THE SECOND YAMABE INVARIANT

B. Ammann and E. Humbert^{1}

Abstract. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3. We define the second Yamabe invariant as the infimum of the second eigenvalue of the Yamabe operator over the metrics conformal to g and of volume 1. We study when it is attained. As an application, we find nodal solutions of the Yamabe equation.

March 15, 2005

MSC 2000: 53A30, 35J60(Primary) 35P30, 58J50, 58C40 (Secondary)

Contents

1. Introduction 1

2. Variational characterization of μ_{2}(M, g) 4

3. Generalized metrics and the Euler-Lagrange equation 5

4. A sharp Sobolev inequality related to μ_{2}(M, g) 9

5. Some properties of μ_{2}(M, g) 11

6. Existence of a minimum of μ_{2}(M, g) 15

7. The invariant μ_{k}(M) for k ≥ 3 22

8. The case of manifolds whose Yamabe invariant is negative 22

A. Appendix: Proof of Lemma 3.1 23

References 25

1. Introduction

Let (M, g) be an n-dimensional compact Riemannian manifold (n ≥ 3). In [Yam60] Yamabe attempted to show that there is a metric g̃ conformal to g such that the scalar curvature Sg̃ of g̃ is constant. However, Trudinger [Tru68] realized that Yamabe’s proof contained a serious gap. The problem is now solved, but it took a very long time to find the good approach. The problem of finding a metric g̃ with constant scalar curvature in the conformal class [g] is called the Yamabe problem. The first step towards a rigorous solution of this problem was achieved by Trudinger [Tru68] who was able to repair the gap of Yamabe’s article in the case that the scalar curvature of g is non-positive. Eight years later, Aubin [Aub76] solved the problem for arbitrary non locally conformally flat manifolds of dimension n ≥ 6. The problem was completely solved another eight years later in an article of Schoen [Sch84] in which the proof was reduced to the positive-mass theorem which had previously been proved by Schoen and Yau [SY79, SY88]. The reader can refer to [LP87], [Aub76] or [Heb97] for more information on this subject. The method to solve the Yamabe problem was the following. Let u ∈ C^{∞}(M), u > 0 be a smooth function and g̃ = u^{N−2}g where N = \frac{2n}{n−2}. Then, multiplying u by a constant, the following equation is satisfied:

1bernd.ammann@gmx.net, humbert@iecn.u-nancy.fr

L_{g}(u) = Sg̃|u|^{N−2}u.

where

L_{g} = c_{n}Δ_{g} + S_{g} = \frac{4(n − 1)}{n − 2}Δ_{g} + S_{g}

is called the Yamabe operator. As a consequence, solving the Yamabe problem is equivalent to finding a positive smooth solution u of

L_{g}(u) = C_{0}|u|^{N−2}u.  (1)

where C_{0} is a constant. In order to obtain solutions of this equation Yamabe defined the quantity

μ(M, g) = inf_{u≠0,u∈C^{∞}(M)} Y (u)

where

Y (u) = \frac{∫_{M} c_{n}|∇u|^{2} + S_{g}u^{2} dv_{g}}{(∫_{M} |u|^{N} dv_{g})^{\frac{2}{N}}} .

Nowadays, μ(M, g) is called the Yamabe invariant, and Y the Yamabe functional. Writing the Euler-Lagrange equation associated to Y, we see that there exists a one to one correspondence between critical points of Y and solutions of equation (1). In particular, if u is a positive smooth function such that Y (u) = μ(M, g), then u is a solution of (1) and g̃ = u^{N−2}g is the desired metric of constant scalar curvature. The key point of the resolution of the Yamabe problem is the following theorem due to Aubin [Aub76]. In the theorem and in the whole article, 𝕊^{N} will always denote the sphere S^{n} with the standard Riemannian structure.

Theorem 1.1. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3. If μ(M, g) < μ(𝕊^{n}), then there exists a positive smooth function u such that Y (u) = μ(M, g).

This strict inequality is used to show that a minimizing sequence does not concentrate in any point. Aubin [Aub76] and Schoen [Sch84] proved the following.

Theorem 1.2. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3. Then μ(M, g) ≤ μ(𝕊^{n}) = n(n − 1)ωn^{\frac{2}{n}} where ω_{n} stands for the volume of the standard sphere S^{n}. Moreover, we have equality in this inequality if and only if (M, g) is conformally diffeomorphic to the sphere.

These theorems solves the Yamabe problem.

In this paper, we introduce and study an invariant that we will call the second Yamabe invariant. It is well known that the operator L_{g} has discrete spectrum

Spec(L_{g}) = {λ_{1}(g), λ_{2}(g), ⋯ }

where the eigenvalues

λ_{1}(g) < λ_{2}(g) ≤ λ_{3}(g) ≤ ⋯ ≤ λ_{k}(g) ⋯ → +∞

appear with their multiplicities. The variational characterization of λ_{1}(g) is given by

λ_{1}(g) = inf_{u≠0,u∈C^{∞}(M)} \frac{∫_{M} c_{n}|∇u|^{2} + S_{g}u^{2} dv_{g}}{∫_{M} |u|^{2} dv_{g}} .

Let [g] be the conformal class of g. Assume now that the Yamabe invariant μ(M, g) ≥ 0. It is easy to check that

μ(M, g) = inf_{g̃∈[g]} λ_{1}(g̃)Vol(M, g̃)^{\frac{2}{n}} ,

where [g] is the conformal class of g. We then enlarge this definition.

Definition 1.3. Let k ∈ ℕ^{∗}. Then, the k^{th} Yamabe invariant is defined by

μ_{k}(M, g) = inf_{g̃∈[g]} λ_{k}(g̃)Vol(M, g̃)^{\frac{2}{n}} .

With these notations, μ_{1}(M, g) equals to Yamabe invariant μ(M, g) in the case μ(M, g) ≥ 0, and μ_{1}(M, g) = −∞ in the case μ(M, g) < 0.

The goal of this article is to study the second Yamabe invariant μ_{2}(M, g) for manifolds whose Yamabe invariant in the case μ(M, g) ≥ 0. As explained in Section 8, the most interesting case is when μ(M, g) > 0. In particular, we discuss whether μ_{2}(M, g) is attained. This question is discussed in Subsection 5.1. In particular, Proposition 5.2 asserts that contrary to the standard Yamabe invariant, μ_{2}(M, g) cannot be attained by a metric if M is connected. In other words, there does not exist g̃ ∈ [g] such that μ_{2}(M, g) = λ_{2}(g̃)Vol(M, g̃)^{\frac{2}{n}}. In order to find minimizers, we enlarge the conformal class [g] to what we call the class of generalized metrics conformal to g. A generalized metric is a “metric” of the form g̃ = u^{N−2}g, where u is no longer necessarily positive and smooth, but u ∈ L^{N}(M), u ≥ 0, u ≢ 0. The definitions of λ_{2}(g̃) and of Vol(M, g̃) can be extended to generalized metrics (see section 3). Then, we are able to prove the following result:

Theorem 1.4. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3 whose Yamabe invariant is non-negative. Then, μ_{2}(M, g) is attained by a generalized metric in the following cases:

• μ_{1}(M, g) > 0 and μ_{2}(M, g) < [μ_{1}(M, g)^{\frac{n}{2}} + μ_{1}(𝕊^{n}))^{\frac{n}{2}} ]^{\frac{2}{n}};

• μ_{1}(M, g) = 0 and μ_{2}(M, g) < μ_{1}(𝕊^{n})

where μ_{1}(𝕊^{n}) = n(n − 1)ωn^{\frac{2}{n}} is the Yamabe invariant of the standard sphere.

The result we obtain in the case μ_{1}(M, g) = 0 is not surprising. Indeed, when μ_{2}(M, g) < μ_{1}(𝕊^{n}), Aubin’s methods [Aub76] can be adapted here and allow to avoid concentration of minimizing sequences. However, when μ_{1}(M, g) > 0 and μ_{2}(M, g) < [μ_{1}(M, g)^{\frac{n}{2}} + μ_{1}(𝕊^{n}))^{\frac{n}{2}} ]^{\frac{2}{n}}, the result is much more difficult to obtain (see Subsection 6). A second result is to find explicit examples for which the assumptions of Theorem 1.4 are satisfied. The method consists in finding an appropriate couple of test functions.

Theorem 1.5. The assumptions of Theorem 1.4 are satisfied in the following cases:

• μ_{1}(M, g) > 0, (M, g) is not locally conformally flat and n ≥ 11;

• μ_{1}(M, g) = 0, (M, g) is not locally conformally flat and n ≥ 9.

One of our motivations is to find solutions of the Yamabe equation (1) with alternating sign, i.e. positive and negative values. If M is connected, alternating sign implies that the zero set u^{−1}(0) of u is not empty. In the following we will use the standard definition to call the zero set u^{−1}(0) of a function u the nodal set of u. A solution with a non-empty nodal set is usually called a nodal solution. If M is connected, then the maximum principle implies that a solution of the Yamabe equation is nodal if and only if it has alternating sign. They are called nodal solutions of the Yamabe equation. The articles [HV94], [DJ02], [Jou99], [Hol99] prove existence of nodal solutions under symmetry assumptions or under some assumptions which allow to use Aubin’s methods, as in Theorem 1.4 when μ_{1}(M, g) = 0 and μ_{2}(M, g) < μ_{1}(𝕊^{n}). If μ(M, g) ≤ 0, another method is given in Section 8. The method we use here is completely different and we obtain solutions on a large class of manifolds. In particular, to our knowledge, there is no work which leads to the existence of such solutions if the Yamabe invariant is positive and if (M, g) is not conformally equivalent to the round sphere. The result we obtain is the following:

Theorem 1.6. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3. Assume that μ_{2}(M, g) is attained by a generalized metric u^{N−2}g where u ∈ L^{N}(M), u ≥ 0 and u ≢ 0. Let Ω be the nodal set of u. Then, there exists a nodal solution w ∈ C^{∞}(M \ Ω) ∩ C^{3,α}(M) ( α ≤ N − 2) of equation (1) such that |w| = u.

A corollary of Theorems 1.4, 1.5 and 1.6 is then

Corollary 1.7. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3 whose Yamabe invariant is non-negative. We assume that one of the following assumptions is true:

• μ_{1}(M, g) > 0, (M, g) is not locally conformally flat and n ≥ 11;

• μ_{1}(M, g) = 0, (M, g) is not locally conformally flat and n ≥ 9.

Then, there exists a nodal solution of Yamabe equation (1).

Acknowledgement The authors want to thank M. Ould Ahmedou for many interesting conversations about nodal solutions of the Yamabe equation. His large knowledge about such problems was a stimulating inspiration for this article. The author are also extremely obliged to Frédéric Robert for having pointed out a little mistake in the first version of this paper.

2. Variational characterization of μ_{2}(M, g)

2.1. Notation. In the whole article we will use the following notations

L_{+}^{N}(M) := {u ∈ L^{N}(M) | u ≥ 0, u ≢ 0} .

2.2. Grassmannians and the min-max principle. Let Gr_{k}(C^{∞}(M)) be the k-dimensional Grassmannian in C^{∞}(M), i.e. the set of all k-dimensional subspaces of C^{∞}(M). The Grassmannian is an important ingredient in the min-max characterization of λ_{k}(g)

λ_{k}(Lg̃) := inf_{V ∈Gr_{k}(C^{∞}(M))} sup_{v∈V \\{0\}} \frac{∫ (Lg̃v)v dvg̃}{∫_{M} v^{2} dvg̃} .

We will also need a slightly modified Grassmannian. For any u ∈ L_{+}^{N}(M) we define Gr_{k}^{u}(C^{∞}) to be the set of all k-dimensional subspaces of C^{∞}(M), such that the restriction operator to M \u^{−1}(0) is injective. More explicitly, we have span(v_{1}, …, v_{k}) ∈ Gr_{k}^{u}(C^{∞}(M)) if and only if v_{1}|_{M\u^{−1}(0)}, …, v_{k}|_{M\u^{−1}(0)} are linearly independent. Sometimes it will be convenient to use the equivalent statement that the functions u^{\frac{N−2}{2}} v_{1}, …, u^{\frac{N−2}{2}} v_{k} are linearly independent.

Similarly, by replacing C^{∞}(M) by H_{1}^{2}(M) we obtain the definitions of Gr_{k}(H_{1}^{2}(M)) and Gr_{k}^{u}(H_{1}^{2}(M)).

2.3. The functionals. For all u ∈ L_{+}^{N}(M), v ∈ H_{1}^{2}(M) such that u^{\frac{N−2}{2}} v ≢ 0, we set

F (u, v) = \frac{∫_{M} c_{n}|∇v|^{2} + S_{g}v^{2} dv_{g}}{∫_{M} v^{2}u^{N−2} dv_{g}} (∫_{M} u^{N} dv_{g})^{\frac{2}{n}} .

2.4. Variational characterization of μ_{2}(M, g). The following characterization will be of central importance for our article.

Proposition 2.1. We have

μ_{k}(M, g) = inf_{{u∈L_{+}^{N} (M) \atop V ∈Gr_{k}^{u}(H_{1}^{2}(M))}} sup_{v∈V \\{0\}} F (u, v)  (2)

Proof. Let u be a smooth positive function on M. For all smooth functions f, f ≢ 0, we set g̃ = u^{N−2}g (N = \frac{2n}{n−2}) and

F′(u, f) = \frac{∫_{M} fLg̃f dvg̃}{∫_{M} f^{2} dvg̃} .

The operator L_{g} is conformally invariant (see [Heb97]) in the following sense:

u^{N−1}Lg̃(u^{−1}f) = L_{g}(f)  (3)

Together with the fact that

dvg̃ = u^{N} dv_{g},  (4)

we get that

F′(u, f) = \frac{∫_{M}(uf)L_{g}(uf) dv_{g}}{∫_{M}(uf)^{2}u^{N−2} dv_{g}} .

Using the min-max principle, we can write that

λ_{k}(g̃) = inf_{V ∈Gr_{k}^{u}(H_{1}^{2}(M))} sup_{f∈V \{0}} F′(u, f)

Now, replacing uf by v, we obtain that

λ_{k}(g̃) = inf_{V ∈Gr_{k}(H_{1}^{2}(M))} sup_{v∈V \\{0\}} \frac{∫_{M} vL_{g}v dv_{g}}{∫_{M} v^{2}u^{N−2} dv_{g}} .  (5)

Using the definition of μ_{2} and Volg̃(M) = ∫_{M} u^{N} dv_{g}, we derive

μ_{k}(M, g) = inf_{{u∈L_{+}^{N} (M) \atop V ∈Gr_{k}^{u}(C^{∞}(M))}} sup_{v∈V \\{0\}} F (u, v)

The result follows immediately.

3. Generalized metrics and the Euler-Lagrange equation

3.1. A regularity result. We will need the following result.

Lemma 3.1. Let u ∈ L^{N}(M) and v ∈ H_{1}^{2}(M). We assume that

L_{g}v = u^{N−2}v

holds in the sense of distributions. Then, v ∈ L^{N+ε}(M) for some ε > 0.

This result is well known for the standard Yamabe equation. Proofs for the standard Yamabe equation can be found in [Tru68] and [Heb97], and the modifications for proving Lemma 3.1 are obvious. Unfortunately,

[Tru68] contains some typos, and the book [Heb97] is difficult to obtain. This is why we included a proof in the appendix for the convenience of the reader.

3.2. The k-th eigenvalue of the Yamabe operator for a generalized metric. On a given Riemannian manifold (M, g) we say that g̃ = u^{N−2}g, u ∈ L_{+}^{N}(M), is a generalized metric conformal to g. For a generalized metric g̃, we can define

λ_{k}(g̃) = inf_{V ∈Gr_{k}^{u}(H_{1}^{2}(M))} sup_{v∈V \\{0\}} \frac{∫_{M} vL_{g}v dv_{g}}{∫_{M} v^{2}u^{N−2} dv_{g}} .  (6)

Proposition 3.2. For any u ∈ L_{+}^{N}, g̃ = u^{N−2} there exist two functions v, w belonging to H_{1}^{2}(M) with v ≥ 0 and such that in the sense of distributions.

L_{g}v = λ_{1}(g̃)u^{N−2}v  (7)

and

L_{g}w = λ_{2}(g̃)u^{N−2}w.  (8)

Moreover, we can normalize v, w by

∫_{M} u^{N−2}v^{2} dv_{g} = ∫_{M} u^{N−2}w^{2} dv_{g} = 1 and ∫_{M} u^{N−2}vw dv_{g} = 0  (9)

For k = 2 the infimum in formula (5) over all subspaces V ∈ Gr_{2}^{u}(H_{1}^{2}(M)) is attained by V = span(v, w) and the supremum over the functions in V \ {0} is attained by w. The reader should pay attention to the fact that the space V is in general non unique. As one can check, if w changes the sign then the supremum over all v ∈ V = span(v, w) \ {0} and the supremum over all v ∈ V_{1} = span(w, |w|) \ {0} coincide.

From section (2), we get

μ_{2}(M, g) = inf_{g̃∈\overline{[g]}} λ_{2}(g̃).

Hence, μ_{2}(M, g) can be attained by a regular metric, or by a generalized metric or it can be not attained at all. These questions are discussed in Section 5. Let us now prove Proposition 3.2.

Proof of Proposition 3.2: Let (v_{m})_{m} be a minimizing sequence for λ_{1}(g̃) i.e. a sequence v_{m} ∈ H_{1}^{2}(M) such that

lim_{m→∞} \frac{∫_{M} c_{n}|∇v_{m}|^{2} + S_{g}v_{m}^{2} dv_{g}}{∫_{M} u^{N−2}v_{m}^{2} dv_{g}} = λ_{1}(g̃).

It is well known that (|v_{m}|)_{m} is also a minimizing sequence. Hence, we can assume that v_{m} ≥ 0. If we normalize v_{m} by ∫_{M} u^{N−2}v_{m}^{2} dv_{g} = 1, then (v_{m})_{m} is bounded in H_{1}^{2}(M) and after restriction to a subsequence we may assume that there exists v ∈ H_{1}^{2}(M), v ≥ 0 such that v_{m} → v weakly in H_{1}^{2}(M), strongly in L^{2}(M) and almost everywhere. If u is smooth, then

∫_{M} u^{N−2}v^{2} dv_{g} = lim_{m} ∫_{M} u^{N−2}v_{m}^{2} dv_{g} = 1  (10)

and by standard arguments, v is a non-negative minimizer of the functional associated to λ_{1}(g̃). We must show that (10) still holds if u ∈ L_{+}^{N}(M). Let A > 0 be a large real number and set u_{A} = inf(u, A). Then, using the Hölder inequality, we write

|∫_{M} u^{N−2} (v_{m}^{2} − v^{2}) dv_{g}| ≤ (∫_{M} u_{A}^{N−2}|v_{m}^{2} − v^{2}| dv_{g} + ∫_{M}(u^{N−2} − u_{A}^{N−2})(|v_{m}| + |v|)^{2} dv_{g})
≤ A ∫_{M} |v_{m}^{2} − v^{2}| dv_{g}

+ (∫_{M}(u^{N−2} − u_{A}^{N−2})^{\frac{N}{N−2}} dv_{g})^{\frac{N−2}{N}} (∫_{M}(|v_{m}| + |v|)^{N} dv_{g})^{\frac{2}{N}} .

By Lebesgue’s theorem we see that

lim_{A→+∞} ∫_{M}(u^{N−2} − u_{A}^{N−2})^{\frac{N}{N−2}} dv_{g} = 0.

Since (v_{m})_{m} is bounded in H_{1}^{2}(M), it is bounded in L^{N}(M) and hence there exists C > 0 such that ∫_{M}(|v_{m}| + |v|)^{N} dv_{g} ≤ C. By strong convergence in L^{2}(M),

lim_{m} ∫_{M} |v_{m}^{2} − v^{2}| dv_{g} = 0.

Equation (10) easily follows and v is a non-negative minimizer of the functional associated to λ_{1}(g̃). Writing the Euler-Lagrange equation of v, we find that v satisfies equation (7). Now, we define

λ′_{2}(g̃) = inf \frac{∫_{M} c_{n}|∇w|^{2} + S_{g}w^{2} dv_{g}}{∫_{M} u^{N−2}|w|^{2} dv_{g}}

where the infimum is taken over smooth functions w such that u^{\frac{N−2}{2}} w ≢ 0 and such that ∫_{M} u^{N−2}vw dv_{g} = 0. With the same method, we find a minimizer w of this problem that satisfies (8) with λ′_{2}(g̃) instead of λ_{2}(g̃). However, it is not difficult to see that λ′_{2}(g̃) = λ_{2}(g̃) and Proposition 3.2 easily follows.

3.3. Euler-Lagrange equation of a minimizer of λ_{2}Vol^{2/n}.

Lemma 3.3. Let u ∈ L_{+}^{N}(M) with ∫ u^{N} = 1. Suppose that w_{1}, w_{2} ∈ H_{1}^{2}(M) \ {0}, w_{1}, w_{2} ≥ 0 satisfy

∫ (c_{n}|∇w_{1}|^{2} + Scal_{g}w_{1}^{2}) dv_{g} ≤ μ_{2}(M, g) ∫ u^{N−2}w_{1}^{2}  (11)
∫ (c_{n}|∇w_{2}|^{2} + Scal_{g}w_{2}^{2}) dv_{g} ≤ μ_{2}(M, g) ∫ u^{N−2}w_{2}^{2}  (12)

and suppose that (M \ w_{1}^{−1}(0)) ∩ (M \ w_{2}^{−1}(0)) has measure zero. Then u is a linear combination of w_{1} and w_{2} and we have equality in (11) and (12).

Proof. We let ū = aw_{1} + bw_{2} where a, b > 0 are chosen such that

\frac{a^{N−2}}{b^{N−2}} \frac{∫_{M} u^{N−2}w_{1}^{2} dv_{g}}{∫_{M} u^{N−2}w_{2}^{2} dv_{g}} = \frac{∫_{M} w_{1}^{N} dv_{g}}{∫_{M} w_{2}^{N} dv_{g}}  (13)

and

∫_{M} ū^{N} dv_{g} = a^{N} ∫_{M} w_{1}^{N} + b^{N} ∫ w_{2}^{N} = 1.  (14)

Because of the variational characterization of μ_{2} we have

μ_{2}(M, g) ≤ sup_{(λ,μ)∈ℝ^{2}\{(0,0)}} F (ū, λw_{1} + μw_{2})  (15)

By (11),(12) and (14), and since (M \ w_{1}^{−1}(0)) ∩ (M \ w_{2}^{−1}(0)) has measure zero

F (ū, λw_{1} + μw_{2}) = \frac{λ^{2} ∫_{M} (c_{n}|∇w_{1}|^{2} + S_{g}w_{1}^{2}) dv_{g} + μ^{2} ∫_{M} (c_{n}|∇w_{2}|^{2} + S_{g}w_{2}^{2}) dv_{g}}{λ^{2} ∫_{M} |ū|^{N−2}w_{1}^{2} dv_{g} + μ^{2} ∫_{M} |ū|^{N−2}w_{2}^{2} dv_{g}}

≤ μ_{2}(M, g)\frac{λ^{2} ∫_{M} u^{N−2}w_{1}^{2} dv_{g} + μ^{2} ∫_{M} u^{N−2}w_{2}^{2} dv_{g}}{λ^{2}a^{N−2} ∫_{M} w_{1}^{N} dv_{g} + μ^{2}b^{N−2} ∫_{M} w_{2}^{N} dv_{g}} .  (16)

As one can check, relation (13) implies that this expression does not depend on λ, μ. Hence, setting λ = a and μ = b, the denominator is 1, and we get

sup_{(λ,μ)∈ℝ^{2}\{(0,0)}} F (ū, λw_{1} + μw_{2}) ≤ μ_{2}(M, g) ∫_{M} u^{N−2}(a^{2}w_{1}^{2} + b^{2}w_{2}^{2}) dv_{g}
= μ_{2}(M, g) ∫_{M} u^{N−2}ū^{2} dv_{g}.

By Hölder inequality,

sup_{(λ,μ)∈ℝ^{2}\\{(0,0)\}} F (ū, λw_{1} + μw_{2}) ≤ μ_{2}(M, g)(∫_{M} u^{N} dv_{g})^{\frac{N−2}{N}} (∫_{M} ū^{N} dv_{g})^{\frac{2}{N}} = μ_{2}(M, g).  (17)

Inequality (15) implies that we have both equality in the Hölder inequality of (17) and in (16). The equality in the Hölder inequality implies that there exists a constant c > 0 such that u = cū almost everywhere. Moreover, since ∫ u^{N} = ∫ ū^{N} = 1, we have u = ū = aw_{1} + bw_{2}. The equality in (16) implies inequality in (11) and (12). □

Theorem 3.4 (Euler-Lagrange equation). Assume that μ_{2}(M, g) ≠ 0 and that μ_{2}(M, g) is attained by a generalized metric g̃ = u^{N−2}g with u ∈ L_{+}^{N}(M). Let v, w be as in Proposition 3.2. Then, u = |w|. In particular,

L_{g}w = μ_{2}(M, g)|w|^{N−2}w  (18)

Moreover, w has alternating sign and w ∈ C^{3,α}(M) (α ≤ N − 2).

Remark 3.5. Assume that μ_{2}(M, g) is equal to 0 and is attained by a generalized metric g′, then, using the conformal invariance of the Yamabe operator, it is easy to check that for all generalized metrics g̃ conformal to g′, we have λ_{2}(g̃) = 0. Consequently, each metric conformal to g is a minimizer for μ_{2}(M, g) and Theorem 3.4 is always false in this case. However, we will still get a nodal solution of (1) if μ_{2}(M, g) = 0. Indeed, by Theorem 1.4 and the remark above, λ_{2}(g) = 0. Let w be an eigenfunction associated to λ_{2}(g). We have L_{g}w = 0. Then, we have a solution of (18).

Remark 3.6. Assume that μ_{2}(M, g) ≠ 0 and that μ_{2}(M, g) is attained by a generalized metric. Let w be the solution of equation (18) given by Theorem 3.4. We let Ω_{+} = {x ∈ M s.t. w(x) > 0} and Ω_{−} = {x ∈ M s.t. w(x) − 0}. Then, a immediate consequence of Lemma 3.3 is that Ω_{+} and Ω_{−} have exactly one connex component.

Proof. Without loss of generality, we can assume that ∫_{M} u^{N} dv_{g} = 1. By assumption we have λ_{2}(g̃) = μ_{2}(M, g). Let v, w ∈ H_{1}^{2}(M) be some functions satisfying equations (7), (8) and relation (9).

Step 1. We have λ_{1}(g̃) < λ_{2}(g̃).

We assume that λ_{1}(g̃) = λ_{2}(g̃). Then, after possibly replacing w by a linear combination of v and w, we can assume that the function u^{\frac{N−2}{2}} w changes the sign. We apply Lemma 3.3 for w_{1} := sup(w, 0) and w_{2} := sup(−w, 0). We obtain the existence of a, b > 0 with u = aw_{1} + bw_{2}. Now, by Lemma 3.1, w ∈ L^{N+ε}(M). By a standard bootstrap argument, equation (8) shows that w ∈ C^{2,α}(M) for all α ∈]0, 1[. It follows that u ∈ C^{0,α}(M) for all α ∈]0, 1[. Now, since λ_{1}(g̃) = λ_{2}(g̃) and by definition of λ_{1}(g̃), w is a minimizer of the functional w̄ ↦ F (u, w̄) among the functions belonging to H_{1}^{2}(M) and such that u^{\frac{N−2}{2}} w̄ ≢ 0. Since F (u, w) = F (u, |w|), we see that |w| is a minimizer for the functional associated to λ_{1}(g̃) and hence, writing the Euler-Lagrange equation of the problem, w satisfies the same equation as w. As a consequence, |w| is C^{2}(M). By the maximum principle, we get |w| > 0 everywhere. This is false. Hence, the step is proved.

Step 2. The function w changes the sign.

Assume that w does not change the sign, i.e. after possibly replacing w by −w, we have w ≥ 0. Using (9) we see that (M \ v^{−1}(0)) ∩ (M \ w^{−1}(0)) has measure zero. Setting w_{1} := v and w_{2} := w we have (11) and (12). While we have equality in (12), Step 1 implies that inequality (11) is strict. However using Lemma 3.3 we can derive equality in (11). Hence we obtain a contradiction, and the step is proved.

Step 3. There exists a, b > 0 such that u = a sup(w, 0) + b sup(−w, 0). Moreover, w ∈ C^{2,α}(M) and u ∈ C^{0,α}(M) for all α ∈]0, 1[.

As in the proof of Step 1 we apply Lemma 3.3 for w_{1} := sup(w, 0) and w_{2} := sup(−w, 0). We obtain the existence of a, b > 0 such that u = aw_{1} + bw_{2}. As in Step 1 we get that w ∈ C^{2,α}(M) and u ∈ C^{0,α}(M) for all α ∈]0, 1[. This proves the present step.

Step 4. Conclusion.

Let h ∈ C^{∞}(M) whose support is contained in M \ {u^{−1}(0)}. For t close to 0, set u_{t} = |u + th|. Since u > 0 on the support of h and since u is continuous (see last step), we have for t close to 0, u_{t} = u + th. As span(v, w) ∈ Gr_{2}^{u}(H_{1}^{2}(M)) we obtain using (2) for all t

μ_{2}(M, g) ≤ sup_{(λ,μ)∈ℝ^{2}\{(0,0)}} F (u_{t}, λv + μw).

Equations (7), (8), and relation (9) yield

F (u_{t}, λv + μw) = \frac{λ^{2}λ_{1}(g̃) ∫_{M} u^{N−2}v^{2} dv_{g} + μ^{2}λ_{2}(g̃) ∫_{M} u^{N−2}w^{2} dv_{g}}{λ^{2} ∫_{M} u_{t}^{N−2}v^{2} dv_{g} + 2λμ ∫_{M} u_{t}^{N−2}vw dv_{g} + μ^{2} ∫_{M} u_{t}^{N−2}w^{2} dv_{g}} (∫_{M} u_{t}^{N} dv_{g})^{\frac{2}{n}}

= \frac{λ^{2}λ_{1}(g̃) + μ^{2}λ_{2}(g̃)}{λ^{2}a_{t} + λμb_{t} + μ^{2}c_{t}} (∫_{M} |u_{t}|^{N} dv_{g})^{\frac{2}{n}} .

where

a_{t} = ∫_{M} u_{t}^{N−2}v^{2} dv_{g},
b_{t} = 2 ∫ u_{t}^{N−2}vw dv_{g}

and

c_{t} = ∫_{M} u_{t}^{N−2}w^{2} dv_{g}.

The functions a_{t}, b_{t} and c_{t} are smooth for t close to 0, furthermore a_{0} = c_{0} = 1 and b_{0} = 0. The function f(t, α) := F (u_{t}, sin(α)v + cos(α)w) is smooth for small t. Using λ_{1}(g̃) < λ_{2}(g̃) one calculates

\frac{∂}{∂α} f(0, α) = 0  ⇔  α ∈ \frac{π}{2} ℤ
\frac{∂^{2}}{∂α^{2}} f(0, α) < 0  for  α ∈ πℤ
\frac{∂^{2}}{∂α^{2}} f(0, α) > 0  for  α ∈ πℤ + \frac{π}{2}

Applying the implicit function theorem to \frac{∂f}{∂α} at the point (0, 0), we see that there is a smooth function t ↦ α(t), defined on a neighborhood of 0 with α(0) = 0 and

f(t, α(t)) = sup_{α∈ℝ} f(t, α) = sup_{(λ,μ)∈ℝ^{2}\{(0,0)}} F (u_{t}, λv + μw).

As a consequence

\frac{d}{dt}|_{t=0} sin^{2} α(t) = \frac{d}{dt}|_{t=0} cos^{2} α(t) = \frac{d}{dt}|_{t=0}(sin^{2} α(t)a_{t}) = \frac{d}{dt}|_{t=0}(sin α(t) cos α(t)b_{t}) = 0.

Hence, \frac{d}{dt}|_{t=0} f(t, α(t)) exists and we have

\frac{d}{dt}|_{t=0} f(t, α(t)) = λ_{2}(M, g̃) (−\frac{d}{dt}|_{t=0}c_{t} + \frac{d}{dt}|_{t=0}(∫_{M} |u_{t}|^{N} dv_{g})^{\frac{2}{n}} )

= λ_{2}(M, g̃)(N − 2) (− ∫_{M} u^{N−3}hw^{2} dv_{g} + ∫_{M} u^{N−1}h dv_{g}) .

By definition of μ_{2}(M, g), f admits a minimum in t = 0. As λ_{2}(M, g̃) = μ_{2}(M, g) ≠ 0 we obtain

∫_{M} u^{N−3}hw^{2} dv_{g} = ∫_{M} u^{N−1}h dv_{g}.

Since h is arbitrary (we just have to ensure that its support is contained in M \ {u^{−1}(0)}), we get that u^{N−3}w^{2} = u^{N−1} on M \ {u^{−1}(0)}, hence u = |w| on M \ {u^{−1}(0)}. Together with Step 3, we get u = |w| everywhere. This proves theorem 3.4. □

4. A sharp Sobolev inequality related to μ_{2}(M, g)

4.1. Statement of the results. For any compact Riemannian manifold (M, g) of dimension n ≥ 3, Hebey and Vaugon have shown in ([HV96]) that there exists B_{0}(M, g) > 0 such that

μ_{1}(𝕊^{n}) = n(n − 1) ωn^{\frac{2}{n}} = inf_{u∈H_{1}^{2}(M)\\{0\}} \frac{∫_{M} c_{n}|∇u|^{2} + B_{0} ∫_{M} u^{2} dv_{g}}{(∫_{M} u^{N} dv_{g})^{\frac{2}{n}}}  (S)

where ω_{n} stands for the volume of the standard n-dimensional sphere 𝕊^{n} and where μ_{1}(𝕊^{n}) is the Yamabe invariant of 𝕊^{n}.

This inequality is strongly related to the resolution of the Yamabe problem. It allows to avoid concentration for the minimizing sequence of μ_{1}(M, g). For the minimization of μ_{2}(M, g), this inequality is not sufficient and another one must be constructed. The following result is adapted to the problem of minimizing μ_{2}(M, g).

Theorem 4.1. On a compact connected Riemannian manifold (M, g) of dimension n ≥ 3 we have

2^{\frac{2}{n}} μ_{1}(𝕊^{n}) = inf_{{u∈L_{+}^{N} (M) \atop V ∈Gr_{k}^{u}(H_{1}^{2}(M))}} sup_{v∈V \\{0\}} \frac{(∫_{M} c_{n}|∇v|^{2} dv_{g} + B_{0}(M, g) ∫_{M} v^{2} dv_{g}) (∫_{M} u^{N} dv_{g})^{\frac{2}{N}}}{∫_{M} u^{N−2}v^{2} dv_{g}}  (S_{1})

where B_{0}(M, g) is given by inequality (S).

We present now two corollaries of Theorem 4.1.

Corollary 4.2. For the standard n-dimensional sphere we have μ_{2}(𝕊^{n}) = 2^{\frac{2}{n}} μ_{1}(𝕊^{n}).

Corollary 4.3. For all u ∈ C_{c}^{∞}(ℝ^{n}) and V ∈ Gr_{2}^{u}(C_{c}^{∞}(ℝ^{n})) we have

2^{2/n}μ_{1}(𝕊^{n}) ≤ sup_{v∈V \\{0\}} \frac{(∫_{ℝ^{n}} c_{n}|∇v|^{2} dv_{g}) (∫_{ℝ^{n}} |u|^{N} dv_{g})^{\frac{2}{N}}}{∫_{ℝ^{n}} |u|^{N−2}v^{2} dv_{g}}

4.2. Proof of theorem 4.1. The functional

G(u, v) := \frac{(∫_{M} c_{n}|∇v|^{2} dv_{g} + B_{0}(M, g) ∫_{M} v^{2} dv_{g}) (∫_{M} u^{N} dv_{g})^{\frac{2}{N}}}{∫_{M} u^{N−2}v^{2} dv_{g}}

is continuous on L_{+}^{N}(M) × (H_{1}^{2}(M) \ {0}). As a consequence I(u, V) := sup_{v∈V \{0}} G(u, v) depends continuously on u ∈ L_{+}^{N}(M) and V ∈ Gr_{2}^{u}(H_{1}^{2}(M)). Thus, in order to show the theorem it is sufficient to show that I(u, V) ≥ 2^{2/n}μ_{1}(𝕊^{n}) for all smooth u > 0 and V ∈ Gr_{2}(C^{∞}(M)). Without loss of generality, we can assume

∫_{M} u^{N} dv_{g} = 1.  (19)

The operator v ↦ P (v) := c_{n}u^{\frac{2−N}{2}} Δ(u^{\frac{2−N}{2}} v) + B_{0}(M, g)u^{2−N}v is an elliptic operator on M, and P is self-adjoint with respect to the L^{2}-scalar product. Hence, P has discrete spectrum λ_{1} ≤ λ_{2} ≤ … and the corresponding eigenfunctions φ_{1}, φ_{2}, … are smooth. Setting v_{i} := u^{\frac{2−N}{2}} φ_{i} we obtain

(c_{n}Δ + B_{0}) (v_{i}) = λ_{i}u^{N−2}v_{i}
∫ u^{N−2}v_{i}v_{j} dv_{g} = 0 if λ_{i} ≠ λ_{j}.

The maximum principle implies that an eigenfunction to the smallest eigenvalue λ_{1} has no zeroes. Hence λ_{1} < λ_{2}, and we can assume v_{1} > 0.

We define w_{+} := a_{+} sup(0, v_{2}) and w_{−} := a_{−} sup(0, −v_{2}), where we choose a_{+}, a_{−} > 0 such that

∫_{M} u^{N−2}w_{−}^{2} dv_{g} = ∫_{M} u^{N−2}w_{+}^{2} dv_{g} = 1.

We let Ω_{−} = {w < 0} and Ω_{+} = {w ≥ 0}. By Hölder inequality,

\begin{aligned}2 & = & ∫_{M} u^{N−2}w_{−}^{2} dv_{g} + ∫_{M} u^{N−2}w_{+}^{2} dv_{g} \\ & ≤ & (∫_{Ω_{−}} u^{N} dv_{g})^{\frac{N−2}{N}} (∫_{M} w_{−}^{N} dv_{g})^{\frac{2}{N}} + (∫_{Ω_{+}} u^{N} dv_{g})^{\frac{N−2}{N}} (∫_{M} w_{+}^{N} dv_{g})^{\frac{2}{N}} . \end{aligned}

Using the sharp Sobolev inequality (S), we get that

2μ_{1}(𝕊^{n}) ≤ (∫_{Ω_{−}} u^{N} dv_{g})^{\frac{N−2}{N}} ∫_{M} w_{−}u^{\frac{N−2}{2}} P (u^{\frac{N−2}{2}} w_{−}) dv_{g}  (20)

+ (∫_{Ω_{+}} u^{N} dv_{g})^{\frac{N−2}{N}} ∫_{M} w_{+}u^{\frac{N−2}{2}} P (u^{\frac{N−2}{2}} w_{+}) dv_{g}  (21)

Since w_{−} resp. w_{+} are some multiples of w on Ω_{−} resp. Ω_{+}, they satisfy the same equation as w. Hence, we get that

\begin{aligned}2 & = & μ_{1}(𝕊^{n})^{−1}λ_{2} ((∫_{Ω_{−}} u^{N} dv_{g})^{\frac{N−2}{N}} ∫_{M} u^{N−2}w_{−}^{2} dv_{g} + (∫_{Ω_{+}} u^{N} dv_{g})^{\frac{N−2}{N}} ∫_{M} u^{N−2}w_{+}^{2} dv_{g}) \\ & = & μ_{1}(𝕊^{n})^{−1}λ_{2} ((∫_{Ω_{−}} u^{N} dv_{g})^{\frac{N−2}{N}} + (∫_{Ω_{+}} u^{N} dv_{g})^{\frac{N−2}{N}} ) . \end{aligned}

Now, for any real non-negative numbers a, b ≥ 0, the Hölder inequality yields

a + b ≤ 2^{\frac{2}{N}} (a^{\frac{N}{N−2}} + b^{\frac{N}{N−2}} )^{\frac{N−2}{N}}

We apply this inequality with a = (∫_{Ω_{−}} u^{N} dv_{g})^{\frac{N−2}{N}} and b = (∫_{Ω_{+}} u^{N} dv_{g})^{\frac{N−2}{N}}. Using (19), we obtain

2 ≤ 2^{\frac{2}{N}} μ_{1}(𝕊^{n})^{−1}λ_{2} (∫_{Ω_{−}} u^{N} dv_{g} + ∫_{Ω_{+}} u^{N} dv_{g}) = 2^{\frac{2}{N}} μ_{1}(𝕊^{n})^{−1}λ_{2}.

We obtain λ_{2} ≥ 2^{\frac{2}{n}} μ(𝕊^{n}). Since λ_{2} = I(u, span(v_{1}, v_{2})), this ends the proof of Theorem 4.1.

4.3. Proof of Corollaries 4.2 and 4.3. It is well known that B_{0}(𝕊^{n}) equals to the scalar curvature of 𝕊^{n}, i.e. B_{0}(𝕊^{n}) = n(n − 1). Replacing B_{0}(𝕊^{n}) by its value and taking the infimum over u, V, the right hand term of inequality (S_{1}) is exactly the variational characterization of μ_{2}(𝕊^{n}) (see equation (2)). This proves that μ_{2}(𝕊^{n}) ≥ 2^{2/n}μ_{1}(𝕊^{n}). Corollary 4.2 then follows from Theorem 5.4. Since ℝ^{n} is conformal to 𝕊^{n} \ {p} (p is any point of 𝕊^{n}), we can use the conformal invariance to prove Corollary 4.3.

5. Some properties of μ_{2}(M, g)

5.1. Is μ_{2}(M, g) attained? Let (M, g) be an n-dimensional compact Riemannian manifold. The Yamabe problem shows that μ_{1}(M, g) is attained by a metric g̃ conformal to g. Some questions arise naturally concerning μ_{2}(M, g):

1- Is μ_{2}(M, g) attained by a metric?

2- Is it possible that μ_{2}(M, g) is attained by a generalized metric?

In this section, we give answers to these questions. The first result we prove is the following:

Proposition 5.1. Let 𝕊^{n} ˙∪𝕊^{n} be the disjoint union of two copies of the sphere equipped with their standard metric. Then, μ_{2}(𝕊^{n} ˙∪𝕊^{n}) = 2^{2/n}μ_{1}(𝕊^{n}) and it is attained by the canonical metric.

Proof. One computes

λ_{2}(𝕊^{n} ˙∪𝕊^{n}) Vol(𝕊^{n} ˙∪𝕊^{n})^{2/n} = 2^{2/n}λ_{1}(𝕊^{n}) Vol(𝕊^{n})^{2/n} = 2^{2/n}μ_{1}(𝕊^{n}).

Hence μ_{2}(𝕊^{n} ˙∪𝕊^{n}) ≤ 2^{2/n}μ_{1}(𝕊^{n}) follows.

Now, let g̃ be an arbitrary smooth metric on S^{n} ˙∪S^{n}. We write S_{1}^{n} for the first S^{n} and S_{2}^{n} for the second S^{n}. Then λ_{2}(S^{n} ˙∪S^{n}, g̃) is the minimum of λ_{2}(S_{1}^{n}, g̃), λ_{2}(S_{2}^{n}, g̃) and max{λ_{1}(S_{1}^{n}, g̃), λ_{1}(S_{2}^{n}, g̃)}.

It follows from Corollary 4.2 that

λ_{2}(S_{1}^{n}, g̃) Vol(S^{n} ˙∪S^{n}, g̃)^{2/n} ≥ λ_{2}(S_{1}^{n}, g̃) Vol(S_{1}^{n}, g̃)^{2/n} ≥ 2^{2/n}μ_{1}(𝕊^{n}),

and obviously we have the same for λ_{2}(S_{2}^{n}, g̃).

Summing

λ_{1}(S_{i}^{n}, g̃)^{n/2} ≥ μ_{1}(𝕊^{n})^{n/2}Vol(S_{i}^{n}, g̃)

over i ∈ {1, 2}, we obtain the remaining inequality

max{λ_{1}(S_{1}^{n}, g̃), λ_{1}(S_{2}^{n}, g̃)} Vol(S^{n} ˙∪S^{n}, g̃)^{2/n} ≥ 2^{2/n}μ_{1}(𝕊^{n}),

and the proposition is proved. □

Question 1 is solved by the following result.

Proposition 5.2. If M is connected, then μ_{2}(M, g) cannot be attained by a metric.

Indeed, otherwise by Theorem 3.4, we would have that u = |w| and hence u cannot be positive. Theorem 1.4 and the following result answer Question 2.

Proposition 5.3. The invariant μ_{2}(𝕊^{n}) is not attained by a generalized metric.

This proposition immediately follows from Proposition 5.6.

5.2. Some bounds of μ_{2}(M, g). At first, we give an upper bound for μ_{2}(M, g).

Theorem 5.4. Let (M, g) be an n-dimensional compact Riemannian manifold with μ_{1}(M, g) ≥ 00. Then,

μ_{2}(M, g) ≤ (μ_{1}(M, g)^{\frac{n}{2}} + μ_{1}(𝕊^{n})^{\frac{n}{2}} )^{\frac{2}{n}} .  (22)

This inequality is strict in the following cases:

• μ_{1}(M, g) > 0, (M, g) is not locally conformally flat and n ≥ 11;

• μ_{1}(M, g) = 0, (M, g) is not locally conformally flat and n ≥ 9.

From the solution of the Yamabe problem by Aubin and Schoen [Aub76, Sch84] we know that if (M, g) is not conformally equivalent to 𝕊^{n}, then μ_{1}(M, g) < μ_{1}(𝕊^{N}). Hence, (22) implies the following corollary.

Corollary 5.5. Let (M, g) be an n-dimensional compact connected Riemannian manifold whose Yamabe invariant is non-negative. Then μ_{2}(M, g) ≤ μ_{2}(𝕊^{n}) with inequality if and only if (M, g) is conformally diffeomorphic to the sphere 𝕊^{n}.

These inequalities are very important, because they can be used to avoid concentration of minimizing sequences for μ_{2}, in a way which is similar to the resolution of the Yamabe problem.

The following proposition gives a lower bound for μ_{2}.

Proposition 5.6. Let (M, g) be a n-dimensional compact Riemannian manifold whose Yamabe invariant is non-negative. Then,

μ_{2}(M, g) ≥ 2^{\frac{2}{n}} μ_{1}(M, g).  (23)

Moreover, if M is connected and if μ_{2}(M, g) is attained by a generalized metric, then this inequality is strict.

When μ_{1}(M, g) = 0, inequality (23) is trivial. If μ_{1}(M, g) > 0, by a possible chande of metric in the conformal class, we can assume that the scalar curvature is positive. The proof of inequality (23) is exactly the same as the one of Theorem 4.1. We just have to replace B_{0}(M, g) by S_{g}. Moreover, if M were connected and if μ_{2}(M, g) were attained by a generalized metric, then inequality (20) would be an equality and we would have that w_{+} or w_{−} is a function for which equality in the Sobolev inequality (S) is attained. By the maximum principle, we would get that w_{+} or w_{−} is positive on M which is impossible.

5.2.1. Proof of theorem 5.4.

Lemma 5.7. For any α > 2, there is a C > 0 such that

|a + b|^{α} ≤ a^{α} + b^{α} + C(a^{α−1}b + ab^{α−1})

for all a, b > 0.

Proof of Lemma 5.7. Without loss of generality, we can assume that a = 1. Then we set for x > 0,

f(x) = \frac{|1 + x|^{α} − (1 + x^{α})}{x^{α−1} + x}.

One checks that lim_{x→0} f(x) = lim_{x→+∞} f(x) = α. Since f is continuous, f is bounded by a constant C on ℝ_{+}. Clearly, this constant is the desired C in inequality of Lemma 5.7.

Proof of Theorem 5.4. For u ∈ H_{1}^{2}(M) \ {0} let

Y (u) = \frac{∫_{M} c_{n}|∇u|^{2} + S_{g}u^{2} dv_{g}}{(∫_{M} |u|^{N} dv_{g})^{\frac{2}{N}}}

be the Yamabe functional of M. The solution of the Yamabe problem provides the existence of a smooth positive minimizer v of Y, and we can assume

∫_{M} v^{N} dv_{g} = 1.  (24)

Then, v satisfies the Yamabe equation

L_{g}v = μ_{1}(M, g)v^{N−1}.  (25)

Let x_{0} ∈ M be fixed and choose a system (x_{1}, ⋯, x_{n}) of normal coordinates at x_{0}. We note r = dist_{g}(x_{0}, .). If δ > 0 is a small fixed number, let η be a smooth cut-off function such that 0 ≤ η ≤ 1, η(B(x_{0}, δ)) = {1} and η(M \ B(x_{0}, 2δ) = {0}, |∇η| ≤ 2/δ. Then, we can define for all ε > 0

v_{ε} = C_{ε}η(ε + r^{2})^{\frac{2−n}{n}} .

where C_{ε} > 0 is such that

∫_{M} v_{ε}^{N} dv_{g} = 1.  (26)

By standard computations (see [Aub76])

lim_{ε→0} Y (v_{ε}) = μ_{1}(𝕊^{n}).  (27)

If (M, g) is not locally conformally flat, if g is well chosen in the conformal class and if x_{0} is well chosen in M, it was also proven in [Aub76] that there exists a constant C(M) > 0 such that

Y (v_{ε}) = | \begin{matrix}μ_{1}(𝕊^{n}) − C(M)ε^{2} + o(ε^{2}) & if n > 6 \\ μ_{1}(𝕊^{n}) − C(M)ε^{2}| ln(ε)| + o(ε^{2}| ln(ε)|) & if n = 6. \end{matrix}  (28)

Moreover, it follows from [Aub76] that

aε^{\frac{n−2}{4}} ≤ C_{ε} ≤ bε^{\frac{n−2}{4}}

where a, b > 0 are independent of ε. If p ≥ 1, standard computations made in [Aub76] show that there exist some constants c, C > 0 independent of ε such that

cα_{p,ε} ≤ ∫_{M} v_{ε}^{p} dv_{g} ≤ Cα_{p,ε}  (29)

where

α_{p,ε} = | \begin{matrix}ε^{\frac{2n−(n−2)p}{4}} & if & p > \frac{n}{n−2}; \\ | ln(ε)|ε^{\frac{n}{4}} & if & p = \frac{n}{n−2}; \\ ε^{\frac{(n−2)p}{4}} & if & p < \frac{n}{n−2} \end{matrix}

Since the large inequality if easier to obtain, we only prove strict inequality. Assume first that μ_{1}(M, g) > 0, that (M, g) is not locally conformally flat and that n ≥ 11. We set,

u_{ε} = Y (v_{ε})^{\frac{1}{N−2}} v_{ε} + μ_{1}(M, g)^{\frac{1}{N−2}} v.

Let us derive estimates for F (u_{ε}, λv_{ε} + μv)). Let (λ, μ) ∈ ℝ^{2} \ {(0, 0)}. Using (24), (26) and the equation (25) of v, we get that

F (u_{ε}, λv_{ε} + μv) = \frac{λ^{2} ∫_{M} v_{ε}L_{g}(v_{ε}) dv_{g} + μ^{2} ∫_{M} vL_{g}(v) dv_{g} + 2λμ ∫_{M} v_{ε}L_{g}v dv_{g}}{λ^{2} ∫_{M} |u_{ε}|^{N−2}(λv_{ε} + μv)^{2} dv_{g}} (∫_{M} u_{ε}^{N} dv_{g})^{\frac{2}{n}} .

= \frac{λ^{2}Y (v_{ε}) + μ^{2}μ_{1}(M, g) + 2λμμ_{1}(M, g) ∫_{M} |v|^{N−2}vv_{ε} dv_{g}}{λ^{2} ∫_{M} |u_{ε}|^{N−2}v_{ε}^{2} dv_{g} + μ^{2} ∫_{M} |u_{ε}|^{N−2}v^{2} dv_{g} + 2λμ ∫_{M} |u_{ε}|^{N−2}vv_{ε} dv_{g}} (∫_{M} u_{ε}^{N} dv_{g})^{\frac{2}{n}} .  (30)

Using the definition of u_{ε}

λ^{2} ∫_{M} |u_{ε}|^{N−2}v_{ε}^{2} dv_{g} + μ^{2} ∫_{M} |u_{ε}|^{N−2}v^{2} dv_{g} + 2λμ ∫_{M} |u_{ε}|^{N−2}vv_{ε} dv_{g}

≥ λ^{2}Y (v_{ε}) ∫_{M} |v_{ε}|^{N} dv_{g} + μ^{2}μ_{1}(M, g) ∫_{M} |v|^{N} dv_{g} + 2λμ ∫_{M} |u_{ε}|^{N−2}vv_{ε} dv_{g}
= λ^{2}Y (v_{ε}) + μ^{2}μ_{1}(M, g) + 2λμ ∫_{M} |u_{ε}|^{N−2}vv_{ε} dv_{g}.

If λμ ≥ 0, we have

2λμ ∫_{M} |u_{ε}|^{N−2}vv_{ε} dv_{g} ≥ 2λμμ_{1}(M, g) ∫_{M} v^{N−2}v_{ε} dv_{g}.

This implies that

\frac{λ^{2}Y (v_{ε}) + μ^{2}μ_{1}(M, g) + 2λμμ_{1}(M, g) ∫_{M} |v|^{N−2}vv_{ε} dv_{g}}{λ^{2} ∫_{M} |u_{ε}|^{N−2}v_{ε}^{2} dv_{g} + μ^{2} ∫_{M} |u_{ε}|^{N−2}v^{2} dv_{g} + 2λμ ∫_{M} |u_{ε}|^{N−2}vv_{ε} dv_{g}} ≤ 1.

If λμ < 0 then, we write that since N − 2 ∈]0, 1[,

|u_{ε}|^{N−2} ≤ Y (v_{ε})v_{ε}^{N−2} + μ_{1}(M, g)v^{N−2}.

We obtain that

λ^{2} ∫_{M} |u_{ε}|^{N−2}v_{ε}^{2} dv_{g} + μ^{2} ∫_{M} |u_{ε}|^{N−2}v^{2} dv_{g} + 2λμ ∫_{M} |u_{ε}|^{N−2}vv_{ε} dv_{g}
≥ λ^{2}Y (v_{ε}) + μ^{2}μ_{1}(M, g) − C (∫_{M} v^{N−1}v_{ε} dv_{g} + ∫_{M} v_{ε}^{N−1}v dv_{g}) .

where C > 0 is as in in the following a positive real number independent of ε. Together with (29), we get that

λ^{2} ∫_{M} |u_{ε}|^{N−2}v_{ε}^{2} dv_{g} + μ^{2} ∫_{M} |u_{ε}|^{N−2}v^{2} dv_{g} + 2λμ ∫_{M} |u_{ε}|^{N−2}vv_{ε} dv_{g} ≥ λ^{2}Y (v_{ε}) + μ^{2}μ_{1}(M, g) + O(ε^{\frac{n−2}{4}} ).

It follows that

sup_{(λ,μ)∈ℝ^{2}\\{(0,0)\}} \frac{λ^{2}Y (v_{ε}) + μ^{2}μ_{1}(M, g) + 2λμμ_{1}(M, g) ∫_{M} |v|^{N−2}vv_{ε} dv_{g}}{λ^{2} ∫_{M} |u_{ε}|^{N−2}v_{ε}^{2} dv_{g} + μ^{2} ∫_{M} |u_{ε}|^{N−2}v^{2} dv_{g} + 2λμ ∫_{M} |u_{ε}|^{N−2}vv_{ε} dv_{g}} ≤ 1 + O(ε^{\frac{n−2}{4}} ). (31)

By Lemma 5.7,

∫_{M} u_{ε}^{N} dv_{g} ≤ (Y (u_{ε}))^{\frac{n}{2}} ∫_{M} v_{ε}^{N} dv_{g} + μ_{1}(M, g)^{\frac{n}{2}} ∫_{M} v^{N} dv_{g}
+C (∫_{M} v^{N−1}v_{ε} dv_{g} + ∫_{M} v_{ε}^{N−1}v dv_{g}) .

By (24), (26), (28) and (29), we obtain

(∫_{M} u_{ε}^{N} dv_{g})^{\frac{2}{n}} ≤ (μ_{1}(M, g)^{\frac{n}{2}} + μ_{1}(𝕊^{n})^{\frac{n}{2}} )^{\frac{2}{n}} − Cε^{2} + O(ε^{\frac{n−2}{4}} ) + o(ε^{2}).  (32)

Since \frac{n−2}{4} > 2, we get from (31) and (32) that for ε small enough

μ_{2}(M, g) ≤ sup_{(λ,μ)∈ℝ^{2}\{(0,0)}} F (u_{ε}, λv_{ε} + μv)
≤ (μ_{1}(M, g)^{\frac{n}{2}} + μ_{1}(𝕊^{n})^{\frac{n}{2}} )^{\frac{2}{n}} − Cε^{2} + O(ε^{\frac{n−2}{4}} ) + o(ε^{2}) < (μ_{1}(M, g)^{\frac{n}{2}} + μ_{1}(𝕊^{n})^{\frac{n}{2}} )^{\frac{2}{n}} .

This proves Theorem 5.4 if μ_{1}(M, g) > 0.

Now, we assume that μ_{1}(M, g) = 0, that (M, g) is not locally conformally flat and that n ≥ 9. For more simplicity, We set u_{ε} = v_{ε} instead of u_{ε} = Y (v_{ε})^{\frac{n−2}{4}} v_{ε} as above. We proceed exactly as in the case μ_{1}(M, g) > 0. We obtain that for (λ, μ) ∈ ℝ^{2} \ {(0, 0)}

F (u_{ε}, λv_{ε} + μv) = \frac{λ^{2}Y (v_{ε})}{λ^{2} ∫_{M} v_{ε}^{N} dv_{g} + μ^{2} ∫_{M} |v_{ε}|^{N−2}v^{2} dv_{g} + 2λμ ∫_{M} |v_{ε}|^{N−1}v dv_{g}} (∫_{M} v_{ε}^{N} dv_{g})^{\frac{2}{n}}
= \frac{λ^{2}Y (v_{ε})}{λ^{2} + μ^{2} ∫_{M} |v_{ε}|^{N−2}v^{2} dv_{g} + 2λμ ∫_{M} |v_{ε}|^{N−1}v dv_{g}} .

Let λ_{ε}, μ_{ε} be such that λ_{ε}^{2} + μ_{ε}^{2} = 1 and such that

F (u_{ε}, λ_{ε}v_{ε} + μ_{ε}v) = sup_{(λ,μ)∈ℝ^{2}\{(0,0)}}(u_{ε}, λv_{ε} + μv).

If λ_{ε} = 0, we obtain that F (u_{ε}, λ_{ε}v_{ε} + μ_{ε}v) = 0 and the theorem would be proven. Then we assume that λ_{ε} ≠ 0 and we write that

F (u_{ε}, λ_{ε}v_{ε} + μ_{ε}v) = \frac{Y (v_{ε})}{1 + 2x_{ε}b_{ε} + x_{ε}^{2}a_{ε}}

where x_{ε} = \frac{μ_{ε}}{λ_{ε}} and where, using (29)

b_{ε} = ∫_{M} v_{ε}^{N−1}v dv_{g} ∼_{ε→0} Cε^{\frac{n−2}{4}}

and

a_{ε} = ∫_{M} v_{ε}^{N−2}v^{2} dv_{g} ∼_{ε→0} Cε.

Maximizing this expression in x_{ε} and using (28), we get that

F (u_{ε}, λ_{ε}v_{ε} + μ_{ε}v) ≤ \frac{μ_{1}(𝕊^{n}) − C(M)ε^{2} + o(ε^{2})}{1 − \frac{b_{ε}^{2}}{a_{ε}}} = \frac{μ_{1}(𝕊^{n}) − C(M)ε^{2} + o(ε^{2})}{1 − O(ε^{\frac{n−4}{2}} )}.

Since n ≥ 9, \frac{n−4}{2} > 2 and we get that for ε small,

F (u_{ε}, λ_{ε}v_{ε} + μ_{ε}v) < μ_{1}(𝕊^{n}).

This proves Theorem 5.4.

6. Existence of a minimum of μ_{2}(M, g)

The aim of this section is to prove Theorem 1.4.

We study a sequence of metrics (g_{m})_{m} = (u_{m}^{N−2}g)_{m} (u_{m} > 0, u_{m} ∈ C^{∞}(M)) which minimizes the infimum in the definition of μ_{2}(M, g) i.e. a sequence of metrics such that

lim_{m} λ_{2}(g_{m})Vol(M, g_{m})^{\frac{2}{n}} = μ_{2}(M, g).

Without loss of generality, we may assume that Vol(M, g_{m}) = 1 i.e. that

∫_{M} u_{m}^{N} dv_{g} = 1.  (33)

In particular, the sequence (u_{m})_{m} is bounded in L^{N}(M) and there exists u ∈ L^{N}(M), u ≥ 0 such that u_{m} ⇀ u weakly in L^{N}(M). We are going to prove that u ≠ 0 and that the generalized metric u^{N−2}g minimizes μ_{2}(M, g). Proposition 3.2 implies the existence of v_{m}, w_{m} ∈ C^{∞}(M), v_{m} ≥ 0 such that

L_{g}v_{m} = λ_{1,m}u_{m}^{N−2}v_{m}  (34)

and

L_{g}w_{m} = λ_{2,m}u_{m}^{N−2}w_{m}.  (35)

where λ_{i,m} = λ_{i}(g_{m}) and such that

∫_{M} u_{m}^{N−2}v_{m}^{2} dv_{g} = ∫_{M} u_{m}^{N−2}w_{m}^{2} dv_{g} = 1 and ∫_{M} u_{m}^{N−2}v_{m}w_{m} dv_{g} = 0  (36)

With these notations and by (33),

lim_{m} λ_{2,m} = μ_{2}(M, g).

Moreover, by the maximum principle, v_{m} > 0. If λ_{1,m} = λ_{2,m} then w_{m} would be a minimizer of the functional associated to λ_{1,m} and by the maximum principle, we would get that w_{m} > 0. This contradicts (36). Hence, λ_{1,m} < λ_{2,m} for all m. The sequences (v_{m})_{m} and (w_{m})_{m} are bounded in H_{1}^{2}(M). We can find v, w ∈ H_{1}^{2}(M), v ≥ 0 such that v_{m} (resp. w_{m}) tends to v (resp. w) weakly in H_{1}^{2}(M). Together with the weak convergence of the (u_{m})_{m} towards u in L^{N}(M), we get that in the sense of distributions

L_{g}v = μ̂_{1}u^{N−2}v  (37)

and

L_{g}w = μ_{2}(M, g) u^{N−2}w.  (38)

where μ̂_{1} = lim_{m} λ_{1,m} ≤ μ_{2}(M, g).

From what we know until now, it is not clear whether v and w are linearly independent, and even if they are, their restrictions to the set M \ u^{−1}(0) might be linearly dependent.

It will take a certain effort to prove the following claim.

Claim 6.1. The functions u^{\frac{N−2}{2}} v and u^{\frac{N−2}{2}} w are linearly independent.

Once the claim is proved, we have span(v, w) ∈ Gr_{2}^{u}(H_{1}^{2}(M)), and this implies that

sup_{(λ,μ)≠(0,0)} F (u, λv + μw) = μ_{2}(M, g).

Hence, by equations (37) and (38), the generalized metric u^{N−2}g minimizes μ_{2}(M, g), i.e. Theorem 1.4 is proved.

The first step in the proof of the claim is an estimate that avoids concentration of w_{m} and v_{m}.

Step 1. Let x ∈ M and ε ∈ ]0, \frac{N−2}{2}[. We choose a cut-off function η ∈ C^{∞} such that 0 ≤ η ≤ 1, η(B_{x}(δ)) ≡ 1 (where δ > 0 is a small number) and η(M \ B_{x}(2δ)) ≡ 0, |∇η| ≤ 2/δ. We define W_{m} = η|w_{m}|^{ε}w_{m}. Then, we have

(∫_{M} |W_{m}|^{N} dv_{g})^{\frac{2}{N}} ≤ μ_{2}(M, g)(1 − α_{ε})^{−1}μ_{1}(𝕊^{n})^{−1}(∫_{B_{x}(2δ)} u_{m}^{N})^{\frac{2}{n}} (∫_{M} |W_{m}|^{N} dv_{g})^{\frac{2}{N}} + C_{δ}. (39)

where C_{δ} is a constant that may depend on δ but not on ε and where lim_{ε→0} α_{ε} = 0. Moreover, the same conclusion is true with V_{m} = η|v_{m}|^{ε}v_{m} instead of W_{m}.

The proof uses classical methods. We will explain the proof for W_{m}. The proof for V_{m} uses exactly the same arguments.

At first, we differentiate the definition of W and obtain

|∇W_{m}|^{2} ≥ |∇(|w_{m}|^{ε}w_{m})|^{2}η^{2} − (2|∇η| |w_{m}|^{1+ε}) (|∇(|w_{m}|^{ε}w_{m})|η) + |∇η|^{2} |w_{m}|^{2+2ε}
≥ |∇(|w_{m}|^{ε}w_{m})|^{2}η^{2} − (\frac{1}{2}|∇(|w_{m}|^{ε}w_{m})|^{2}η^{2} + 2|∇η|^{2}|w_{m}|^{2+2ε}) + |∇η|^{2}|w_{m}|^{2+2ε}

This leads to

η^{2}|∇(|w_{m}|^{ε}w_{m})|^{2} ≤ 2|∇W_{m}|^{2} + 2|∇η|^{2}|w_{m}|^{2+2ε}.  (40)

Now, we want to derive lower bound for

(∇(η^{2}|w_{m}|^{2ε}w_{m}), ∇w_{m}) = |∇W_{m}|^{2} − |∇(η|w_{m}|^{ε})|^{2} |w_{m}|^{2}  (41)

For the second summand on the right hand side in (41) we have the bound

|∇(η|w_{m}|^{ε})|^{2}|w_{m}|^{2} = |∇η|^{2}|w_{m}|^{2+2ε} + 2(∇η, ∇|w_{m}|^{ε}) η|w_{m}|^{2+ε} + η^{2}|∇(|w_{m}|^{ε})|^{2}w_{m}^{2}
≤ 2|∇η|^{2}|w_{m}|^{2+2ε} + 2η^{2}|∇(|w_{m}|^{ε})|^{2}w_{m}^{2}
≤ 2|∇η|^{2}|w_{m}|^{2+2ε} + \frac{2η^{2}ε^{2}}{(1 + ε)^{2}} |∇(|w_{m}|^{ε}w_{m})|^{2}
≤ (2 + \frac{4ε^{2}}{(1 + ε)^{2}} )|∇η|^{2}|w_{m}|^{2+2ε} + \frac{4ε^{2}}{(1 + ε)^{2}} |∇W_{m}|^{2}.

Here, we used (40) in the last line. Coming back to (41), we obtain that

(∇(η^{2}|w_{m}|^{2ε}w_{m}), ∇w_{m}) ≥ (1 − α_{ε})|∇W_{m}|^{2} − C|∇η|^{2}|w_{m}|^{2+2ε}.

where α_{ε} → 0 when ε → 0 and where C > 0 is a constant independent of ε. This relations shows that

∫_{M} η^{2}|w_{m}|^{2ε}w_{m}L_{g}(w_{m}) dv_{g} ≥ (1 − α_{ε}) ∫_{M} c_{n}|∇W_{m}|^{2} dv_{g} − C ∫_{M} |∇η|^{2}|w_{m}|^{2+2ε} dv_{g} + min Scal ∫ W_{m}^{2} dv_{g}.

Now, since ε < \frac{N−2}{2},the sequence (w_{m})_{m} is bounded in L^{2+2ε}(M) (and hence the sequence (W_{m})_{m} is bounded in L^{2}(M)). As a consequence, there exists a constant C_{δ} possibly depending on δ but not on ε, and such that

∫_{M} η^{2}|w_{m}|^{2ε}w_{m}L_{g}(w_{m}) dv_{g} ≥ (1 − α_{ε}) ∫_{M} (c_{n}|∇W_{m}|^{2} + B_{0}(M, g)W_{m}^{2}) dv_{g} − C_{δ}.  (42)

Using equation (35) in the left hand side of (42) and applying Sobolev inequality (S) to the right hand side, we get that

μ_{2}(M, g) ∫_{M} u_{m}^{N−2}W_{m}^{2} dv_{g} ≥ (1 − α_{ε})μ_{1}(𝕊^{n})(∫_{M} |W_{m}|^{N} dv_{g})^{\frac{2}{N}} − C_{δ}.

By the Hölder inequality, we obtain

(∫_{M} |W_{m}|^{N} dv_{g})^{\frac{2}{N}} ≤ μ_{2}(M, g)(1 − α_{ε})^{−1}μ_{1}(𝕊^{n})^{−1}(∫_{B_{x}(2δ)} u_{m}^{N})^{\frac{2}{n}} (∫_{M} |W_{m}|^{N} dv_{g})^{\frac{2}{N}} + C_{δ}.

This ends the proof of the step.

Step 2. If μ_{2}(M, g) < μ_{1}(𝕊^{n}), then the generalized metric u^{N−2}g minimizes μ_{2}(M, g).

From (39), and the fact μ_{2}(M, g) < μ_{1}(𝕊^{n}), we get that for ε small enough, there exists a constant K < 1 such that

(∫_{M} |W_{m}|^{N} dv_{g})^{\frac{2}{N}} ≤ K(∫_{B_{x}(2δ)} u_{m}^{N})^{\frac{2}{n}} (∫_{M} |W_{m}|^{N} dv_{g})^{\frac{2}{N}} + C_{δ}.

Since ∫_{B_{x}(2δ)} u_{m}^{N} ≤ 1, the sequence ∫_{M} |W_{m}|^{N} dv_{g} is bounded. This implies that (w_{m})_{m} is bounded in L^{N+ε}(B_{x}(δ)) and since x is arbitrary in L^{N+ε}(M). Weak convergences w_{m} → w in H_{1}^{2}(M) implies strong convergence w_{m} → w in L^{N−ε}(M). The Hölder inequality yields then strong convergence in L^{N}(M). After passing to a subsequence we obtain that (w_{m})_{m} tends to w strongly in L^{N}(M). This implies that we can pass to the limit in (36) and hence that u^{\frac{N−2}{2}} v and u^{\frac{N−2}{2}} w are linearly independent. The claim follows in this case.

In the following, we assume that μ_{1}(M, g) > 0 and that

μ_{2}(M, g) < (μ_{1}(M, g)^{\frac{n}{2}} + μ_{1}(𝕊^{n})^{\frac{n}{2}} )^{\frac{2}{n}} .

We define the set of concentration points

Ω = \{x ∈ M | ∀δ > 0, lim sup_{m} ∫_{B_{x}(δ)} u_{m}^{N} dv_{g} > \frac{1}{2}\}.

Since ∫_{M} u_{m}^{N} dv_{g} = 1, we can assume — after passing to a subsequence — that Ω contains at most one point.

We now prove that:

Step 3. Let U be an open set such that \overline{U} ⊂ M \ Ω. Then, the sequence (v_{m})_{m} (and (w_{m})_{m} resp.) converges towards v (and w resp.) strongly in H_{1}^{2}(\overline{U}).

Without loss of generality, we prove the result only for w. For any x ∈ M \ Ω we can find δ > 0 with

lim sup_{m} ∫_{B_{x}(2δ)} u_{m}^{N} dv_{g} ≤ \frac{1}{2}.

Using μ_{2}(M, g) < (μ_{1}(M, g)^{\frac{n}{2}} + μ_{1}(𝕊^{n})^{\frac{n}{2}} )^{\frac{2}{n}} ≤ 2^{\frac{2}{n}} μ_{1}(𝕊^{n}) we obtain for a small ε > 0

μ_{2}(M, g)(1 − α_{ε})^{−1}μ_{1}(𝕊^{n})^{−1}(∫_{B_{x}(2δ)} u_{m}^{N})^{\frac{2}{n}} ≤ K < 1

for almost all m. Together with inequality (39), this proves that ∫_{M} |W_{m}|^{N} dv_{g} is bounded. This implies that (w_{m})_{m} is bounded in L^{N+ε}(B_{x}(δ)). As in last step, this proves that up to a subsequence, (w_{m})_{m} tends to w strongly in L^{N}(U). Using equation (35) and (38), we easily obtain that

lim_{m} ∫_{U} |∇w_{m}|^{2}dv_{g} = ∫_{U} |∇w|^{2}dv_{g}.

Together with the weak convergence of (w_{m})_{m} to w, this proves the step.

Now, we set for all m,

S_{m} = {λv_{m} + μw_{m}|λ^{2} + μ^{2} = 1} and S = {λv + μw|λ^{2} + μ^{2} = 1}.

Step 4. There exists a sequence (\overline{w}_{m})_{m} (\overline{w}_{m} ∈ S_{m}) and \overline{w} ∈ S such that \overline{w}_{m} tends to \overline{w} strongly in H_{1}^{2}(M).

By theorem 4.1, there exists λ_{m}, μ_{m} such that λ_{m}^{2} + μ_{m}^{2} = 1 and such that

2^{2/n}μ_{1}(𝕊^{n}) ∫_{M} u_{m}^{N−2}(λ_{m}(v_{m} − v) + μ_{m}(w_{m} − w))^{2} dv_{g}
≤ ∫_{M} c_{n}|∇(λ_{m}(v_{m} − v) + μ_{m}(w_{m} − w))|^{2} dv_{g}  (43)
+ ∫_{M} B_{0}(M, g)(λ_{m}(v_{m} − v) + μ_{m}(w_{m} − w))^{2} dv_{g}

Up to a subsequence, there exists λ, μ such that λ^{2}+μ^{2} = 1 and such that lim_{m} λ_{m} = λ and lim_{m} μ_{m} = μ. We set \overline{w}_{m} = λ_{m}v_{m} + μ_{m}w_{m} ∈ S_{m} and \overline{w} = λv + μw. Then, \overline{w}_{m} tends to \overline{w} weakly in H_{1}^{2}(M). A first remark is that by strong convergence in L^{2}(M)

lim_{m} ∫_{M}(λ_{m}(v_{m} − v) + μ_{m}(w_{m} − w))^{2} dv_{g} = 0.  (44)

Using the weak convergence of \overline{w}_{n} to \overline{w} in H_{1}^{2}(M) and the weak convergence of u_{m} to u in L^{N}(M), it is easy to compute that

∫_{M} u_{m}^{N−2}(λ_{m}(v_{m} − v) + μ_{m}(w_{m} − w))^{2} dv_{g} = ∫_{M} u_{m}^{N−2}\overline{w}_{m}^{2} dv_{g} − ∫_{M} u^{N−2}\overline{w}^{2} dv_{g} + o(1)  (45)

and that

∫_{M} c_{n}|∇(λ_{m}(v_{m} − v) + μ_{m}(w_{m} − w))|^{2} dv_{g} = λ^{2} (∫_{M} c_{n}|∇v_{m}|^{2} dv_{g} − ∫_{M} c_{n}|∇v|^{2} dv_{g})
+ μ^{2} (∫_{M} c_{n}|∇w_{m}|^{2} dv_{g} − ∫_{M} c_{n}|∇w|^{2} dv_{g})
+ 2λμ (∫_{M} c_{n}(∇v_{m}, ∇w_{m}) dv_{g} − ∫_{M} c_{n}(∇v, ∇w) dv_{g}) + o(1).

Using equations (34), (35), (37) and (38), we get that

∫_{M} c_{n}|∇(λ_{m}(v_{m} − v) + μ_{m}(w_{m} − w))|^{2} dv_{g} = λ^{2} μ̂_{1} (∫_{M} u_{m}^{N−2}v_{m}^{2} dv_{g} − ∫_{M} u^{N−2}v^{2}) dv_{g})
+ μ^{2} μ_{2}(M, g) (∫_{M} u_{m}^{N−2}w_{m}^{2} dv_{g} − ∫_{M} u^{N−2}w^{2}) dv_{g})
+ 2λμ μ_{2}(M, g) (∫_{M} u_{m}^{N−2}v_{m}w_{m} dv_{g} − ∫_{M} u^{N−2}vw dv_{g}) + o(1).

Since μ̂_{1} ≤ μ_{2}(M, g) and since, by weak convergence

lim inf_{m} ∫_{M} u_{m}^{N−2}v_{m}^{2} dv_{g} − ∫_{M} u^{N−2}v^{2}) dv_{g} ≥ 0,

we get that

∫_{M} c_{n}|∇(λ_{m}(v_{m} − v) + μ_{m}(w_{m} − w))|^{2} dv_{g} ≤ λ^{2} μ_{2}(M, g) (∫_{M} u_{m}^{N−2}v_{m}^{2} dv_{g} − ∫_{M} u^{N−2}v^{2} dv_{g})
+ μ^{2} μ_{2}(M, g) (∫_{M} u_{m}^{N−2}w_{m}^{2} dv_{g} − ∫_{M} u^{N−2}w^{2} dv_{g})
+ 2λμ μ_{2}(M, g) (∫_{M} u_{m}^{N−2}v_{m}w_{m} dv_{g} − ∫_{M} u^{N−2}vw dv_{g}) ,

and hence,

∫_{M} c_{n}|∇(λ_{m}(v_{m} − v) + μ_{m}(w_{m} − w))|^{2} dv_{g} ≤ μ_{2}(M, g) (∫_{M} u_{m}^{N−2}\overline{w}_{m}^{2} dv_{g} − ∫_{M} u^{N−2}\overline{w}^{2} dv_{g}) + o(1). (46)

Together with (43), (44) and (45), we obtain that

2^{2/n}μ_{1}(𝕊^{n}) (∫_{M} u_{m}^{N−2}\overline{w}_{m}^{2} dv_{g} − ∫_{M} u^{N−2}\overline{w}^{2} dv_{g})
≤ μ_{2}(M, g) (∫_{M} u_{m}^{N−2}\overline{w}_{m}^{2} dv_{g} − ∫_{M} u^{N−2}\overline{w}^{2} dv_{g}) + o(1).

Since μ_{2}(M, g) < (μ_{1}(M, g)^{\frac{n}{2}} + μ_{1}(𝕊^{n})^{\frac{n}{2}} )^{\frac{2}{n}} ≤ 2^{\frac{2}{n}} μ_{1}(𝕊^{n}), we get that

(∫_{M} u_{m}^{N−2}\overline{w}_{m}^{2} dv_{g} − ∫_{M} u^{N−2}\overline{w}^{2} dv_{g}) ≤ K_{0} (∫_{M} u_{m}^{N−2}\overline{w}_{m}^{2} dv_{g} − ∫_{M} u^{N−2}\overline{w}^{2} dv_{g}) + o(1)

where K_{0} < 1. This implies that

1 = lim_{m} ∫_{M} u_{m}^{N−2}\overline{w}_{m}^{2} dv_{g} = ∫_{M} u^{N−2}\overline{w}^{2} dv_{g}  (47)

and hence by (46).

lim_{m} ∫_{M} c_{n}|∇(λ_{m}(v_{m} − v) + μ_{m}(w_{m} − w))|^{2} dv_{g} = 0.

The step easily follows.

As a remark, (47) implies that u^{\frac{N−2}{2}} \overline{w} ≢ 0. Now, we set \overline{v}_{m} = −μ_{m}v_{m} + λ_{m}w_{m} and \overline{v} = −μv + λw. We prove that Step 5. There exists x ∈ M such that

lim sup_{m} ∫_{B_{x}δ} u_{m}^{2}(\overline{v}_{m} − \overline{v})^{2} dv_{g} = 1

for all δ > 0.

The sequence (\overline{v}_{m})_{m} tends to \overline{v} weakly in H_{1}^{2}(M). If Ω = ∅, then we know from Step 3 that (\overline{v}_{m})_{m} tends to \overline{v} strongly in H_{1}^{2}(M), which implies ∫ u^{N−2}v̄w̄ = 0. Hence, in the case Ω = ∅, the functions u^{\frac{N−2}{2}} v̄ and u^{\frac{N−2}{2}} w̄ are linearly independent, and the claim follows. Hence, without loss of generality let Ω = {x} where x is some point of M. We assume that the claim is false, i.e. u^{\frac{N−2}{2}} v and u^{\frac{N−2}{2}} w are linearly dependent. As u^{\frac{N−2}{2}} \overline{w} ≢ 0, there exists b ∈ ℝ with u^{\frac{N−2}{2}} \overline{v} = bu^{\frac{N−2}{2}} \overline{w}. Hence,

0 = ∫_{M} u^{N−2}\overline{v}^{2} dv_{g} + b^{2} ∫_{M} u^{N−2}\overline{w}^{2} dv_{g} − 2b ∫_{M} u^{N−2}\overline{v} \overline{w} dv_{g}.

By strong convergence of (\overline{w}_{m})_{m} to \overline{w} in H_{1}^{2}(M), weak convergence of (\overline{v}_{m})_{m} to \overline{v} in H_{1}^{2}(M) and weak convergence of (u_{m})_{m} to u in L^{N}(M), we have ∫_{M} u^{N−2}\overline{w}^{2} dv_{g} = 1 and ∫_{M} u^{N−2}\overline{v} \overline{w} dv_{g} = 0. We obtain ∫_{M} u^{N−2}\overline{v}^{2} dv_{g} + b^{2} = 0. As a consequence, u^{\frac{N−2}{2}} \overline{v} ≡ 0. Let now δ > 0. We write that

∫_{B_{x}(δ)} u_{m}^{N−2}(\overline{v}_{m} − \overline{v})^{2} dv_{g} = ∫_{B_{x}(δ)} u_{m}^{N−2}\overline{v}_{m}^{2} dv_{g}
= 1 − ∫_{M\B_{x}(δ)} u_{m}^{N−2}\overline{v}_{m}^{2} dv_{g}.

By step 3,

lim_{m} ∫_{M\B_{x}(δ)} u_{m}^{N−2}\overline{v}_{m}^{2} dv_{g} = ∫_{M\B_{x}(δ)} u^{N−2}\overline{v}^{2} dv_{g} = 0.

This proves the step. Step 6. Conclusion.

THE SECOND YAMABE INVARIANT 21

Let δ > 0 be a small fixed number. In the following, o(1) denotes a sequence of real numbers which tends to 0, however we do not claim that the convergence is uniform in δ. By step 5 and the Hölder inequality,

1 = ∫_{B_{x}(δ)} u_{m}^{N−2}(\overline{v}_{m} − \overline{v})^{2} dv_{g} + o(1)
≤ (∫_{B_{x}(δ)} u_{m}^{N} dv_{g})^{\frac{2}{n}} (∫_{M} |\overline{v}_{m} − \overline{v}|^{N} dv_{g})^{\frac{2}{n}} + o(1).

Applying Sobolev inequality (S), we get that

1 ≤ (∫_{B_{x}(δ)} u_{m}^{N} dv_{g})^{\frac{2}{n}} μ_{1}(𝕊^{n})^{−1} (∫_{M} c_{n}|∇(\overline{v}_{m} − \overline{v})|^{2} dv_{g} + B_{0}(M, g) ∫_{M}(\overline{v}_{m} − \overline{v})^{2} dv_{g}) + o(1).

By strong convergence of (\overline{v}_{m} − \overline{v})_{m} to 0 in L^{2}(M),

1 ≤ (∫_{B_{x}(δ)} u_{m}^{N} dv_{g})^{\frac{2}{n}} μ_{1}(𝕊^{n})^{−1} (∫_{M} c_{n}|∇(\overline{v}_{m} − \overline{v})|^{2} + S_{g}(\overline{v}_{m} − \overline{v})^{2} dv_{g}) + o(1)

Using equations (34), (35), (37), (38) and the fact that μ̂_{1} ≤ μ_{2}(M, g), we get that

1 ≤ (∫_{B_{x}(δ)} u_{m}^{N} dv_{g})^{\frac{2}{n}} μ_{1}(𝕊^{n})^{−1}μ_{2}(M, g) ∫_{M} u_{m}^{N−2}(\overline{v}_{m} − \overline{v})^{2} dv_{g}
= (∫_{B_{x}(δ)} u_{m}^{N} dv_{g})^{\frac{2}{n}} μ_{1}(𝕊^{n})^{−1}μ_{2}(M, g).

Since μ_{2}(M, g) < (μ_{1}(M, g)^{\frac{n}{2}} + μ_{1}(𝕊^{n})^{\frac{n}{2}} )^{\frac{2}{n}}, we obtain that

∫_{B_{x}(δ)} u_{m}^{N} dv_{g} > \frac{μ_{1}(𝕊^{n})^{\frac{n}{2}}}{μ_{1}(M, g)^{\frac{n}{2}} + μ_{1}(𝕊^{n})^{\frac{n}{2}}} .

and since ∫_{M} u_{m}^{N} dv_{g} = 1,

∫_{M\B_{x}(δ)} u_{m}^{N} dv_{g} < \frac{μ_{1}(M, g)^{\frac{n}{2}}}{μ_{1}(M, g)^{\frac{n}{2}} + μ_{1}(𝕊^{n})^{\frac{n}{2}}} .  (48)

Now, we write that by strong convergence of (\overline{w}_{m})_{m} in H_{1}^{2}(M),

a_{δ} = ∫_{B_{x}(δ)} u_{m}^{N−2}\overline{w}_{m}^{2} dv_{g}
1 − a_{δ} = ∫_{M\B_{x}(δ)} u_{m}^{N−2}\overline{w}_{m}^{2} dv_{g}

where a_{δ} does not depend of m and tends to 0 when δ tends to 0. By Hölder inequality,

1 − a_{δ} ≤ (∫_{M\B_{x}(δ)} u_{m}^{N} dv_{g})^{\frac{2}{n}} (∫_{M} \overline{w}^{N} dv_{g})^{\frac{2}{n}} .

Since μ_{1}(M, g) is the minimum of Yamabe functional, we get that

1 − a_{δ} ≤ (∫_{M\B_{x}(δ)} u_{m}^{N} dv_{g})^{\frac{2}{n}} μ_{1}(M, g)^{−1} ∫_{M} (c_{n}|∇\overline{w}_{m}|^{2} + S_{g}\overline{w}_{m}^{2}) dv_{g}.

As we did for \overline{v}, we obtain

1 − a_{δ} ≤ (∫_{M\B_{x}(δ)} u_{m}^{N} dv_{g})^{\frac{2}{n}} μ_{1}(M, g)^{−1}μ_{2}(M, g) \underbrace{∫_{M} u_{m}^{N−2}\overline{w}_{m}^{2} dv_{g}}_{1}

By (48), in the limit δ → 0, this gives

μ_{2}(M, g) ≥ (μ_{1}(M, g)^{\frac{n}{2}} + μ_{1}(𝕊^{n})^{\frac{n}{2}} )^{\frac{2}{n}} .

This is false by assumption. Hence, the claim is proved, and Theorem 1.4 follows.

7. The invariant μ_{k}(M) for k ≥ 3

A natural question is: Can we do the same work for μ_{k}(M) with k ≥ 3? This problem is still open but seems to be hard. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3. Using the variational characterization of μ_{k}(M), one can check that μ_{k}(M) ≤ k^{\frac{2}{n}} μ_{1}(𝕊^{n}). It is natural to conjecture that one has equality if M is the round sphere i.e. that μ_{k}(𝕊^{n}) = k^{\frac{2}{n}} μ_{1}(𝕊^{n}). However, the following result shows that is false:

Proposition 7.1. Let n ∈ ℕ^{∗}. Then, for n ≥ 7

μ_{n+2}(𝕊^{n}) < (n + 2)^{\frac{2}{n}} μ_{1}(𝕊^{n}).

Proof: Let us study 𝕊^{n} with its natural embedding into ℝ^{n+1}. We have L_{g}(1) = n(n − 1). Hence, λ_{1}(𝕊^{n}) ≤ n(n − 1). Let also x_{i} (i ∈ [1, ⋯, n + 1]) be the canonical coordinates on ℝ^{n+1}. As one can check,

L_{g}(x_{i}) = \frac{n(n − 1)(n + 2)}{n − 2}x_{i}

and hence λ_{n+2}(𝕊^{n}) ≤ \frac{n(n−1)(n+2)}{n−2}. This shows that

μ_{n+2}(𝕊^{n}) ≤ \frac{n(n − 1)(n + 2)}{n − 2}ωn^{\frac{2}{n}} .

As one can check, for n ≥ 7

\frac{n(n − 1)(n + 2)}{n − 2}ωn^{\frac{2}{n}} < (n + 2)^{\frac{2}{n}} n(n − 1)ωn^{\frac{2}{n}} = (n + 2)^{\frac{2}{n}} μ_{1}(𝕊^{n}).

This ends the proof of Proposition 7.1.

8. The case of manifolds whose Yamabe invariant is negative

We let (M, g) be a compact Riemannian manifold of dimension n ≥ 3. Then, we have:

Proposition 8.1. Let k ∈ ℕ^{∗}. Assume that μ_{k}(M, g) < 0. Then, μ_{k}(M, g) = −∞.

Proof: After a possible change of metric in the conformal class, we can assume that λ_{k}(g) < 0. This implies that we can find some smooth functions v_{1}, ⋯, v_{k} satisfying

L_{g}v_{i} = λ_{i}(g)v_{i}

for all i ∈ {1, ⋯, k} and such that

∫_{M} v_{i}v_{j}dv_{g} = 0

for all i, j ∈ {1, ⋯, k}, i ≠ j. Let v_{ε} be defined as in the proof of Theorem 5.4. We define u_{ε} = v_{ε} + ε to obtain a positive function. We set V = {v_{1}, ⋯, v_{k}}. It is easy to check that, uniformly in v ∈ V

lim_{εto0} ∫_{M} v_{ε}^{N−2}v^{2}dv_{g} = 0.

Since λ_{i} < 0, it is then easy to see that sup_{v∈V} F (v_{ε}, v) = −∞. Together with the variational characterization of μ_{k}(M, g), we get that μ_{k}(M, g) = −∞.

This result proves for example that if the Yamabe invariant of (M, g) is negative, then μ_{1}(M, g) = −∞. This is the reason why we restricted in this article to the case of non-negative Yamabe invariant. Many of our results and proofs remain valid in the case μ_{2}(M) ≥ 0. However, if the Yamabe invariant of (M, g) is non-positive, there are other ways to find nodal solutions of Yamabe equation. Indeed, Aubin’s methods [Aub76] can be applied to avoid concentration phenomenom. See for example [DJ02], [Jou99], [Hol99] for such methods. Here, we present very briefly one new method in this case. We just sketch it since it is not the purpose of our paper to find solutions of Yamabe equation with Aubin’s type methods.

At first, for any metric g̃ conformal to g, we let λ_{1}^{+}(g̃) be the first positive eigenvalue of Yamabe operator. We then define λ^{+} = inf λ_{1}^{+}(g̃)Vol(M, g̃)^{\frac{2}{n}} where the infimum is taken over the conformal class of g. Then, proceeding in a way analogous to [Amm03a, Amm04], one shows that

0 < λ^{+} = inf \frac{(∫_{M} |L_{g}u|^{\frac{2n}{n+2}} dv_{g})^{\frac{n+2}{n}}}{∫_{M} uL_{g}u dv_{g}}

where the infimum is taken over the smooth functions u such that

∫_{M} uL_{g}u dv_{g} > 0.

Then, one shows using test functions that λ_{+} ≤ μ_{1}(𝕊^{n}). If the inequality is strict, then we can find a minimizer for the functional above which is a solution of the Yamabe equation. If the Yamabe invariant is positive, this solution is a Yamabe metric and hence is positive. However, if the Yamabe invariant is non-positive, this solution has an alternating sign.

A. Appendix: Proof of Lemma 3.1

Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3 and let v ∈ H_{1}^{2}(M), v ≢ 0 and u ∈ L_{+}^{N}(M) be two functions which satisfy in the sense of distributions

L_{g}v = u^{N−2}v.  (Eq)

We define v_{+} = sup(v, 0). We let q ∈]1, \frac{n}{n−2}] be a fixed number and l > 0 be a large real number which will tend to +∞. We let β = 2q − 1. We then define the following functions for x ∈ ℝ:

G_{l}(x) = | \begin{matrix}0 & if & x < 0 \\ x^{β} & if & x ∈ [0, l[ \\ l^{q−1}(ql^{q−1}x − (q − 1)l^{q}) & if & x ≥ l \end{matrix}

and

F_{l}(x) = | \begin{matrix}0 & if & x < 0 \\ x^{q} & if & x ∈ [0, l[ \\ ql^{q−1}x − (q − 1)l^{q} & if & x ≥ l \end{matrix}

It is easy to check that for all x ∈ ℝ,

(F′_{l}(x))^{2} ≤ qG′_{l}(x),  (49)

(F_{l}(x))^{2} ≥ xG_{l}(x)  (50)

and

xG′(x) ≤ βG_{l}(x).  (51)

Since F_{l} and G_{l} are uniformly lipschitz continuous functions, F_{l}(v_{+}) and G_{l}(v_{+}) belong to H_{1}^{2}(M). Now, let x_{0} ∈ M be any point of M. We denote by η a C^{2} non-negative function supported in B_{x_{0}}(2δ) (δ > 0 being a small number to be fixed) such that 0 ≤ η ≤ 1 and such that η(B_{x_{0}}(δ)) = {1}. Multiply equation (Eq) by η^{2}G_{l}(v_{+}) and integrate over M. Since the supports of v_{+} and G_{l}(v_{+}) coincide, we get:

c_{n} ∫_{M}(∇v_{+}, ∇η^{2}G_{l}(v_{+}))dv_{g} + ∫_{M} S_{g}v_{+}η^{2}G_{l}(v_{+})dv_{g} = ∫_{M} u^{N−2}v_{+}η^{2}G_{l}(v_{+})dv_{g}.  (52)

Let us deal with the first term of the left hand side of (52). In the following, C will denote a positive constant depending possibly on η, q, β, δ but not on l. We have

∫_{M}(∇v_{+}, ∇η^{2}G_{l}(v_{+}))dv_{g} = ∫_{M} G_{l}(v_{+})(∇v_{+}, ∇η^{2})dv_{g} + ∫_{M} G′_{l}(v_{+})η^{2}|∇v_{+}|^{2}dv_{g}
= ∫_{M} G_{l}(v_{+})v_{+}Δ(η^{2}) − 2 ∫_{M} v_{+}G′_{l}(v_{+})η(∇v_{+}, ∇η)dv_{g} + ∫_{M} G′_{l}(v_{+})η^{2}|∇v_{+}|^{2}dv_{g}
≥ −C ∫_{M} v_{+}G_{l}(v_{+})dv_{g} − 2 ∫_{M} v_{+}^{2}G′_{l}(v_{+})|∇η|^{2}dv_{g} + \frac{1}{2} ∫_{M} G′_{l}(v_{+})η^{2}|∇v_{+}|^{2}dv_{g}.

Using (49), (50) and (51), we get

∫_{M}(∇v_{+}, ∇η^{2}G_{l}(v_{+}))dv_{g} ≥ −C ∫_{M}(F_{l}(v_{+}))^{2}dv_{g} + \frac{1}{2q} ∫_{M}(F′_{l}(v_{+}))^{2}η^{2}|∇v_{+}|^{2}dv_{g}
≥ −C ∫_{M}(F_{l}(v_{+}))^{2}dv_{g} + \frac{1}{2q} ∫_{M} η^{2}|∇F_{l}(v_{+})|^{2}dv_{g}
≥ −C ∫_{M}(F_{l}(v_{+}))^{2}dv_{g} + \frac{1}{4q} ∫_{M} |∇(ηF (v_{+}))|^{2}dv_{g} − \frac{1}{2q} ∫_{M} |∇η|^{2}(F_{l}(v_{+}))^{2}dv_{g}
≥ −C ∫_{M}(F_{l}(v_{+}))^{2}dv_{g} + \frac{1}{4q} ∫_{M} |∇(ηF (v_{+}))|^{2}dv_{g}.  (53)

Using the Sobolev embedding H_{1}^{2}(M) into L^{N}(M), there exists a constant A > 0 depending only on (M, g) such that

∫_{M} |∇(ηF (v_{+}))|^{2}dv_{g} ≥ A(∫_{M}(ηF (v_{+}))^{N}dv_{g})^{\frac{2}{N}} − ∫_{M}(ηF (v_{+}))^{2}dv_{g}.

Together with (53), we obtain

∫_{M}(∇v_{+}, ∇η^{2}G_{l}(v_{+}))dv_{g} ≥ −C ∫_{M}(F_{l}(v_{+}))^{2}dv_{g} + \frac{A}{4q} (∫_{M}(ηF (v_{+}))^{N}dv_{g})^{\frac{2}{N}}  (54)

Independently, we choose δ > 0 small enough such that

∫_{B_{x_{0}} (2δ)} u^{N}dv_{g} ≤ (c_{n} \frac{A}{8q} )^{\frac{n}{2}} .

Relation (50) and Hölder inequality then lead to

∫_{M} u^{N−2}v_{+}η^{2}G_{l}(v_{+})dv_{g} ≤ ∫_{M} u^{N−2}η^{2}(F_{l}(v_{+}))^{2}dv_{g} ≤ c_{n} \frac{A}{8q} (∫_{M}(ηF (v_{+}))^{N}dv_{g})^{\frac{2}{N}} .  (55)

Since, by (50),

∫_{M} S_{g}v_{+}η^{2}G_{l}(v_{+})dv_{g} ≥ −C ∫_{M}(F_{l}(v_{+})^{2})dv_{g},

we get from (52), (54) and (55) that

c_{n} \frac{A}{8q} (∫_{M}(ηF (v_{+}))^{N}dv_{g})^{\frac{2}{N}} ≤ C ∫_{M}(F_{l}(v_{+}))^{2}dv_{g}.

Now, by Sobolew embedding, v_{+} ∈ L^{N}(M). Since 2q ≤ N and since C does not depend on l, the right hand side of this inequality is bounded when l tends to +∞. We obtain that

lim sup_{l→+∞} ∫_{M}(ηF (v_{+}))^{N}dv_{g} < +∞.

This proves that v_{+} ∈ L^{qN}(B_{x_{0}}(δ)). Since x_{0} is arbitrary, we get that v_{+} ∈ L^{qN}(M). Doing the same with sup(−v, 0) instead of v_{+}, we get that v ∈ L^{qN}(M). This proves Lemma 3.1.

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Authors’ address:

Bernd Ammann and Emmanuel Humbert Institut Élie Cartan BP 239 Université de Nancy 1 54506 Vandoeuvre-lès -Nancy Cedex France

E-Mail:

bernd.ammann at gmx.net and humbert at iecn.u-nancy.fr