, Eigenvalue, Upper bound

In this paper, we study the spectrum of the weighted Laplacian (also called Bakry-Emery or Witten Laplacian) Lσ on a compact, connected, smooth Riemannian manifold (M, g) endowed with a measure σdvg. First, we obtain upper bounds for the k-th eigenvalue of Lσ which are consistent with the power of k in Weyl's formula. These bounds depend on integral norms of the density σ, and in the second part of the article, we give examples showing that this dependence is, in some sense, sharp. As a corollary, we get bounds for the eigenvalues of Laplace type operators, such as the Schrödinger operator or the Hodge Laplacian on p-forms. In the special case of the weighted Laplacian on the sphere, we get a sharp inequality for the first nonzero eigenvalue which extends Hersch's inequality.

Note that L σ is self-adjoint as an operator on L 2 (σdv g ) and is unitarily equivalent (through the transform

is nothing but the restriction to functions of the Witten Laplacian associated to f . That is why L σ itself is sometimes called Witten Laplacian.

Weighted manifolds arise naturally in several situations in the context of geometric analysis and their study has been very active in recent years. Their

Introduction

In this article, our main aim is to study the spectrum of the weighted Laplacian (also called Bakry-Emery Laplacian) L σ on a compact, connected, smooth Riemannian manifold (M, g) endowed with a measure σdv g , where σ = e -f ∈ C 2 (M ) is a positive density and dv g is the Riemannian measure induced by the metric g. Such a triple (M, g, σ) is known in literature as a weighted Riemannian manifold, a manifold with density, a smooth metric measure space or a Bakry-Emery manifold. Denoting by ∇ g and ∆ g the gradient and the Laplacian with respect to the metric g, the operator L σ is defined by

L σ = ∆ g - 1 σ ∇ g σ • ∇ g = ∆ g + ∇ g f • ∇ g
so that, for any function u ∈ C 2 (M ), satisfying Neumann boundary conditions if ∂M = ∅, Bakry-Emery curvature Ric σ = Ric g + Hessf plays a role which is similar in many respects to that played by the Ricci curvature for Riemannian manifolds, and appears as a centerpiece in the analysis of singularities of the Ricci flow in Perelman's work (see [START_REF] Morgan | Manifolds with density[END_REF][START_REF] Morgan | Manifolds with density and Perelman's proof of the Poincaré conjecture[END_REF]). The weighted Laplacian L σ appears naturally in the study of diffusion processes (see e.g., the pioneering work of Bakry and Emery [START_REF] Bakry | Diffusions hypercontractives[END_REF]). Eigenvalues of L σ are strongly related to asymptotic properties of mm-spaces, such as the study of Levy families (see [START_REF] Funano | Concentration, Ricci Curvature, and Eigenvalues of Laplacian[END_REF][START_REF] Gromov | A topological application of the isoperimetric inequality[END_REF][START_REF] Milman | On the role of convexity in isoperimetry, spectral gap and concentration[END_REF]). Without being exhaustive, we refer to the following articles and the references therein: [START_REF] Lott | Some geometric properties of the Bakry-Émery-Ricci tensor[END_REF][START_REF] Lott | Optimal transport and Ricci curvature for metric-measure spaces[END_REF][START_REF] Lott | Ricci curvature for metric-measure spaces via optimal transport[END_REF][START_REF] Munteanu | Smooth metric measure spaces with nonnegative curvature[END_REF][START_REF] Munteanu | Analysis of weighted Laplacian and applications to Ricci solitons[END_REF][START_REF] Munteanu | The curvature of gradient Ricci solitons[END_REF][START_REF] Wei | Comparison geometry for the Bakry-Emery Ricci tensor[END_REF] and, closely related to our topic, [START_REF] Andrews | Eigenvalue comparison on Bakry-Emery manifolds[END_REF][START_REF] Futaki | On the first eigenvalue of the Witten-Laplacian and the diameter of compact shrinking ricci solitons[END_REF][START_REF] Futaki | Lower diameter bounds for compact shrinking Ricci solitons[END_REF][START_REF] Hassannezhad | Eigenvalues of perturbed Laplace operators on compact manifolds[END_REF][START_REF] Lu | Eigenvalues of collapsing domains and drift Laplacians[END_REF][START_REF] Ma | Extension of Reilly formula with applications to eigenvalue estimates for drifting Laplacians[END_REF][START_REF] Setti | Eigenvalue estimates for the weighted Laplacian on a Riemannian manifold[END_REF][START_REF] Wu | Upper bounds on the first eigenvalue for a diffusion operator via Bakry-Émery Ricci curvature[END_REF][START_REF] Wu | Upper Bounds on the First Eigenvalue for a Diffusion Operator via Bakry-Émery Ricci Curvature II[END_REF] The spectrum of L σ , with Neumann boundary conditions if ∂M = ∅, consists of an unbounded sequence of eigenvalues

Spec(L σ ) = {0 = λ 1 (L σ ) < λ 2 (L σ ) ≤ λ 3 (L σ ) ≤ • • • ≤ λ k (L σ ) ≤ • • • }
which satisfies the Weyl's asymptotic formula

λ k (L σ ) ∼ 4π 2 ω -2 n n k V g (M ) 2 n , as k → ∞
where V g (M ) is the Riemannian volume of (M, g) and ω n is the volume of the unit ball in R n . The first aim of this paper is to obtain bounds for λ k (L σ ) which are consistent with the power of k in Weyl's formula. Before stating our results, let us recall some known facts about the eigenvalues (λ k (g)) k≥1 of the usual Laplacian ∆ g (case σ = 1). Firstly, the well-known Hersch's isoperimetric inequality (see [START_REF] Hersch | Quatre propriétés isopérimétriques de membranes sphériques homogènes[END_REF]) asserts that on the 2dimensional sphere S 2 , the first positive normalized eigenvalue λ 2 (g)V g (M ) is maximal when g is a "round" metric (see [START_REF] Soufi | A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle[END_REF][START_REF] Soufi | Sur la première valeur propre des tores[END_REF][START_REF] Jakobson | Extremal metric for the first eigenvalue on a Klein bottle[END_REF][START_REF] Li | A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces[END_REF][START_REF] Nadirashvili | Berger's isoperimetric problem and minimal immersions of surfaces[END_REF] for similar results on other surfaces). Korevaar [START_REF] Korevaar | Upper bounds for eigenvalues of conformal metrics[END_REF] proved that on any compact manifold of dimension n = 2, λ k (g)V g (M ) is bounded above independently of g. More precisely, if M is a compact orientable surface of genus γ, then

λ k (g)V g (M ) ≤ C(γ + 1)k ( 1
)
where C is an absolute constant (see [START_REF] Hassannezhad | Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem[END_REF] for an improved version of this inequality). On the other hand, on any compact manifold M of dimension n ≥ 3, the normalized first positive eigenvalue λ 2 (g)V g (M ) 2/n can be made arbitrarily large when g runs over the set of all Riemannian metrics on M (see [START_REF] Colbois | Riemannian metrics with large λ1[END_REF][START_REF] Lohkamp | Discontinuity of geometric expansions[END_REF]). However, the situation changes as soon as we restrict ourselves to a fixed conformal class of metrics. Indeed, on the sphere S n , round metrics maximize λ 2 (g)V g (S n ) 2/n among all metrics g which are conformally equivalent to the standard one (see [START_REF] Soufi | Immersions minimales, première valeur propre du laplacien et volume conforme[END_REF]Proposition 3.1]). Furthermore, Korevaar [START_REF] Korevaar | Upper bounds for eigenvalues of conformal metrics[END_REF] proved that for any compact Riemannian manifold (M, g) one has

λ k (g)V g (M ) 2/n ≤ C([g])k 2/n (2)
where C([g]) is a constant depending only on the conformal class [g] of the metric g. Korevaar's approach has been revisited and placed in the context of metric measure spaces by Grigor'yan and Yau [START_REF] Grigor ′ Yan | Decomposition of a metric space by capacitors[END_REF] and, then, by Grigor'yan, Netrusov and Yau [START_REF] Grigor | Eigenvalues of elliptic operators and geometric applications[END_REF].

The first observation we can make about possible extensions of these results to weighted Laplacians is that, given any compact Riemannian manifold (M, g), the eigenvalues λ k (L σ ) cannot be bounded above independently of σ. Indeed, from the semi-classical analysis of the Witten Laplacian (see [START_REF] Helffer | Puits multiples en mécanique semi-classique. IV. Étude du complexe de Witten[END_REF]), we can easily deduce that (Proposition 2.1) if f is any smooth Morse function on M with m 0 stable critical points, then the family of densities

σ ε = e -f /ε satisfies for k > m 0 , λ k (L σε ) -→ ε→0 +∞.
Therefore, any extension of the inequalities (1) and ( 2) to L σ must necessarily have a density dependence in the right-hand side. The following theorem gives such an extension in which the upper bound depends on the ratio between the L n n-2 -norm and the L 1 -norm of σ. In all the sequel, the L p norm of σ with respect to dv g will be denoted by σ g p .

Theorem 1.1. Let (M, g, σ) be a compact weighted Riemannian manifold. The eigenvalues of the operator L σ , with Neumann boundary conditions if ∂M = ∅, satisfy :

(I) If n ≥ 3, then, ∀k ≥ 1, λ k (L σ ) ≤ C([g]) σ g n n-2 σ g 1 k 2/n
where C([g]) is a constant depending only on the conformal class of g.

(II) if n = 2 and M is orientable of genus γ, then, ∀k ≥ 1, λ k (L σ ) ≤ C σ ∞ σ g 1 (γ + 1)
k
where C is an absolute constant.

It is clear that taking σ = 1 in Theorem 1.1, we recover the inequalities (1) and [START_REF] Bakry | Diffusions hypercontractives[END_REF]. Moreover, as we will see in the next section, if M is boundaryless and the conformal class [g] contains a metric g 0 with nonnegative Ricci curvature, then C([g]) ≤ C(n), where C(n) is a constant which depends only on the dimension.

Notice that there already exist upper bounds for the eigenvalues of L σ in the literature, but they usually depend on derivatives of σ, either directly or indirectly, through the Bakry-Emery curvature. The main feature of our result is that the upper bounds we obtain depend only on L p -norms of the density. The proof of Theorem 1.1 relies in an essential way on the technique developed by Grigor'yan, Netrusov and Yau [START_REF] Grigor | Eigenvalues of elliptic operators and geometric applications[END_REF]. Regarding Hersch's isoperimetric type inequalities, they extend to our context as follows (see Corollary 3.2): Given any metric g on S n which is conformally equivalent to the standard metric g 0 , and any positive density σ ∈ C 2 (S n ), one has

λ 2 (L σ ) ≤ n|S n | n 2 σ g n n-2 σ g 1
where |S n | is the volume of the standard n-sphere and with the convention that σ g n n-2 = σ ∞ when n = 2. Moreover, the equality holds in the inequality if and only if σ is constant and g is a round metric.

Next, let us consider a Riemannian vector bundle E over a Riemannian manifold (M, g) and a Laplace type operator

H = D * D + T
acting on smooth sections of the bundle. Here D is a connection on E which is compatible with the Riemannian metric and T is a symmetric bundle endomorphism (see e.g., [START_REF] Bérard | From vanishing theorems to estimating theorems: the Bochner technique revisited[END_REF]Section E]). The operator H is self-adjoint and elliptic and we will list its eigenvalues as:

λ 1 (H) ≤ λ 2 (H) ≤ • • • ≤ λ k (H) ≤ • • • .
Important examples of such operators are given by Schrödinger operators acting on functions (here T is just the potential), the Hodge Laplacian acting on differential forms (in which case T is the curvature term in Bochner's formula), and the square of the Dirac operator (T being in this case a multiple of the scalar curvature). Another important example is the Witten Laplacian acting on differential forms, whose restriction to functions is precisely given by a weighted Laplacian, the main object of study of this paper.

In Section 4 we will prove (Theorem 4.1) an upper bound for the gap between the k-th eigenvalue and the first eigenvalue of H involving integral norms of a first eigensection ψ. For example, if n ≥ 3, then

λ k (H) -λ 1 (H) ≤ C([g])   ψ g 2n n-2 ψ g 2   2 k 2/n , (3)
where C[g] is a constant depending only on the conformal class of g. The reason why we bound the gap instead of λ k (H) itself is due to the fact that, even when σ = 1 and H is the standard Hodge Laplacian acting on p-forms, the first positive eigenvalue is not bounded on any conformal class of metrics (see [START_REF] Colbois | Eigenvalues of the Laplacian acting on p-forms and metric conformal deformations[END_REF]). For estimates on the gap when a finite group of isometries is acting, we refer to [START_REF] Colbois | Involutive isometries, eigenvalue bounds and a spectral property of Clifford tori[END_REF]. Inequality (3) should be regarded as an extension of Theorem 1.1. Indeed, if

H σ = √ σL σ 1
√ σ is the Schrödinger operator which is unitarily equivalent to the operator L σ , then λ 1 (H σ ) = λ 1 (L σ ) = 0 and any first eigenfunction of H σ is a scalar multiple of √ σ. Thus, taking ψ = √ σ in (3) we recover the first estimate in Theorem 1.1. Our main aim in section 5 is to discuss the accuracy of the upper bounds given in Theorem 1.1 regarding the way they depend on the density σ. That is why we exhibit an explicit family of compact manifolds (M, g), each endowed with a sequence of densities {σ j }, and give a sharp lower estimate of the first positive eigenvalue λ 2 (L σ j ) in terms of j. This enables us to see that both λ 2 (L σ j ) and the ratio σ j g n n-2 / σ j g 1 tend to infinity linearly with respect to j. Thus, we have

A σ j g n n-2 σ j g 1 ≤ λ 2 (L σ j ) ≤ B σ j g n n-2 σ j g 1
with λ 2 (L σ j ) -→ +∞ as j → +∞, and A and B are two positive constants which do not depend on j.

The examples of densities we give are modeled on Gaussian densities (i.e. σ j (x) = e -j|x| 2 ) on R n . For example, if Ω is a bounded convex domain in R n , we observe that λ 2 (L σ j ) ≥ 2j for all j. We then extend the lower bound to manifolds of revolution (at least asymptotically as j → ∞). However, in the case of a closed manifold of revolution, there is an additional difficulty coming from the fact that we need to extend smoothly this kind of density to the whole manifold in such a way as to preserve the estimates on both the eigenvalues and the L p -norms.

2. Upper bounds for weighted eigenvalues in a smooth metric measure space and proof of Theorem 1.1

Let (M, g) be a compact connected Riemannian manifold, possibly with a non-empty boundary. Let σ ∈ L ∞ (M ) be a bounded nonnegative function on M and let ν be a non-atomic Radon measure on M with 0 < ν(M ) < ∞.

To such a pair (σ, ν), we associate the sequence of non-negative numbers {µ k (σ, ν)} k∈N given by

µ k (σ, ν) = inf E∈S k sup u∈E\{0} R σ,ν (u)
where S k is the set of all k-dimensional vector subspaces of H 1 (M ) and

R σ,ν (u) = M |∇ g u| 2 g σdv g M u 2 dν .
In the case where σ is of class C 2 and ν = σdv g , the variational characterization of eigenvalues of the weighted Laplacian L σ = ∆ g -1 σ ∇ g σ • ∇ g gives (see e.g. [START_REF] Grigor | Heat kernels on weighted manifolds and applications[END_REF])

λ k (L σ ) = µ k (σ, σdv g ).
(4) Theorem 1.1 is a direct consequence of the following Theorem 2.1. Let (M, g) be a compact Riemannian manifold possibly with nonempty boundary. Let σ ∈ L ∞ (M ) be a nonnegative function and let ν be a non-atomic Radon measure with 0 < ν(M ) < ∞. (I) If n ≥ 3, then for every k ≥ 1, we have

µ k (σ, ν) ≤ C([g]) σ g n n-2 ν(M ) k 2/n ,
where C([g]) is a constant depending only on the conformal class of g. Moreover, if M is closed and [g] contains a metric with nonpositive Ricci curvature, then

C([g]) ≤ C(n) where C(n) is a constant depending only on n. (II) If M is a compact orientable surface of genus γ, then for every k ≥ 1, we have µ k (σ, ν) ≤ C σ ∞ ν(M ) (γ + 1) k.
where C is an absolute constant.

The proof of this theorem is based on the method described by Grigor'yan, Netrusov and Yau in [START_REF] Grigor | Eigenvalues of elliptic operators and geometric applications[END_REF] and follows the same lines as the proof they have given in the case σ = 1. The main step consists in the construction of a family of disjointly supported functions with controlled Rayleigh quotient.

Let us fix a reference metric g 0 ∈ [g] and denote by d 0 the distance associated to g 0 . An annulus A ⊂ M is a subset of M of the form {x ∈ M : r < d 0 (x, a) < R} where a ∈ M and 0 ≤ r < R (if necessary, we will denote it A(a, r, R)). The annulus 2A is by definition the annulus {x ∈ M : r/2 < d 0 (x, a) < 2R}.

To such an annulus we associate the function u A supported in 2A and such that

u A (x) =    1 -2 r d 0 (x, A) if r 2 ≤ d 0 (x, a) ≤ r 1 if x ∈ A 1 -1 R d 0 (x, A) if R ≤ d 0 (x, a) ≤ 2R
We introduce the following constant:

Γ(g 0 ) = sup x∈M,r>0 V g 0 (B(x, r)) r n
where B(x, r) stands for the ball of radius r centered at x in (M, d 0 ). Notice that since M is compact, the constant Γ(g 0 ) is finite and depends only on g 0 . This constant can be bounded from above in terms of a lower bound of the Ricci curvature Ric g 0 and an upper bound of the diameter diam(M, g 0 ) (Bishop-Gromov inequality). In particular, if the Ricci curvature of g 0 is nonnegative, then Γ(g 0 ) is bounded above by a constant depending only on the dimension n.

Lemma 2.1. For every annulus A ⊂ (M, d 0 ) one has

M |∇ g u A | 2 σdv g ≤ 8 Γ(g 0 ) 2 n 2A σ n n-2 dv g 1-2 n .
Proof. Let A = A(a, r, R) be an annulus of (M, d 0 ). Since u A is supported in 2A we get, using Hölder inequality,

M |∇ g u A | 2 σdv g = 2A |∇ g u A | 2 σdv g ≤ 2A |∇ g u A | n dv g 2 n 2A σ n n-2 dv g 1-2 n .
From the conformal invariance of 2A |∇ g u A | n dv g we have

2A |∇ g u A | n dv g = 2A |∇ g 0 u A | n dv g 0 with |∇ g 0 u A | a.e. =    2 r if r 2 ≤ d 0 (x, a) ≤ r 0 if r ≤ d 0 (x, a) ≤ R 1 R if R ≤ d 0 (x, a) ≤ 2R. Hence, 2A |∇ g 0 u A | n dv g 0 ≤ 2 r n V g 0 (B(a, r)) + 1 R n V g 0 (B(a, 2R)) ≤ 2 n+1 Γ(g 0 )
where the last inequality follows from the definition of Γ(g 0 ). Putting together all the previous inequalities, we obtain the result of the Lemma.

Proof of part (I) of Theorem 2.1: Let us introduce the constant N (M, d 0 ), that we call the covering constant, defined to be the infimum of the set of all integers N such that, for all r > 0, any ball of radius 2r in (M, d 0 ) can be covered by N balls of radius r. Again, the compactness of M ensures that N (M, d 0 ) is finite, and Bishop-Gromov inequality allows us to bound it from above in terms of the dimension when the Ricci curvature of (M, g 0 ) is nonnegative.

Since the metric measure space (M, d 0 , ν) has a finite covering constant and a non atomic measure, one can apply Theorem 1.1 of [START_REF] Grigor | Eigenvalues of elliptic operators and geometric applications[END_REF] and conclude that there exists a constant c(N ) depending only on N (M, d 0 ) such that for each positive integer k, there exists a family of 2k annuli

A 1 , • • • , A 2k on M such that the annuli 2A 1 , • • • , 2A 2k are mutually disjoint and, ∀i ≤ 2k, ν(A i ) ≥ c(N ) ν(M ) k . (5)
Since the annuli 2A i are mutually disjoint, one has

i≤2k 2A i σ n n-2 dv g ≤ M σ n n-2 dv g . Thus, at most k annuli among 2A 1 , • • • , 2A 2k satisfy 2A i σ n n-2 dv g > 1 k M σ
n n-2 dv g . Therefore, we can assume without loss of generality that the

k annuli A 1 , • • • , A k satisfy 2A i σ n n-2 dv g 1-2 n ≤ 1 k 1-2 n σ g n n-2 .

The corresponding functions u

A 1 • • • , u A k are such that (Lemma 2.1) R σ,ν (u A i ) = M |∇ g u A i | 2 g σdv g M u 2 A i dν ≤ 1 ν(A i ) 8Γ(g 0 ) 2 n 2A i σ n n-2 dv g 1-2 n ≤ 1 ν(A i ) 8Γ(g 0 ) 2 n σ g n n-2 k 1-2 n .
Using inequality (5), we get

R σ,ν (u A i ) ≤ 8 Γ(g 0 ) 2 n c(N (M, d 0 )) σ g n n-2 ν(M ) k 2 n .
Since the functions u A 1 , • • • , u A k are disjointly supported, they form a kdimensional subspace on which the Rayleigh quotient is bounded above by the right hand side of the last inequality. We set C([g]) = 8 Γ(g 0 )

2 n c(N (M,d 0 )) and conclude using the min-max formula.

As we mentioned above, if M is closed and the Ricci curvature of g 0 is non negative, then the constants Γ(g 0 ), N (M, d 0 ) and, hence, C([g]) are bounded in terms of the dimension n.

Proof of part (II) of Theorem 2.1. Assume now that (M, g) is a compact orientable surface of genus γ, possibly with boundary, and let ρ ∈ C ∞ (M ) be a positive function on M . If M has nonempty boundary, then we glue a disk on each boundary component of M and extend ν by 0. This closed surface admits a conformal branched cover ψ over S 2 with degree deg(ψ) ≤ ⌊ γ+3 2 ⌋We endow S 2 with the usual spherical distance d 0 and the pushforward measure µ = ψ * (ν), We apply Theorem 1.1 of [START_REF] Grigor | Eigenvalues of elliptic operators and geometric applications[END_REF] to the metric measure space (S 2 , d 0 , µ) and deduce that there exist an absolute constant c = c N (S 2 , d 0 ) and k annuli A 1 , . . . , A k ⊂ S 2 such that the annuli 2A 1 , . . . , 2A k are mutually disjoint and, ∀i ≤ k,

µ(A i ) ≥ c µ(S 2 ) k . (6)
We set for each i ≤ k, v i = u A i • ψ. From the conformal invariance of the energy and Lemma 2.1, one deduces that, for every i ≤ k,

M |∇ g v i | 2 dv g = deg(ψ) S 2 |∇ g 0 u A i | 2 dv g 0 ≤ 8Γ(S 2 , g 0 )⌊ γ + 3 2 ⌋, while, since u A i is equal to 1 on A i , M v 2 i dν ≥ ν ψ -1 (A i ) = µ(A i ) ≥ c µ(S 2 ) k = c ν(M ) k .
Therefore,

R σ,ν (v i ) ≤ 8Γ(S 2 , g 0 ) cν(M ) ⌊ γ + 3 2 ⌋ k ≤ C(γ + 1) ν(M ) k
where C is an absolute constant. Noting that the k functions v 1 , . . . v k are disjointly supported in M , we deduce the desired inequality for µ k (σ, ν).

We end this section with the following observation showing that the presence of the density in the RHS of the inequalities of Theorem 1.1 is essential. This question will also be discussed in Section 5.

Proposition 2.1. Let (M, g) be a compact Riemannian manifold and let f be a smooth Morse function on M . For every ε > 0 we set σ ε = e -f /ε . If m 0 denotes the number of stable critical points of f , then there exists ε 0 > 0 such that, ∀ε ∈ (0, ε 0 ),

λ m 0 +1 (L σε ) ≥ 1 √ ε .
Proof. First observe that for any density σ = e -f , the operator L σ is unitarily equivalent to the Schrödinger operator

H σ = ∆ g - ∆ g √ σ √ σ = ∆ g + 1 4 |∇ g f | 2 + 1 2 ∆ g f acting on L 2 (dv g ) (indeed, L σ = 1 √ σ H σ √ σ, where multiplication by √ σ is a unitary transform from L 2 (σdv g ) to L 2 (dv g )).
Consequently, denoting by (λ k (ε)) k≥1 the eigenvalues of the semiclassical Schrödinger operator

ε 2 ∆ g + 1 4 |∇ g f | 2 + ε 2 ∆ g f , we get λ k (L σε ) = 1 ε 2 λ k (ε).
According to [START_REF] Helffer | Puits multiples en mécanique semi-classique. IV. Étude du complexe de Witten[END_REF] (see also [23, proposition 2.2]), there exists ε 0 > 0 such that, ∀ε < ε 0 , the number of eigenvalues λ k (ε) contained in the interval [0, ε 3/2 ) is exactly m 0 , that is, λ m 0 (ε) < ε 3/2 and λ m 0 +1 (ε) ≥ ε 3/2 . Therefore,

λ m 0 +1 (L σε ) = 1 ε 2 λ m 0 +1 (ε) ≥ 1 √ ε .

Sharp estimates for the first positive eigenvalue

Let (M, g) be a compact connected Riemannian n-dimensional manifold, possibly with a non-empty boundary. Li and Yau introduced in [START_REF] Li | A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces[END_REF] the notion of conformal volume as follows : Given any immersion φ from M to the standard sphere (S p , g S p ) of dimension p, we denote by V (φ) the volume of M with respect to the metric φ * g S p , and by V c (φ) the supremum of V (γ • φ) as γ runs over the group of conformal diffeomorphisms of (S p , g S p ). The conformal volume of (M, [g]) is

V c (M, [g]) = inf p>n inf {V c (φ) : φ ∈ conf ((M, g), S p )}
where conf ((M, g), S p ) is the set of all conformal immersions from (M, g) to S p .

With the same notations as in the previous section, we have the following Theorem 3.1. Let (M, g) be a compact Riemannian manifold possibly with nonempty boundary. Let σ ∈ L ∞ (M ) be a nonnegative function and let ν be a non-atomic Radon measure on M with 0 < ν(M ) < ∞. One has

µ 2 (σ, ν) ≤ nV c (M, [g]) 2/n σ g n n-2 ν(M ) , (7)
with the convention that

σ n n-2 = σ ∞ if n = 2. Proof. From the definition of µ 2 (σ, ν), it is clear that if u ∈ C 1 (M )
is any nonzero function such that M u dν = 0, then, taking for E the 2-dimensional vector space generated by constant functions and u, we have

µ 2 (σ, ν) ≤ sup w∈E R σ,ν (w) ≤ R σ,ν (u).
Let φ ∈ conf ((M, g), S p ). Using standard center of mass lemma (see e.g., [15, Proposition 4.1.5]), there exists a conformal diffeomorphism γ of S p so that the Euclidean components of the map φ = γ • φ satisfy

M φj dν = 0, j ≤ p + 1.
Thus, for every j ≤ p + 1,

µ 2 (σ, ν) M φ2 j dν ≤ M |∇ g φj | 2 σdv g . (8)
From the fact that φ is conformal one has φ * g

S p = 1 n j≤p+1 |∇ g φj | 2 g and V ( φ) = M   1 n j≤p+1 |∇ g φj | 2   n/2 dv g .
We sum up in [START_REF] Soufi | Immersions minimales, première valeur propre du laplacien et volume conforme[END_REF] and use Hölder's inequality to get

µ 2 (σ, ν) ν(M ) ≤ M j≤p+1 |∇ g φj | 2 σdv g ≤ nV ( φ) 2/n σ g n n-2 ≤ nV c (φ) 2/n σ g n n-2 .
The proof of the theorem follows immediately.

An immediate consequence of Theorem 3.1 is the following Corollary 3.1. Let (M, g) be a compact Riemannian manifold possibly with nonempty boundary and let σ ∈ C 2 (M ) be a positive function. One has

λ 2 (L σ ) ≤ nV c (M, [g]) 2/n σ g n n-2 σ g 1 . (9)
In [START_REF] Soufi | Immersions minimales, première valeur propre du laplacien et volume conforme[END_REF], Ilias and the second author proved that if the Riemannian manifold (M, g) admits an isometric immersion into a Euclidean space R p whose Euclidean components are first non-constant eigenfunctions of the Laplacian ∆ g , then the following equality holds

λ 2 (∆ g )V g (M ) 2/n = nV c (M, [g]) 2/n . ( 10
)
In this case, the inequality (9) reads

λ 2 (L σ ) ≤ λ 2 (∆ g )V g (M ) 2/n σ g n n-2 σ g 1 (11)
where the equality holds whenever σ is a constant function. This proves the sharpness of the inequality of Corollary 3.1.

Notice that all compact rank one symmetric spaces satisfy [START_REF] Soufi | Sur la première valeur propre des tores[END_REF] and hence [START_REF] Funano | Concentration, Ricci Curvature, and Eigenvalues of Laplacian[END_REF]. In particular, we have the following result that extends Hersch's isoperimetric inequality [START_REF] Hersch | Quatre propriétés isopérimétriques de membranes sphériques homogènes[END_REF] and its generalization to higher dimensions [START_REF] Soufi | Immersions minimales, première valeur propre du laplacien et volume conforme[END_REF].

Corollary 3.2. Let g be a Riemannian metric on S n which is conformal to the standard metric g S n , and let σ ∈ C 2 (S n ) be a positive function. One has

λ 2 (L σ ) ≤ n|S n | 2 n σ g n n-2 σ g 1 (12)
where |S n | is the volume of the Euclidean unit sphere. Moreover, the equality holds in [START_REF] Futaki | On the first eigenvalue of the Witten-Laplacian and the diameter of compact shrinking ricci solitons[END_REF] if and only if σ is constant and g is homothetically equivalent to g S n .

Proof. Let γ = (γ 1 , ..., γ n+1 ) : S n → S n be conformal transformation of S n such that, for every i ≤ n + 1, we have

S n γ i σ dv g = 0.
Using the same arguments as in the proof of Theorem 3.1, we get

λ 2 (L σ ) S n γ 2 i σ dv g ≤ S n |∇ g γ i | 2 σ dv g , i ≤ n + 1, λ 2 (L σ ) σ g 1 ≤ S n i≤n+1 |∇ g γ i | 2 σ dv g ≤ n    S n   1 n i≤n+1 |∇ g γ i | 2   n/2 dv g    2/n σ g n n-2
with (since γ is conformal from (S n , g) to (S n , g S n ) )

S n   1 n i≤n+1 |∇ g γ i | 2   n/2 dv g = V γ * g S n (S n ) = |S n |.
The inequality [START_REF] Futaki | On the first eigenvalue of the Witten-Laplacian and the diameter of compact shrinking ricci solitons[END_REF] follows immediately. Assume that the equality holds in [START_REF] Futaki | On the first eigenvalue of the Witten-Laplacian and the diameter of compact shrinking ricci solitons[END_REF]. This implies that

• L σ γ i = λ 2 (L σ )γ i , i ≤ n + 1, and • the function σ is constant if n = 2 and, if n ≥ 3, the functions n+1 i=1 |∇ g γ i | 2 n/2
and σ n n-2 are proportional.

An elementary computation gives

0 = i≤n+1 L σ γ 2 i = 2 i≤n+1 γ i L σ γ i -|∇ g γ i | 2 = 2   λ 2 (L σ ) - i≤n+1 |∇ g γ i | 2   .
From the previous facts we see that σ is constant in all cases and, since

γ * g S n = 1 n j≤n+1 |∇ g γ j | 2 g = λ 2 (Lσ)
n g, the metric g is homothetically equivalent to g S n .

Eigenvalues of Laplace type operators

In this section we show that Theorem 1.1 extends to a much more general framework to give upper bounds of the eigenvalues of certain operators acting on sections of vector bundles, precisely the Laplace-type operators defined in the introduction. Throughout this section, (M, g) denotes a compact Riemannian manifold without boundary. We will use the notations introduced in Section 2 and refer to [START_REF] Bérard | From vanishing theorems to estimating theorems: the Bochner technique revisited[END_REF][START_REF] Peter | Spectral geometry, Riemannian submersions, and the Gromov-Lawson conjecture[END_REF] for details on Laplace type operators.

Theorem 4.1. Let H = D ⋆ D + T be an operator of Laplace type acting on sections of a Riemannian vector bundle E over (M, g), and let ψ be an eigensection associated to λ 1 (H). One has for all k ≥ 1

λ k (H) -λ 1 (H) ≤ µ k (σ, ν) with σ = |ψ| 2 and ν = |ψ| 2 dv g . Thus, a) If n ≥ 3 then λ k (H) -λ 1 (H) ≤ C([g])   ψ g 2n n-2 ψ g 2   2 k 2/n .
where C([g]) is a constant depending only on the conformal class of g. b) If M is a compact, orientable surface of genus γ then :

λ k (H) -λ 1 (H) ≤ C ψ ∞ ψ g 2 2 (γ + 1)k
where C is an absolute constant.

Proof. As H = D ⋆ D + T , the quadratic form associated to H is given by:

Q(ψ) = M Hψ, ψ dv g = M |Dψ| 2 + T ψ, ψ dv g ,
where ψ denotes a generic smooth section. If u is any Lipschitz function on M , then an integration by parts gives (see [START_REF] Colbois | Large eigenvalues and concentration[END_REF]Lemma 8])

Q(uψ) = M u 2 Hψ, ψ + |∇u| 2 |ψ| 2 dv g .
Now assume that ψ is a first eigensection: Hψ = λ 1 (H)ψ. Then we obtain:

Q(uψ) M u 2 |ψ| 2 dv g = λ 1 (H) + M |∇u| 2 |ψ| 2 dv g M u 2
|ψ| 2 dv g for all Lipschitz functions u. Let σ = |ψ| 2 and ν = |ψ| 2 dv g . Restricting the test-sections to sections of type uψ, where u is Lipschitz (hence in 1 (M )) and ψ is a fixed first eigensection, an obvious application of the min-max principle gives:

λ k (H) ≤ λ 1 (H) + µ k (σ, ν).
The remaining part of the theorem is an immediate consequence of the last inequality and Theorem 1.1.

Corollary 4.1. Assume that a Laplace type operator H acting on sections of a Riemannian vector bundle E over (M, g), admits a first eigensection of constant length. Then, for all k ≥ 1

λ k (H) -λ 1 (H) ≤ λ k (∆ g ).
Indeed, when σ is constant, µ k (σ, σdv g ) is nothing but the k-th eigenvalue of the Laplacian ∆ g acting on functions. In the particular case where H (p) is the Hodge Laplacian acting on p-forms, Corollary 4.1 says that the existence of a nonzero harmonic p-form of constant length on M leads to

λ k (H (p) ) ≤ λ k (∆ g )
for every positive integer k, which extends the result of Takahashi [START_REF] Takahashi | The gap of the eigenvalues for p-forms and harmonic p-forms of constant length[END_REF].

Lower bounds for eigenvalues on weighted Euclidean domains and manifolds of revolution

The scope of this section is to show that our main upper bound is asymptotically sharp for some special weighted manifolds, namely convex Euclidean domains and revolution manifolds endowed with a Gaussian density. By a Gaussian density we mean a function of type σ j (x) = e -jd(x,x 0 ) 2 where x 0 is a fixed point and j is a positive integer (a slight modification is needed for closed revolution manifolds). We will give a lower bound of λ 2 (L σ j ) and verify that in all these cases both λ 2 (L σ j ) and our main upper bound grow to infinity linearly in j as j → ∞.

In what follows, we make use of the Reilly formula, recently extended to weighted manifolds (see [START_REF] Ma | Extension of Reilly formula with applications to eigenvalue estimates for drifting Laplacians[END_REF]). Here M is a compact Riemannian manifold of dimension n with smooth boundary ∂M ; we let σ = e -f be a positive smooth density and u ∈ C ∞ (M ). Then we have:

M (L σ u) 2 σ = M |∇ 2 u| 2 σ + Ric(∇u, ∇u)σ + ∇ 2 f (∇u, ∇u)σ + ∂M 2 ∂u ∂N L ∂M σ u • σ + B(∇ ∂M u, ∇ ∂M u)σ + ∂M (n -1)H + ∂f ∂N ∂u ∂N 2 σ ( 13
)
where N is the inner unit normal, B the second fundamental form of ∂M with respect to N and (n -1)H = trB is the mean curvature. By L ∂M σ we denote the induced operator on ∂M , naturally defined as

L ∂M σ u = ∆ ∂M u + ∇ ∂M f , ∇ ∂M u .
When not explicitly indicated, integration is taken with respect to the canonical Riemannian measure.

Convex Euclidean domains.

Here is the main statement of this section.

Theorem 5.1. Let Ω be a convex domain of R n containing the origin, endowed with the Gaussian density σ j (x) = e -j|x| 2 . There exists a constant K n,R depending only on n and R = dist(O, ∂Ω) such that, for all j ≥ K n,R ,

σ j n n-2 σ j 1 ≤ λ 2 (L σ j ) ≤ B n σ j n n-2 σ j 1 ( 14
)
where B n is a constant that depends only on n. Moreover, λ 2 (L σ j ) tends to infinity with a linear growth as j → ∞.

The proof follows from Theorem 1.1 and the following estimates.

Proposition 5.1. Let Ω be a convex domain in R n . Then, for all j > 0,

λ 2 (L σ j ) ≥ 2j.
Proof. We apply the Reilly formula [START_REF] Futaki | Lower diameter bounds for compact shrinking Ricci solitons[END_REF] to an eigenfunction u associated

to λ = λ 2 (L σ j ). By definition, f = j|x| 2 so that ∇ 2 f = 2jI. As ∂u ∂N = 0
(Neumann condition) and B ≥ 0 by the convexity assumption, we arrive at

λ 2 Ω u 2 σ ≥ 2j Ω |∇u| 2 σ = 2jλ Ω u 2 σ,
and the inequality follows. . In particular

σ j n n-2 ≤ C n j n-2 2 .
b) There exists a constant K n,R depending only on n and R such that, if

j ≥ K n,R , then σ j 1 ≥ C n 2j n 2
.

The constants C n , C n,p will be given explicitely in the proof. From the above facts we obtain for j ≥ K n,R ,

σ j n n-2 σ j 1 ≤ 2j ≤ λ 2 (L σ j ).
Proof of Lemma 5.1. We use the well-known formula

∞ 0 r n-1 e -jr 2 dr = Γ( n 2 ) 2j n 2
, where Γ is the Gamma function. Then

R n e -j|x| 2 dx = |S n-1 | ∞ 0 r n-1 e -jr 2 dr = C n j n 2
.

with

C n = 1 2 |S n-1 |Γ( n 2
). This implies that

σ j p = Ω e -jp|x| 2 dx 1 p ≤ R n e -jp|x| 2 dx 1 p = C n,p j n 2p with C n,p = (C n ) 1 p p n 2p
, which proves a). Now, one has

σ j 1 = Ω e -j|x| 2 dx = R n e -j|x| 2 dx - Ω c e -j|x| 2 dx = C n j n 2 - Ω c
e -j|x| 2 dx.

On the complement of Ω one has |x| ≥ R by the definition of R. Then,

Ω c e -j|x| 2 dx = Ω c e -j 2 |x| 2 e -j 2 |x| 2 dx ≤ e -jR 2 2 R n e -|x| 2 2 dx = D n e -jR 2 2 .
As R is fixed and j n 2 e -jR 2 2 tends to zero when j tends to infinity, one sees immediately that there exists K n,R such that, for j ≥ K n,R ,

Ω c e -j|x| 2 dx ≤ D n e -jR 2 2 ≤ C n 2j n 2
.

In that range of j one indeed has

σ j 1 ≥ C n 2j n 2
.

Revolution manifolds with boundary.

A revolution manifold with boundary of dimension n is a Riemannian manifold (M, g) with a distinguished point N such that (M \{N }, g) is isometric to (0, R]×S n-1 endowed with the metric g = dr 2 + θ 2 (r)g S n-1 , g S n-1 denoting the standard metric on the sphere. Here θ(r) is a smooth function on [0, R] which is positive on (0, R] (note that we assume in particular θ(R) > 0) and is such that:

θ(0) = θ ′′ (0) = 0, θ ′ (0) = 1.
We notice that, as R < +∞, M is compact, connected and has a smooth boundary ∂M = {R} × S n-1 isometric to the (n -1)-dimensional sphere of radius θ(R). If we make the stronger assumption that θ has vanishing even derivatives at zero then the metric is C ∞ -smooth everywhere. Given a Gaussian radial density of the form σ j (r) = e -jr 2 centered at the pole N , we wish to study the first non-zero eigenvalue λ 2 (L σ j ) of L σ j , with Neumann boundary conditions.

Theorem 5.2. Let (M, g) be a revolution manifold with boundary endowed with the Gaussian density σ j (r) = e -jr 2 , j ≥ 1. Then there is a (possibly negative) constant C, not depending on j, such that

λ 2 (L σ j ) ≥ 2j + C. Moreover, if (M, g) has non-negative Ricci curvature, then λ 2 (L σ j ) ≥ 2j.
We start the proof by making general considerations which are valid for any radial density σ(r) = e -f (r) (Lemma 5.2 and 5.3). As for the usual Laplacian, we can separate variables and prove that there is an orthonormal basis of L 2 (σ) made of eigenfunctions of type

u(r, x) = φ(r)ξ(x) (15)
where φ is a smooth function of r ∈ [0, R] and ξ(x) is an eigenfunction of the Laplacian on S n-1 . Listing the eigenvalues of S n-1 as {µ k }, where k = 1, 2, . . . , and computing the Laplacian, one arrives at

∆u(r, x) = -φ ′′ (r) -(n -1) θ ′ (r) θ(r) φ ′ (r) + µ k θ(r) 2 φ(r) ξ(x). As f = f (r) is radial one has ∇f , ∇u (r, x) = f ′ (r)φ ′ (r)ξ(x) so that L σ u(r, x) = -φ ′′ (r)-(n-1) θ ′ (r) θ(r) φ ′ (r)+f ′ (r)φ ′ (r)+ µ k θ(r) 2 φ(r) ξ(x) (16)
Let us now focus on the first positive eigenvalue λ 2 (L σ ) with associated eigenfunction u of the form [START_REF] Girouard | Maximization of the second positive Neumann eigenvalue for planar domains[END_REF]. There are only two cases to examine:

• either µ k = µ 1 = 0, so that ξ(x) is constant and the eigenfunction u is radial,

• or µ k = µ 2 = n -1, the first positive eigenvalue of S n-1 . In fact, higher eigenvalues of S n-1 do not occur, otherwise u would have too many nodal domains. We summarize this alternative in the following lemma.

Lemma 5.2. Let M = (0, R] × S n-1 be a manifold of revolution as above, endowed with a radial density σ(r) = e -f (r) . Then, either L σ admits a (Neumann) radial eigenfunction associated to λ 2 (L σ ), or λ 2 (L σ ) is the first positive eigenvalue of the problem

   φ ′′ + (n -1) θ ′ θ -f ′ φ ′ + λ - n -1 θ 2 φ = 0 φ(0) = φ ′ (R) = 0
Note that the condition φ ′ (R) = 0 follows from the Neumann boundary condition, while the condition φ(0) = 0 is imposed to insure that the eigenfunction is continuous at the pole N .

We then prove the following lower bound for the "radial spectrum".

Lemma 5.3. In the hypothesis of Lemma 5.2, assume that L σ admits a a (Neumann) radial eigenfunction associated to the eigenvalue λ. Then

λ ≥ inf (0,R) (n -1) θ ′ θ 2 + Ric 0 + f ′′ ,
where Ric 0 is a lower bound of the Ricci curvature of M .

Proof. Let u = u(r) be a radial eigenfunction associated to λ. We apply the Reilly formula [START_REF] Futaki | Lower diameter bounds for compact shrinking Ricci solitons[END_REF] to obtain:

λ M u 2 σ = M |∇ 2 u| 2 σ + Ric(∇u, ∇u)σ + ∇ 2 f (∇u, ∇u)σ.
In fact, the boundary terms vanish because on ∂M one has ∂u ∂N = 0 and, as u is radial (hence constant on ∂M ), one also has ∇ ∂M u = 0. We now wish to bound from below the terms involving the hessians. For that we need to use a suitable orthonormal frame. So, fix a point p = (r, x) and consider a local frame (ē 1 , . . . , ēn-1 ) around x which is orthonormal for the canonical metric of S n-1 . We can assume that this frame is geodesic at x. Taking e i = 1 θ ēi it is clear that (e 1 , . . . , e n-1 , ∂ ∂r ) is a local orthonormal frame on (M, g). If ∇ denotes (as usual) the Levi-Civita connection of (M, g) one sees easily that at p,

∇ e i e j = -δ ij θ ′ θ ∂ ∂r , ∇ e i ∂ ∂r = θ ′ θ e i , ∇ ∂ ∂r e i = ∇ ∂ ∂r ∂ ∂r = 0.
We are now ready to prove Theorem 5.2. Let σ j = e -jr 2 so that f ′′ (r) = 2j. If λ 2 (L σ j ) is associated to a radial eigenfunction, Lemma 5.3 gives

λ 2 (L σ j ) ≥ 2j + C 1 ,
with C 1 independent of j. It is also clear that if M has non-negative Ricci curvature then λ 2 (L σ j ) ≥ 2j.

Taking into account Lemma 5.2, Theorem 5.2 will now follow from Lemma 5.4. Let λ 2 be the first positive eigenvalue of the problem

   φ ′′ + (n -1) θ ′ θ -2jr φ ′ + λ - n -1 θ 2 φ = 0 φ(0) = φ ′ (R) = 0 Then λ 2 ≥ 2j + C 2 where C 2 = (n -1) inf r -θ ′ θ rθ 2 : r ∈ (0, R) . Moreover, if M has non- negative Ricci curvature, then C 2 ≥ 0.
We observe that C 2 is always finite because since r → rθ ′ θ rθ 2 approaches -2 3 θ ′′′ (0) as r → 0, and then is bounded on [0, R]. Proof. Set φ(r) = ry(r) so that φ ′ = y + ry ′ and φ ′′ = 2y ′ + ry ′′ . Substituting in the equation we obtain

y ′′ + 2 r + (n -1) θ ′ θ -2jr y ′ + λ -2j + (n -1) θ ′ θ -r rθ 2 y = 0. If β = r 2 θ n-1 e -jr
(βy ′ ) ′ + λ -2j + (n -1) θ ′ θ -r rθ 2 βy = 0.
Multiplying by y and integrating on (0, R) we end-up with

(λ -2j) R 0 βy 2 dr = - R 0 (βy ′ ) ′ y dr + R 0 (n -1) r -θ ′ θ rθ 2 βy 2 dr.
We observe that β(0) = 0 and, as φ ′ (R) = 0, we have y(R) = -Ry ′ (R).

Then, integrating by parts:

- R 0 (βy ′ ) ′ y dr = -β(R)y ′ (R)y(R) + R 0 βy ′2 dr = β(R)y ′ (R) 2 R + R 0 βy ′2 dr ≥ 0.
From the definition of the constant C 2 we obtain

(λ -2j) R 0 βy 2 dr ≥ C 2 R 0 βy 2 dr
which gives the assertion.

Finally, it is well-known that Ric( ∂ ∂r , ∂ ∂r ) = -(n -1)

θ ′′ θ . If M has non-
negative Ricci curvature then θ ′′ ≤ 0; in turn we have θ ′ ≤ 1 and θ ≤ r, which implies that rθ ′ θ ≥ 0 and, then, C 2 ≥ 0.

5.3. Closed revolution manifolds. A closed revolution manifold of dimension n is a compact manifold (M, g) without boundary, having two distinguished points N, S such that (M \{N, S}, g) is isometric to (0, R)× S n-1 endowed with the metric dr 2 + θ 2 (r)g S n , where θ : [0, R] → R is smooth and

θ(0) = θ(R) = 0, θ ′ (0) = -θ ′ (R) = 1, θ ′′ (0) = θ ′′ (R) = 0.
Under these assumptions, the metric g is C 2 . To have a C ∞ metric it is enough to assume that θ (2i) (0) = θ (2i) (R) = 0 for all i = 0, 1, 2, . . . .

Our aim is to construct a sequence of radial densities {σ j } on M such that λ 2 (L σ j ) grows linearly with j. These densities will be Gaussian functions centered at the pole N of M , suitably smoothened near the pole S so that the resulting function is globally C 1 (this is enough for our purpose). Thus, let us define

σ j (r) = e -f j (r) (17)
where f j (r) = jh 2 j (r) and

h j (r) =      r if 0 ≤ r ≤ r j . = R - 1 αj r - αj 2 (r -r j ) 2 if r j ≤ r ≤ R (18)
Here α ≥ 1 is a fixed number, which is large enough so that

(n -1)α 2 16 -2αR ≥ 2. The function h j is of class C 1 on [0, R]. Since σ ′ j (0) = σ ′ j (R) = 0, we see that σ j is a C 1 function on M .
Theorem 5.3. Let (M, g) be a closed revolution manifold endowed with the density σ j defined in [START_REF] Grigor | Heat kernels on weighted manifolds and applications[END_REF] and [START_REF] Grigor | Eigenvalues of elliptic operators and geometric applications[END_REF]. Then there exist an integer j 0 and a constant C, not depending on j, such that, for all j ≥ j 0 , one has:

λ 2 (L σ j ) ≥ 2j + C.
We give the proof of Theorem 5.3 in the next subsection. We just want to mention here the analogue of Theorem 5.1.

Theorem 5.4. Let (M, g) be a revolution manifold with boundary (resp. a closed revolution manifold) endowed with the density σ j as in Theorem 5.2 (resp. Theorem 5.3). Then, for j sufficiently large,

A σ j n n-2 σ j 1 ≤ λ 2 (L σ j ) ≤ B σ j n n-2 σ j 1
where A and B are positive constants depending on M , but not on j. Moreover, λ 2 (L σ j ) tends to infinity with a linear growth as j → ∞.

The proof of Theorem 5.4 is quite similar to that of the corresponding statement in Theorem 5.1, and we only sketch it. Assume that M has a boundary and n ≥ 3. We start from:

σ j p p = M σ p j (r)dv g = |S n-1 | R 0 θ n-1 (r)σ p j (r) dr.
Now |θ(r) -r| ≤ cr 3 and θ n-1 (r) ∼ r n-1 as r → 0; since, for p = n n-2 and j large: R 0 r n-1 e -jpr 2 dr

1 p R 0 r n-1 e -jr 2 dr ≤ C ′ j, we get in turn σ j n n-2 σ j 1 ≤ C ′′ j.
The assertion follows from the last inequality and the estimate of λ 2 (L σ j ) we have obtained in Theorem 5.2. If M is closed the only change is in the definition of σ j . However, this change occurs far from the pole and thus contributes with exponentially decreasing terms, which, modulo a change in the constants, do not show up in the final estimate. We omit further details.

5.4. Proof of Theorem 5.3. We start the proof by observing, as in the previous section, that there is an eigenfunction associated to λ 2 (L σ j ) of type u(r, x) = φ(r)ξ(x) with φ satisying a suitable Sturm-Liouville problem on the interval [0, R] and ξ being an eigenfunction of the Laplacian on S n-1 . For a closed manifold, Lemma 5.2 takes the following form.

Lemma 5.5. Let (M, g) be a closed revolution manifold endowed with a radial density σ(r) = e -f (r) . Then, either L σ admits a radial eigenfunction associated to λ 2 (L σ ), or λ 2 (L σ ) is the first positive eigenvalue of the following Sturm-Liouville problem on [0, R]:

   φ ′′ + (n -1) θ ′ θ -f ′ φ ′ + λ - n -1 θ 2 φ = 0 φ(0) = φ(R) = 0
Note the boundary conditions on φ, which are imposed so that the corresponding eigenfunction u is continuous at both poles N and S.

By definition, Rr j = 1 αj and, as j → ∞ we have r j → R. Since θ ′ is continuous and θ ′ (R) = -1, there exists an integer j 1 such that, for all j ≥ j 1 , -2 ≤ θ ′ (r) ≤ -1 2 for all r ∈ [r j , R]. Consequently, on that interval we also have

1 2 (R -r) ≤ θ(r) ≤ 2(R -r) and | θ ′ (r) θ(r) | ≥ 1 4(R -r) . (19)
According to Lemma 5.5 there are two cases to discuss.

First case: there is a radial eigenfunction associated to λ = λ 2 (L σ j ). Then, we apply Lemma 5.3 (which holds without change) and for the lower bound it suffices to verify the inequality:

(n -1) θ ′ (r) θ(r)

2 + f ′′ j (r) ≥ 2j (20)
for all r ∈ (0, R). Indeed, one has f ′′ j = 2j(h ′2 j + h j h ′′ j ). Thus, on the interval (0, r j ) one gets f ′′ j = 2j and ( 20) is immediate. On the interval (r j , R) one has h j ≤ r ≤ R and h ′′ j = -αj. Then:

f ′′ j ≥ 2jh j h ′′ j ≥ -2αj 2 R.
On the other hand, as Rr ≤ Rr j = 1 αj , we see from [START_REF] Gromov | A topological application of the isoperimetric inequality[END_REF]:

θ ′ (r) θ(r) 2 ≥ 1 16(R -r) 2 ≥ α 2 j 2 16
.

Recalling the definition of α we see that, for j ≥ j 1 :

(n -1) θ ′ (r) θ(r) 2 + f ′′ j (r) ≥ (n -1)

α 2 16 -2αR j 2 ≥ 2j 2 ≥ 2j,
and the assertion follows.

Second case: λ 2 (L σ j ) is the first positive eigenvalue of the problem:

   φ ′′ + (n -1) θ ′ θ -f ′ j φ ′ + λ - n -1 θ 2 φ = 0 φ(0) = φ(R) = 0
First observe that there exists R ∈ (0, R) such that φ ′ ( R) = 0 (note that R depends on j). It follows that u is a Neumann eigenfunction associated to λ 2 (L σ j ) for both of the following domains: Ω 1 = {(r, x) ∈ M : r ≤ R}, Ω 2 = {(r, x) ∈ M : r ≥ R}.

We first assume that R < r j and focus our attention on Ω 1 . As f ′ j = 2jr on [0, R], we see that λ 2 (L σ j ) is bounded below by the first positive eigenvalue of the problem

   φ ′′ + (n -1) θ ′ θ -2jr φ ′ + λ - n -1 θ 2 φ = 0 φ(0) = φ ′ ( R) = 0.
By Lemma 5.4 we have λ 2 (L σ j ) ≥ 2j + C2 , where C2 = (n -1) inf 0>r> R rθ ′ θ rθ 2 ≥ (n -1) inf 0>r>R rθ ′ θ rθ 2 , a constant which does not depend on j, and we are done.

Finally, it remains to examine the case where R ≥ r j . In this case, we view u as a Neumann eigenfunction on the domain Ω 2 . Let us briefly sketch the argument. As j → ∞, Ω 2 is quasi-isometric to a Euclidean ball of small radius (of the order of 1/j) and has a density with uniformly controlled variation. Therefore, its first positive eigenvalue must be large (of the order of j 2 ). Let us clarify the details. We know that λ 2 (L σ j ) = Ω 2 |∇u| 2 σ j dv g Ω 2 u 2 σ j dv g .

Since h j (r) is increasing in r ∈ (r j , R), one has r j = h j (r j ) ≤ h j (r) ≤ h j (R) ≤ R. Thus, for all r ∈ ( R, R), one has (recall that r j = R -1 αj )

e -jR 2 ≤ σ j (r) ≤ e -j(R-1 αj ) 2

.

Consequently,

λ 2 (L σ j ) ≥ Ω 2 |∇u| 2 dv g Ω 2 u 2 dv g e -2R α . (21)
As u(r, x) = φ(r)ξ(x) and S n-1 ξ = 0, we see that Ω 2 udv g = 0. Hence, by the min-max principle:

Ω 2 |∇u| 2 dv g Ω 2 u 2 dv g ≥ µ 2 (Ω 2 , g) (22)
where µ 2 (Ω 2 , g) denotes the first positive Neumann eigenvalue of the Laplacian on the domain (Ω 2 , g). On the other hand, the first inequality in [START_REF] Gromov | A topological application of the isoperimetric inequality[END_REF] shows that the metric g is quasi-isometric, on Ω 2 , to the standard Euclidean metric g euc = dr 2 + r 2 g S n-1 , with quasi-isometry ratio bounded by 4. Thus, (Ω 2 , g) is quasi-isometric to the Euclidean ball (Ω 2 , g euc ) of radius R -R. Therefore:

µ 2 (Ω 2 , g) ≥ 4 -(n+2) µ 2 (Ω 2 , g euc ) ≥ 4 -(n+2) µ(n + 1) (R -R) 2 (23)
where µ(n+1) is the first positive Neumann eigenvalue of the unit Euclidean ball. Since R -R ≤ Rr j = 1 αj , we conclude from ( 21), ( 22), ( 23) that:

λ 2 (L σ j ) ≥ C 3 j 2
where C 3 is a constant depending only on n, α and R. By taking j larger than a suitable integer j 2 we see that λ 2 (L σ j ) ≥ 2j as asserted. The conclusion is that, if j ≥ j 0 . = max{j 1 , j 2 }, then the inequality of Theorem 5.3 is verified. This ends the proof.

Lemma 5 . 1 .

 51 Let j ∈ N * . a) For all p > 1 one has σ j p ≤ C n,p j n 2p