Extremal domains for the first eigenvalue in a general Riemannian manifold

We prove the existence of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in any compact Riemannian manifold. This result generalizes a results of F. Pacard and the second author where the existence of a nondegenerate critical point of the scalar curvature of the Riemannian manifold was required.

in Ω u > 0 in Ω u = 0 on ∂Ω g(∇u, ν) = constant on ∂Ω can be solved for some positive constant λ, where ν denotes the outward unit normal vector about ∂Ω for the metric g.
In R n the only extremal domains are balls. This is a consequence of a very well known result of J. Serrin: if there exists a solution u to the overdetermined elliptic problem for a given bounded domain Ω ⊂ R n and a given Lipschitz function f , where ν denotes the outward unit normal vector about ∂Ω and ·, · the scalar product in R n , then Ω must be a ball, [19]. In the Euclidean space, round balls are in fact not only extremal domains, but also minimizers for the first eigenvalue of the Laplacian with 0 Dirichlet boundary condition in the class of domains with the same volume. This follows from the Faber-Krähn inequality, where B n (Ω) is a ball of R n with the same volume as Ω, because equality holds in (3) if and only if Ω = B n (Ω), see [3] and [9].
Nevertheless, very few results are known about extremal domains in a Riemannian manifold. The result of J. Serrin, based on the moving plane argument introduced by A. D. Alexandrof in [1], uses strongly the symmetry of the Euclidean space, and naturally it fails in other geometries. The classification of extremal domains is then achieved in the Euclidean space, but it is completely open in a general Riemannian manifold.
For small volumes, a method to build new examples of extremal domains in some Riemannian manifolds has been developed in [12] by F. Pacard and P. Sicbaldi. They proved that when the Riemannian manifold has a nondegenerate critical point of the scalar curvature, then it is possible to build extremal domains of any given volume small enough, and such domains are close to geodesic balls centered at the nondegenerate critical point of the scalar curvature. The method fails if the Riemannian method does not have a nondegenerate critical point of the scalar curvature.
In this paper we improve the result of F. Pacard and P. Sicbaldi by eliminating the hypothesis of the existence of a nondegenerate critical point for the scalar curvature. In particular, we are able to build extremal domains of small volume in every compact Riemannian manifold.
For ǫ > 0, we denote by B g ǫ (p) ⊂ M the geodesic ball of center p ∈ M and radius ǫ. We denote by B ǫ ⊂ R n the Euclidean ball of radius ǫ centered at the origin. The main result of the paper is the following: Theorem 1.1. Let M be a Riemannian manifold of dimension n ≥ 2. There exist ǫ 0 > 0 and a smooth function Φ : M × (0, ǫ 0 ) −→ R such that: (1) For all ǫ ∈ (0, ǫ 0 ), if p is a critical point of the function Φ(·, ǫ) then there exists an extremal domain Ω ǫ ⊂ M whose volume is equal to the Euclidean volume of B ǫ . Moreover, there exists c > 0 and, for all ǫ ∈ (0, ǫ 0 ), the boundary of Ω ǫ is a normal graph over ∂B g ǫ (p) for some function v(p, ǫ) with v(p, ǫ) C 2,α (∂B g ǫ (p)) ≤ c ǫ 3 . (2) There exists a function r defined on M that can be written as where Riem, Ric, R denote respectively the Riemann curvature tensor, the Ricci curvature tensor and the scalar curvature of (M, g), and K 1 , K 2 , K 3 and K 4 are constants depending only on n, such that for all k ≥ 0 Φ(p, ǫ) − R p − ǫ 2 r p C k (M ) ≤ c k ǫ 3 for some constant c k > 0 which does not depend on ǫ ∈ (0, ǫ 0 ) (the subscript p means that we evaluate the function at p).
(3) The following expansion holds: where λ 1 is the first Dirichlet eigenvalue of the unit Euclidean ball.
The explicit computation of the constants K i is given in section 7. We remark that if M is compact, then there exists always a critical point of Φ(·, ǫ), and then we have small extremal domains obtained as perturbation of small geodesic balls in every compact Riemannian manifold without boundary.
It is clear that this theorem generalizes the result of [12] because the construction of extremal domains does not require the existence of a nondegenerate critical point of the scalar curvature. In fact, if the scalar curvature function R has a nondegenerate critical point p 0 , then for all ǫ small enough there exists a critical point p = p(ǫ) of Φ(·, ǫ) such that dist(p, p 0 ) ≤ c ǫ 2 . and then the geodesic ball B g ǫ (p) can be perturbed in order to obtain an extremal domain. We recover in this case the result of [12], but with a better estimation of the distance of p to p 0 (in [12] the distance between p and p 0 is bounded by c ǫ). In particular, we have the p-independent expansion The result of [12] can not be applied to some natural metrics such that a Einstein one, i.e when Ric = k g for some constant k, or simply a constant scalar curvature one. In the case where R is a constant function, one gets the existence of extremal domains close to any nondegenerate critical point of the function r. In the particular case where the metric g is Einstein we obtain extremal domains close to any nondegenerate critical point of the function (we will see that In order to put the result in perspective let us digress slightly. The solutions of the isoperimetric problem I κ := min Ω⊂M : Volg Ω=κ Vol g in ∂Ω are (where they are smooth enough) constant mean curvature hypersurfaces (here g in denotes the induced metric on the boundary of Ω). In fact, constant mean curvature are the critical points of the area functional Ω → Vol g in ∂Ω under a volume constraint Vol g Ω = κ. Now, it is well known (see [3], [9] and [10]) that the determination of the isoperimetric profile I κ is related to the Faber-Krähn profile, where one looks for the least value of the first eigenvalue of the Laplace-Beltrami operator amongst domains with prescribed volume The result of F. Pacard and P. Sicbaldi [12] had been inspired by some parallel results on the existence of constant mean curvature hypersurfaces in a Riemannian manifold M. In fact, R. Ye built in [22] constant mean curvature topological spheres which are close to geodesic spheres of small radius centered at a nondegenerate critical point of the scalar curvature, and the result of F. Pacard and P. Sicbaldi can be considered the parallel of the result of R. Ye in the context of extremal domains. The method used in [12] is based on the study of the operator that to a domain associates the Neumann value of its first eigenfunction, which is a nonlocal first order elliptic operator. This represents a big difference with respect to the result of R. Ye, where the operator to study was a local second order elliptic operator.
In a recent paper, [13], F. Pacard and X. Xu generalize the result of R. Ye by eliminating the hypothesis of the existence of a nondegenerate critical point of the scalar curvature function. For every ǫ small enough, they are able to build a small topological sphere of constant mean curvature equal to n−1 ǫ by perturbing a small geodesic ball centered at a critical point of a certain function defined on M which is close to the scalar curvature function. For this, they use the variational characterization of constant H 0 mean curvature hypersurfaces as critical points of the functional in the class of topological sphere, where D S is the domain enclosed by S, see [13].
Our construction is based on some ideas of [13]. For this, we use the variational characterization of extremal domains. The main difference and difficulties with respect to the result of F. Pacard and X. Xu arise in the fact that there does not exist an explicit formulation to compute the first eigenvalue of a domain while there exists an explicit formulation to compute the volume of a surface.
Our result shows once more the similarity between constant mean curvature hypersurfaces and extremal domains. The deep link between such two objects has been underlined also in [16] and [17].
It is important to remark that P. Sicbaldi was able to build extremal domains of big volume in some compact Riemannian manifold without boundary by perturbing the complement of a small geodesic ball centered at a nondegenerate critical point of the scalar curvature function, see [20]. As in the case of small volume domains, the existence of a nondegenerate critical point of the scalar curvature function is mandatory (and such result requires also that the dimension of the manifold is at least 4). It would be interesting to adapt our result in order to build extremal domains of big volume in any compact Riemannian manifold without boundary by perturbing the complement of small geodesic balls of radius ǫ centered at a critical point of the function Φ(·, ǫ) or some other similar function. This result would allow for example to obtain extremal domains Ω ǫ that are given by the complement of a small topological ball in a flat 2-dimensional torus, and by the characterization of extremal domains this would lead to a nontrivial solution of (2), with f (t) = λ t, in the universal coveringΩ ǫ of Ω ǫ , which is a nontrivial unbounded domain of R 2 . Up to our knowledge the existence of this unbounded domain is not known. Remark thatΩ ǫ is a double periodic domain, made by the complement of a infinitely countable union of topological balls. The existence ofΩ ǫ would establish once more the strong link between extremal domains and constant mean curvature surfaces, via the double periodic constant mean curvature surfaces (see [6], [15] and [14]).
Acknowledgement. Both authors are grateful to Philippe Delanoë for his pleasant "Séminaire commun d'analyse géométrique" that took place at CIRM (Marseille) in september 2012, where they met and started the collaboration. This work was done from september 2012 to january 2013, when the first author was member of the Laboratoire d'Analyse Topologie et Probabilité of the Aix-Marseille University as "chercheur CNRS en délégation", and he his grateful to the member of such research laboratory for their warm hospitality. The first author is partially supported by the ANR-10-BLAN 0105 ACG and the ANR SIMI-1-003-01.
In this case we say that Ξ is the vector field that generates the deformation. The deformation is said to be volume preserving if the volume of Ω t does not depend on t.
is a deformation of Ω 0 , and λ Ωt and u t are respectively the first eigenvalue and the first eigenfunction (normalized to be positive and have L 2 (Ω t ) norm equal to 1) of −∆ g on Ω t with 0 Dirichlet boundary condition, both applications t −→ λ Ωt and t −→ u t inherit the regularity of the deformation of Ω 0 . These facts are standard and follow at once from the implicit function theorem together with the fact that the least eigenvalue of the Laplace-Beltrami operator with 0 Dirichlet boundary condition is simple.
A domain Ω 0 is an extremal domain (for the first eigenvalue of −∆ g with 0 Dirichlet boundary condition) if for any volume preserving deformation Assume that {Ω t } t is a perturbation of a domain Ω 0 generated by the vector field Ξ. The outward unit normal vector field to ∂Ω t is denoted by ν t . We have the following result, whose proof can be found in [2] or in [12]: The derivative of the first eigenvalue with respect to the deformation of the domain is given by This result allows to characterize extremal domains as the domains where there exists a positive solution to the overdetermined elliptic problem for a positive constant λ, where ν is the outward unit normal vector about ∂Ω. The proof of this fact follows directly from Proposition 2.1, but can be found also in [12].
Given a point p ∈ M we denote by E 1 , . . . , E n an orthonormal basis of the tangent plane T p M. Geodesic normal coordinates x := (x 1 , . . . , x n ) ∈ R n at p are defined by where Exp g p is the exponential map at p for the metric g. It will be convenient to identify R n with T p M and S n−1 with the unit sphere in T p M. If It corresponds to the vector of T p M whose geodesic normal coodinates are x. Given a continuous function f : S n−1 −→ (0, +∞) whose L ∞ -norm is small (say less than the cut locus of p) we define For notational convenience, given a continuous function f : S n−1 → (0, ∞), we set When we do not indicate the metric as a superscript, we understand that we are using the Euclidean one. Similarly, we denote by Vol g the volume in the metric g, by dvol g the volume element in the metric g to integrate over a domain, by dvol g in the volume element in the induced metric g in to integrate over the boundary of a domain. When we do not indicate anything we understand that we are considering the Euclidean volume, or the Euclidean measure, or the measure induced by the Euclidean one on boundaries.
Our aim is to show that, for all ǫ > 0 small enough, we can find a point p ∈ M and a function v : S n−1 −→ R such that where ω n is the Euclidean volume of the unit sphere S n−1 ) and the overdetermined problem g(∇φ, ν) = constant on ∂B g ǫ(1+v) (p) has a non trivial positive solution for some positive constant λ, where ν is the unit normal vector field about ∂B g ǫ(1+v) (p). Clearly, this problem does not make sense when ǫ = 0. In order to bypass this problem, we observe that, considering the dilated metricḡ := ǫ −2 g, the above problem is equivalent to finding a point p ∈ M and a function v : and for which the overdetermined problem has a non trivial positive solution for some positive constantλ, whereν is the unit normal vector field about ∂Bḡ 1+v (p). Taking in account that the functions φ andφ have L 2 -norm equal to 1, we have that the relation between the solutions of the two problems is simply given by

Some expansions in normal geodesic coordinates
We precise that through this paper we consider the following definition of the Riemann curvature tensor: where ∇ denotes the Levi-Civita connection on the manifold M.
Geodesic normal coordinates are very useful because there exists a well known formula for the expansion of the coefficients of a metric near the center of such coordinates, see [21], [11] or [18]. At the point of coordinate x, the following expansion holds 1 : and the subscript p means that we evaluate the quantity at p. In (8) the Einstein notation is used (i.e., we do a summation on every index appearing up and down). Such notation will be always used through this paper.
This expansion allows to obtain other expansions, as those of the volume of a geodesic ball, or the first eigenvalue and the first eigenfunction on a geodesic ball. In order to recall such expansions, let us introduce some notations. Let us denote by λ 1 the first eigenvalue of the Laplacian in the unit ball B 1 with 0 Dirichlet boundary condition. We denote by φ 1 the associated eigenfunction normalized to be positive and have L 2 (B 1 ) norm equal to 1. It is clear that φ 1 is a radial function φ 1 (x) = φ 1 (|x|). We denote r = |x|.
We recall now some expansions we will need later, whose proofs can be deduced from (8). We remind to [13] and [8] for the proofs. For the volume of a geodesic ball of radius ǫ we have: For the first eigenvalue of the Laplace-Beltrami operator with 0 Dirichlet boundary condition on a geodesic ball of radius ǫ we have: and the constant c 2 is given by For the associate eigenfunction φ in the geodesic ball B g ǫ (p) normalized to be positive and with L 2 -norm equal to 1, we have where q is the point of M whose geodesic coordinates are ǫ y for y ∈ B 1 , and G 2 is defined implicitly as a solution of an ODE in [8]. Although we do not need its expression, for completeness we recall it: if we solve such ODE we found (15) G 2 (r) = 1 12 n r 2 φ 1 (r) − c 2 ω n 6n (n + 2) φ 1 (r) .

Known results
Our aim is to perturbe the boundary of a small ball Bḡ 1 (p) with a function v in order to obtained an extremal domain Bḡ 1+v (p). The natural space for the function v is C 2,α (S n−1 ) but not all functions in this space are admissible because v must satisfy also the condition Volḡ Bḡ 1+v (p) = Vol B 1 In order to have a space of admissible functions not depending on the point p, we use a result proved in [12], that allows to use as space of admissible function the space The result is the following: For all ǫ small enough and all functionv ∈ C 2,α m (S n−1 ) whose C 2,α -norm is small enough there exist a unique positive functionφ =φ(p, ǫ,v) ∈ C 2,α (Bḡ 1+v (p)), a constantλ =λ(p, ǫ,v) ∈ R and a constant v 0 = v 0 (p, ǫ,v) ∈ R such that Volḡ Bḡ 1+v (p) = Vol B 1 where v := v 0 +v andφ is a solution to the problem In additionφ,λ and v 0 depend smoothly on the functionv and the parameter ǫ andφ = φ 1 , Instead of working on a domain depending on the function v = v 0 +v, it will be more convenient to work on a fixed domain B 1 endowed with a metric depending on both ǫ and the function v. This can be achieved by considering the parametrization of Bḡ 1+v (p) given by where χ is a cutoff function identically equal to 0 when |y| ≤ 1/2 and identically equal to 1 when |y| ≥ 3/4. Hence the coordinates we consider from now on are y ∈ B 1 with the metricĝ := Y * ḡ .
Up to some multiplicative constant, the problem we want to solve can now be rewritten in the form When ǫ = 0 andv ≡ 0, a solution of (17) is given byφ = φ 1 ,λ = λ 1 and v 0 = 0. In the general case, the relation between the functionφ and the functionφ is simply given by After canonical identification of ∂Bḡ 1+v (p) with S n−1 , we define the operator whereν denotes the unit normal vector field to ∂Bḡ 1+v and (φ, v 0 ) is the solution of (16) provided by the Proposition 4.1. Recall that v = v 0 +v. Schauder's estimates imply that F is well defined from a neighbourhood of is naturally the space of functions in C 1,α (S n−1 ) whose mean is 0). Our aim is to find (p, ǫ,v) such that F (p, ǫ,v) = 0. Observe that, with this condition,φ will be the solution to problem (7).
We also have the alternative expression for F using the coordinates of B 1 and the metriĉ g: where this timeν is the the unit normal vector field to ∂B 1 using the metricĝ.
For allv ∈ C 2,α m (S n−1 ) let ψ be the (unique) solution of (20) We recall that the eigenvalues of the operator −∆ S n−1 are given by µ j = j (n − 2 + j) for j ∈ N, and we denote by V j the eigenspace associated to µ j .
The following result shows that H is the linearization of F with respect tov at ǫ = 0 andv = 0: Using the previous proposition, the implicit function theorem gives directy the following: [12]) There exists ǫ 0 > 0 such that, for all ǫ ∈ [0, ǫ 0 ] and for all p ∈ M, there exists a unique functionv =v(p, ǫ) ∈ C 2,α m (S n−1 ), orthogonal to V 0 ⊕ V 1 , and a vector a = a(p, ǫ) ∈ R n such that (22) F (p, ǫ,v) + a, · = 0 The functionv and the vector a depend smoothly on p and ǫ and we have In other word, for every point p ∈ M it is possible to perturbe the small ball Bḡ 1 (p) in a domain Bḡ 1+v (p), whose volume did not change, but with the (strong) property that F (p, ǫ,v) (i.e. the Neumann data of its first eigenfunction minus its mean) is the restriction of a linear function a, · on S n−1 . It is important to underline that this result does not depend on the geometry of the manifold, because it is true for every point p. Now, we have to find the good point p for which such linear function a, · is the 0 function. And in this research we will see the geometry of the manifold.
Let us now compute the differential of Ψ ǫ . Let Ξ ∈ T p M and q := Exp p (tΞ).
For t small enough, the boundary of Bḡ 1+v(q,ǫ) (q) can be written as a normal graph over the boundary of Bḡ 1+v(p,ǫ) (p) for some function f , depending on p, ǫ, t and Ξ, and smooth on t. This defines a vector field on ∂Bḡ 1+v(p,ǫ) (p) by whereν is the normal of ∂Bḡ 1+v(p,ǫ) (p). Let X be the parallel transported of Ξ from geodesic issued from p. As the metricḡ is close to the Euclidean one for ǫ small, there exists a constant c such that for all ǫ small enough and any Ξ the estimation holds. The variation of the first eigenvalue, see Proposition 2.1, gives We thus obtain Recall that the variation we made is volume preserving, i.e. where |A| ≤ c ba g . Using this equality in equation (24), we deduce that for all ǫ small enought there exists a constant C independent on ǫ and a such that Now the left hand side is bounded by below by b 2 a 2 , so finally we obtain Observe that we cannot have b = 0 because it would imply that butφ > 0 on B 1 soλ = 0 which is not possible because it is the first eigenvalue of the Laplace-Beltrami operator with 0 Dirichlet boundary condition. As a = O(ǫ 2 ), then for ǫ small (recall b = 0) we obtain that a = 0 and this concludes the proof of the proposition.
We now define Φ(p, ǫ) = − 6 n (n + 2) n (n + 2) + 2λ 1 Propositions 4.3 and 5.1 completes the proof of the first part of Theorem 1.1. In the following sections, we will prove the second and the third parts of Theorem 1.1, and for this we have to find an expansion in power of ǫ for Ψ ǫ (p). Such expansion will involve the geometry of the manifold.

Expansion of the first eigenvalue on perturbations of small geodesic balls
In this section we want to find an expansion of the first eigenvalueλ =λ(p, ǫ,v) in power of ǫ andv, where p is fixed in M. In a second time, we will use the functionv =v(p, ǫ) given by Proposition 4.3 in order to find an expansion ofλ(p, ǫ,v(p, ǫ)) in power of ǫ. Keeping in mind that we will havev = O(ǫ 2 ) we write formallŷ We thus study all of theses terms.
Proof. It suffices to find the expansion ofλ(p, ǫ, 0) in power of ǫ. First we have to expand v 0 (p, ǫ, 0) and this can be done by using expansion (10), keeping in mind the definition of the metricĝ and the fact that whenv = 0 the constant v 0 is given by the relation Now we use expansion (12) replacing ǫ by ǫ(1 + v 0 ). We obtain This concludes the proof of the result.
Proof. Let Ω 0 = B 1 be the unit ball of R n , and let Ω t = B (1+v 0 +tv) , where we recall that S n−1v = 0 and v 0 = v 0 (t) is chosen in order that Vol Ω t = Vol Ω 0 = ωn n . We have where λ Ωt is the first Dirichlet eigenvalue of Ω t . The expansion of Vol Ω t directly prove that v 0 = O(t 2 ). In fact, in polar coordinates, we have Differentiating this expression with respect to t, and keeping in mind that v 0 (0) = 0, we obtain that v 0 (t) = O(t 2 ). For y ∈ Ω 0 and t small, let where χ is a cutoff function identically equal to 0 when |y| ≤ 1/2 and identically equal to 1 when |y| ≥ 3/4, so that h(t, Ω 0 ) = Ω t . We will denote the t-derivative with a dot. Let V (t, h(t, y)) =ḣ(t, y) be the first variation of the domain Ω t . Let ν be the unit normal to ∂Ω t and let σ = V, ν the normal variation about ∂Ω t . Let λ be the first eigenvalue and φ the first eigenfunction of the Dirichlet Laplacian over Ω t normalized in order to have L 2 norm equal to 1. From Proposition 2.1 we havė where ∂ ν φ = ∇φ, ν . At t = 0 and on the boundary, we have φ = φ 1 , ∂ ν φ = ∂ r φ 1 = c 1 , σ =v. Thenλ(0) = 0. This proves the first part of the Lemma.
We can use now equality (33) of Proposition 10.1 of the Appendix (with f = (∂ ν φ) 2 σ) in order to derivate this formula with respect to t. We obtain whereH is the mean curvature of ∂Ω t . Now the second variation of the volume of Ω t is  (20). We obtain The proof of the Lemma follows at once.
where Θ is the vector of T p M whose geodesic coordinates are y ∈ S n−1 , according with (5), and c 1 := ∂ r φ 1 | r=1 is the constant defined in (20).
In order to prove this lemma, we start with a preliminary result. The formulas for the geometric quantities we will consider are potentially complicated, and to keep notations short, we agree on the following: any expression of the form L p (v) denotes a linear combination of the function v together with its derivatives up to order 1, whose coefficients can depend on ǫ and there exists a positive constant c independent on ǫ ∈ (0, 1) and on p such that similarly, given a ∈ N, any expression of the form Q (a) p (v) denotes a nonlinear operator in the function v together with its derivatives up to order 1, whose coefficients can depend on ǫ and there exists a positive constant c independent on ǫ ∈ (0, 1) and on p such that provided v i C 2,α (S n−1 ) ≤ 1, for i = 1, 2.
Lemma 6.4. We have where Θ is the vector of T p M whose geodesic coordinates are y ∈ S n−1 , according with (5).
Proof. The expansion in ǫ and v for the volume of the perturbed geodesic ball B g ǫ(1+v) (p) is given in the Appendix of [13] (the corresponding notations with respect to [13] are B g ǫ (1+v) (p) = B p,ǫ (−v) and n = m + 1). We have: where Θ is the vector in T p M whose coordinates are y ∈ S n−1 , and W 0 , W are given by (11). Putting v = v 0 +v in expansion (25), where S n−1v = 0 and v 0 is chosen in order that the volume of B g ǫ (1+v) (p) is equal to the volume of B ǫ , we obtain is the traceless Ricci curvature. In order to compute the expansion oḟ we derivate with respect to s, at s = 0, equality Vol g B g ǫ(1+v 0 (p,ǫ,sv)+sv) (p) = Vol B ǫ using the expansion above. Recall that we know v 0 (p, ǫ, 0) = O(ǫ 2 ). We find This completes the proof of the Lemma.
We are now able to prove Lemma 6.3.
Proof. (Lemma 6.3). We make a development up to power 2 in ǫ, of the function d ds s=0λ (p, ǫ, sv) .
In that formula, the termĝ(∇φ,ν) is computed with s = 0 or equivalentlyv = 0. From the definition ofĝ and the expansion of the metric g, whenv = 0 we havê The expansion ofφ(p, ǫ, 0) is almost known: it suffices to replace ǫ by ǫ(1 + v 0 ) in formula (14). We haveφ Using the notation R j k m l = g ja g mb R akbl we thus have on ∂B 1 where on the boundary Now we have to expand the measure on the boundary. This is classical and can be done directly from expansion (8). We have where Θ is the vector in T p M whose coordinates are y ∈ S n−1 and dvol| S n−1 is the Euclidean volume element induced on S n−1 . For the term ∂vv 0 (p, ǫ, 0)(v) appearing in V we use Lemma 6.4. We have We finally obtain d ds s=0λ (p, ǫ, sv) = C ǫ 2 The proof of the Lemma follows at once.
Summarizing the results of Lemmas 6.1, 6.2 and 6.3 we obtain the following: Proposition 6.5. Let p ∈ M, let ǫ andv be small enough. Then: where Θ is the vector of T p M whose coordinates are x ∈ S n−1 according with (5), and we agree with the convention about L p (v), Q p (v) and Q p (v) we gave before. Proof. It suffices to put together the results of Lemmas 6.1, 6.2 and 6.3.

Localisation of the obtained extremal domains
Now we want to find the expansion of the function Ψ ǫ (p) in power of ǫ. Recall that In order to find such expansion we will relate the first term in the expansion ofv(p, ǫ) to the curvature of the manifold at p.
The first term of the expansion ofv(p, ǫ) is related to the traceless Ricci curvature at p, as stated by the following: where Θ is the vector of T p M whose geodesic coordinates are y ∈ S n−1 , according with (5), and α 2 is the eigenvalue of the operator H defined in Proposition 4.2 associated to the eigenspace V 2 .
Proof. Let us recall that F (p, ǫ,v(p, ǫ)) + a(p, ǫ), · = 0. where v(p, ǫ) C 2,α (S n−1 ) + a(p, ǫ) ≤ c ǫ 2 Now, because F (p, ǫ, 0) = O(ǫ 2 ) and becausev(p, ǫ) = O(ǫ 2 ), we can write In the computation of the mixed derivatives ofλ in the proof of Lemma 6.3 we have already computed the expansion ofĝ(∇φ,ν) forv = 0, so we directly deduce Then we have Writing a = a p ǫ 2 + O(ǫ 3 ) andv =v p ǫ 2 + O(ǫ 3 ) and considering the expansion of F , from equation (22) we obtain We know thatv, and hencev p , is L 2 -orthogonal to V 0 ⊕ V 1 (see Propositions 4.3). Observe that Ric(Θ, Θ) is L 2 (S n−1 )-orthogonal to V 1 since the function Θ → Ric(Θ, Θ) is invariant when Θ is changed into −Θ and hence its L 2 -projection over elements of the form g(Ξ, Θ) is 0 for every Ξ. ThenRic(Θ, Θ) is L 2 (S n−1 )-orthogonal to V 0 ⊕ V 1 . In factRic(Θ, Θ) is the restriction on S n−1 of a homogeneous polynomial of degree 2 which has mean 0, and then it is an eigenfunction for −∆ S n−1 with eigenvalue 2n. As H preserves the eigenspaces of −∆ S n−1 and his kernel is given by V 1 (see Proposition 4.2), we have that there exists a constant α 2 = 0 such that i.e. a p , · is in the image of H. But it belongs also to the kernel of H, and then a p = 0 and In order to complete the proof of the proposition we use equation (31) and Lemma 8.1 of the Appendix. where (30) K 1 = 1 n (n + 2) + 2 λ 1 18 c 2 + λ 1 10(n + 4) and formula (29) follows at once. The fact that the constants K i depend only on n comes immediately from the computation of c 2 by Lemma 8.2 in the Appendix: This completes the proof of the proposition.
Remark 1. We remark that K 1 > 0 in order to justify our discussion about critical point of Riem for Einstein metrics in the introduction.
the proof of the second and third part of Theorem 1.1 follows at once.

Appendix I : On the first eigenfunction in the unit Euclidean ball
In this Appendix we state and prove some relations between the first eigenfunction and the first eigenvalue of the Dirichlet Laplacian on the unit ball.
The proof of the Lemma follows at once from Lemma 8.1.

Appendix II: The second eigenvalue of the operator H
Here we compute the eigenvalue α 2 of the operator H associated to the eigenspace V 2 . When w is an homogeneous polynomial harmonic of degree 2 (abusively identified with its restriction to the unit sphere) we have ∆ S n−1 w = −µ 2 w = −2n w and H(w) = α 2 w. We recall that H(w) = (∂ r ψ)| ∂B 1 + c 2 w = (∂ r ψ)| In this Appendix we recall a useful result that allows to derivate the integral of a function with respect to a parameter t that appears in the function and also in the domain of integration. The proof of the such a result can be found in [7], page 14. Let Ω t = h(t, Ω 0 ), V (t, h(t, p)) = ∂h ∂t (t, p) and N(t, q) the unit outward normal at q ∈ ∂Ω t . Then where ·, · denote the scalar product in R n , s denote the area element of ∂Ω t and H is the mean curvature of ∂Ω t .