The impact of the flatness condition on the prescribed Webster scalar curvature

In this paper, we give existence and multiplicity results for the problem of prescribing the Webster scalar curvature on the three CR sphere of C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{C}^{2}$$\end{document} under mixed conditions: non-degenerancy and flatness.


Introduction
Let S 3 be the unit sphere of C 2 endowed with its standard contact form θ and K be a given C 2 positive function on S 3 . We will give conditions on the set of critical points of K to ensure to this function to be the Webster scalar curvature for a contact formθ conformal to θ . This problem is equivalent to solve the following differential equation where L θ is the conformal laplacian, L θ = 4 θ + R θ , θ is the sublaplacian operator on (S 3 , θ) and R θ is the Webster scalar curvature of (S 3 , θ). The problem (P K ) has a variational structure, however the associated Euler functional does not satisfy the Palais-Smale condition. Therefore, the compactness methods cannot be applied in the present case instead we will use a method due to Bahri [2].
To state our result, we set the following notations. Let K be a C 2 positive function on S 3 such that the set of its critical points which we denote by K is finite. We say that K satisfies condition (H 1 ) if for all y ∈ K we have either: y is a nondegenerate critical point with K (y) = 0 (1. 1) or there exist β = β(y) ∈]2, 4[ and b 1 = b 1 (y), b 2 = b 2 (y), b 0 = b 0 (y) ∈ R * such that 2 k=1 b k + τ b 0 = 0, 2 k=1 b k + τ b 0 = 0 and in some pseudohermitian normal coordinates centered at y, K has the following expansion: ∇ s denotes all possible derivatives of order s and [β] is the integer part of β. We denote by I 1 = {y ∈ K; K (y) < 0}, I 2 = y ∈ K; y satisfies (1.2) and Fo each p-tuple of critical points of K , (y i 1 , . . . , y i p ) ∈ I p 1 , we associate the matrix M(y i 1 , . . . , where G is the Green's function of the conformal Laplacian L θ = 4 θ + R θ . We say that K satisfies (H 2 ) if for each (y i 1 , . . . , y i p ) ∈ I p 1 the corresponding matrix M(y i 1 , . . . , y i p ) is nondegenerate.
For each (y i 1 , . . . , y i p ) ∈ I p 1 we denote by (y i 1 , . . . , y i p ) the least eigenvalue of M(y i 1 , . . . , y i p ). To give our multiplicity result, we introduce the following definition and notations: • We recall that a solution u of (P K ) is said to be nondegnerate if the linearized operator L(ϕ) := L θ (ϕ) − 3u 2 ϕ does not have zero as an eigenvalue. • Let l + := max p ∈ N s.t ∃ (y i 1 , . . . , y i p ) ∈ I p 1 , (y i 1 , . . . , y i p ) > 0 and where for y ∈ I 1 , Morse(K , y) denotes the Morse index of K at y and for y ∈ I 2 , m(y) = #{b k (y); b k (y) < 0}. Now we are ready to state our multiplicity result.
The rest of this paper is organized as follows. In Sect. 2, we introduce the Heisenberg group and the Cayley transform which give an equivalence between S 3 minus a point and the Heisenberg group H 1 (for details see [8,12]). We give in Sect. 3 the Euler-Lagrange functional J associated to the problem (P K ) and some connected facts. The characterization of the critical points at infinity of J is the main tool of Sect. 4. Finally, Sect. 5 is devoted to the proof of Theorem 1.1.

The Heisenberg group and Cayley transform
The Heisenberg group H 1 is the Lie group whose underlying manifold is C × R = R 3 with coordinates (z, t) = (x 1 , x 2 , t) and whose group law is given by The H 1 -dilations are the H 1 -transformations given by: The Jacobian determinant of d λ is λ 4 , so that the homogeneous dimension of the group is equal to 4. The homogeneous norm of the space is given by 1 4 and the natural distance is accordingly defined by The space T 1,0 = span ∂ ∂z + iz ∂ ∂t gives the CR structure of H 1 .
The tangent bundle of H 1 has the following natural decomposition T H 1 = H ⊕ RT, where T = ∂ ∂t . The complex structure on H 1 is given by and (H, J ) gives a real CR stucture on H 1 .
The real 1-form annihilates T 1,0 , we take it to be the contact form for the CR structure.

Variational structure and preliminaries
In this section, we recall the associated Euler-Lagrange functional of (P K ) where u is on the unit sphere of S 2 1 (S 3 ) equipped with the norm is a Folland-Stein space (see [13]).
To investigate a solution for Problem (P K ) is equivalent to search a critical point of J subjected to the constraint u ∈ + , where For a ∈ S 3 and λ > 0, let are the coordinates of x and a, respectively, in some pseudohermitian normal coordinates and the constant c 0 is such that the following equation is satisfied We define now the set of potential critical points at infinity associated to the functional J . For ε > 0 and p ∈ N * , let denotes the interaction between the two masses a i and a j , respectively, the components of a i and a j in some pseudohermitian normal coordinates.
For w a solution of (P K ), let also The failure of the Palais-Smale condition for the functional J is characterized as follows.
Proposition 3.1 Let (u k ) ∈ + be a sequence such that ∂ J (u k ) tends to zero and J (u k ) is bounded. Then, there exist an integer p ∈ N * , a sequence (ε k ), ε k tends to zero, and an extracted subsequence of (u k )'s, again denoted (u k ), such that u k ∈ V ( p, ε k , w) where w is zero or a solution of (P K ).
If a function u belongs to V ( p, ε), we consider the following minimization problem for u ∈ V ( p, ε) with ε small Then we have the following results.
has a unique solution (up to permutation). In particular, we can write u ∈ V ( p, ε) as follows satisfying the following condition Here , L θ denotes the scalar product defined on S 2 and where (a i ) 1 , (a i ) 2 , (a i ) 0 are the coordinates of a i in some pseudohermitian normal coordinates.

Proposition 3.3
There exists a C 1 map which, to each (α 1 , . . . , α p , a 1 , . . . , a p , λ 1 , . . . , λ p ) such that Moreover, there exists c > 0 such that the following holds has a unique solution (ᾱ,λ,ā,h). Thus, we write u as follows: where v belongs to S 2 1 (S 3 )∩ T w (W s (w)) and satisfies (V 0 ), T w (W u (w)) and T w (W s (w)) are the tangent spaces at w to the unstable and stable manifolds of w.

Critical points at infinity of the variational problem
In the sequel, ∂ J designates the gradient of J with respect to the scalar product , L θ , that is [2]), we set the following definitions and notations. Here w is either zero or a solution of (P K ) and ε(s) is some function tending to zero when s −→ ∞. Using Proposition 3.4, u(s) can be written as: Denoting a i := lim s−→∞ a i (s) and

Definition 4.2 A critical point at infinity z ∞ is said to be dominated by another critical point at infinity
If we assume that the intersection is transverse, then we obtain index(z ∞ ) ≥ index(z ∞ ) + 1.
4.1 Ruling out the existence of critical points at infinity in V ( p, ε, w) for w = 0 Given a solution w of (P K ), following the lines of the method given in [16] we prove a non exitence result for critical points or critical points at infinity of J in V ( p, ε, w) for p ∈ N * and ε > 0. Proposition 4.3 Let K be a C 2 positive function on S 3 and let w be a nondegenerate critical point of J in + . Then, for each p ∈ N * , there is no critical points or critical points at infinity in the set V ( p, ε, w), that means we can construct a pseudogradient of J so that the Palais-Smale condition is satisfied along the decreasing flow lines.

Morse lemma at infinity in
Proof of Proposition 4.4. Without loss of generality, we can assume that λ 1 ≤ · · · ≤ λ p . Let M be a large positive constant and define Note that the set I contains the indices i such that λ i and λ 1 are of the same order. We divide the set V ( p, ε) into four subsets: where C is a large positive constant. In each subset we will define a pseudogradient and the vector-field W required in the proposition will be defined as a convex combination of all the pseudogradients. Let us denote by where ψ 1 is a cut off function defined by ψ 1 (t) = 0 if t ≤ ρ and ψ 1 (t) = 1 if t ≥ 2ρ, where ρ is a large positive constant (2ρ ≤ C). First, by an easy computation we obtain Hence using Proposition A.1. of [16], for each q ≤ p since C is large, it is easy to obtain (4.7) • In F 1 , we define . Using (4.6), (4.7) and the fact that there exists an index i ∈ I such that λ i |∇ K (a i )| ≥ C, we derive the estimate of claim (i) in this case.
• In F 2 , we remark that we can write u as Observe that u 1 ∈ V (#I, ε). Since I = ∅, we can apply the vector field defined in [11] in this set, we will denote it by Y (u 1 ). Hence we define where M is defined in 4.1. Using [11] and (4.6), (4.7), we obtain • In F 3 , we have I = I = {1}. Let y 1 be the critical point which is close to a 1 , for simplicity, we can assume that F(y 1 ) = 0, so the Cayley transform gives a pseudohermitian normal coordinates centered at y 1 .
In this case we define and ψ 2 is a cut off function defined by ψ 2 (t) = 1 if t ≤ μ and ψ 2 (t) = 0 if t ≥ 2μ, where μ is a small positive constant.
From [16], we have In this region, the pseudogradient will be defined as where C 2 is a large positive constant. Using (4.6), (4.7), (4.10) and the fact that λ −2 i = o(ε i j ) for each i / ∈ I and j ∈ I , we derive Thus the estimate of claim (i) follows in this case.
• In F 4 , we decrease all the variables λ i with different speeds. First, observe that λ −2 i ≤ c M ε i j for each i, j ∈ I and for each i ∈ I we have K (a i ) = o (1). In this region, the pseudogradient will be defined by where m 1 and m 2 are small positive constants (m 1 c M and m 2 /m 1 are small).
Using Proposition A.1. of [16], formulas (4.6) and (4.7) and the fact that λ −2 i = o(ε i j ) for each i / ∈ I and j ∈ I , we derive which implies the estimate of claim (i) in this case. Finally, the pseudogradient W will be defined by a convex combination of W 1 , W 2 , W 3 , W 4 . This vector field satisfies claim (i).
Regarding claim (ii), as in [11], it follows from claim (i) and the estimates of v 2 and ∂ J (u +v) v . Concerning claim (iii), it is easy to get that |W | is bounded. Furthermore, from the definition of the W i 's, we remark that the maximum of the λ i 's is a decreasing function on the sets F 1 , F 3 and F 4 , However, in F 2 , if I = {1, . . . , p}, the maximum of the λ i 's is a decreasing function. But if I = {1, . . . , p}, we have the same case as [11]. Thus claim (iii) follows. The proof of the proposition is thereby complete.

Morse Lemma at infinity in V (1, ε)
In this section, we will characterize the critical points at infinity in V (1, ε). Proposition 4.6 Let K be a C 2 positive function on S 3 satisfying (H 1 ) and (H 2 ) and let β = max{β(y), y ∈ K}. Then, there exists a pseudogradient W so that the following holds: there is a positive constant c > 0 independent of u = αδ (a,λ) ∈ V (1, ε), such that Furthermore, λ is an increasing function along the flow lines generated by W only if a is close to a critical point y ∈ I 1 ∪ I 2 .
Proof The construction of W depends on the variables a and λ. We will divide the set V (1, ε) into three subsets: where C is a large positive constant.
• In F 1 , we define and using Proposition A.1 from [16], we get (4.11) • In F 2 , we define (a,λ) ∂λ and using Proposition A.1 from [16], we derive that (4.12) • In F 3 , we set W 3 = W 1 3 + W 2 3 where W 1 3 and W 2 3 are defined by (4.8) and (4.9), respectively. We have The required pseudogradient W will be defined by convex combination of W 1 , W 2 , W 3 . Using (4.11), (4.12) and (4.13), claim (i) follows. Regarding claim (ii), it follows from claim (i) and the estimate of v 2 . Concerning claim (iii), it follows from the definition of W . Hence the proof of the proposition is completed.
Once the pseudogradient is constructed, following [1,11], one can find a change of variables which gives the normal form of the functional J on the subset F y = {αδ (a,λ) + v : a is close to y}, y ∈ K. More precisely, we have Proposition 4.7 For y ∈ K in F y = {αδ (a,λ) + v : a is close to y}, there exists a change of variables: where μ is a small positive constant.
The proof of Proposition 4.7 is exactly the same as in [1,4,11]. As a consequence of the above proposition, we have the following result:

Proof of the theorem
First, we will prove a topological argument which provides a lower bound on the number of critical points of J . For this purpose let K ∞ be the set of critical points at infinity of J and let L 0 be their maximal Morse index. We set: where W u (z ∞ ) is the unstable manifold of the critical point at infinity z ∞ . By a theorem of Bahri-Rabinowitz [3], we have It follows, that X is a stratified set of top dimension ≤ L 0 . Moreover it is contractible in + , by taking its suspension, that is, for a ∈ + , we consider Let U be such a contraction which is a stratified set of top dimension L 0 + 1. U can also be deformed using the flow of −∂ J . For dimension's reason the stable manifold of any critical point of Morse index ≥ L 0 + 2 can be avoided during such a deformation (see [15]). Therefore, U is deformed onto some set where x is a critical point or critical point at infinity.
We have the following result.
where i(z ∞ ) denotes the Morse index of the critical point at Infinity z ∞ .
Proof Since X is contractible in Z , we have from the exact sequence in homology that: where H k (X ) := H k (X, Q) is the kth homology group with rational coefficients. Therefore, it follows that Now observe that the pair (Z , X ) is built by adding to X the unstable manifold of other critical points of index ≤ L 0 + 1. Namely each time we add one of these unstable manifolds, starting from X and going with increasing index. At each step the new object we obtain has a total dimension of homology increased at most by one. Therefore, the total homology of (Z , X ) has its dimension upper bounded by the number of critical points of index ≤ L 0 + 1, not dominated by X . Therefore, we have the following Lemma: It follows from formula 5.1 that Using now the upperbound of Lemma 5.2 on the homology on the right hand side of the above formula, our claim in Proposition 5.1 follows.
Once Proposition 5.1 is proved, Theorem 1.1 follows immediately from Proposition 5.1.
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