Methods of the theory of critical points at infinity on Cauchy Riemann manifolds

Arabian Journal of Mathematics, Sep 2017

Sub-Riemannian spaces are spaces whose metric structure may be viewed as a constrained geometry, where motion is only possible along a given set of directions, changing from point to point. The simplest example of such spaces is given by the so-called Heisenberg group. The characteristic constrained motion of sub-Riemannian spaces has numerous applications in robotic control in engineering and neurobiology where it arises naturally in the functional magnetic resonance imaging (FMRI). It also arises naturally in other branches of pure mathematics as Cauchy Riemann geometry, complex hyperbolic spaces, and jet spaces. In this paper, we review the use of the relationship between Heisenberg geometry and Cauchy Riemann (CR) geometry. More precisely, we focus on the problem of the prescription of the scalar curvature using techniques related to the theory of critical points at infinity. These techniques were first introduced by Bahri, Bahri and Brezis for the Yamabe conjecture in the Riemannian settings.

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Methods of the theory of critical points at infinity on Cauchy Riemann manifolds

Methods of the theory of critical points at infinity on Cauchy Riemann manifolds Dedicated to the Memory of Professor Abbas Bahri 0 Mathematics Subject Classification 0 0 N. Gamara College of Science, Taibah University , Medina , Saudi Arabia Sub-Riemannian spaces are spaces whose metric structure may be viewed as a constrained geometry, where motion is only possible along a given set of directions, changing from point to point. The simplest example of such spaces is given by the so-called Heisenberg group. The characteristic constrained motion of sub-Riemannian spaces has numerous applications in robotic control in engineering and neurobiology where it arises naturally in the functional magnetic resonance imaging (FMRI). It also arises naturally in other branches of pure mathematics as Cauchy Riemann geometry, complex hyperbolic spaces, and jet spaces. In this paper, we review the use of the relationship between Heisenberg geometry and Cauchy Riemann (CR) geometry. More precisely, we focus on the problem of the prescription of the scalar curvature using techniques related to the theory of critical points at infinity. These techniques were first introduced by Bahri, Bahri and Brezis for the Yamabe conjecture in the Riemannian settings. 1 Introduction In 1995, Professor Bahri proposed to Yacoub and Gamara, to solve the remaining cases left open by Jerison and Lee of the Cauchy Riemann Yamabe Conjecture [56–58]. In 1987, Jerison and Lee formulated in [56] the CR Yamabe conjecture and developed the analogy with the Yamabe problem in conformal Riemannian geometry, which had already been solved by Aubin [ 2 ] and Schoen [66]. Besides the proof of Aubin and Schoen, another proof by Bahri [ 6 ], Bahri and Brezis [ 17 ] was available using methods related to the theory of critical points at infinity. This theory has been widely developed by Bahri at mid-1980s. Bahri introduced and performed the theory of critical points at infinity by establishing several methods which are a fundamental step in the calculus of variation. Based on his experience and his numerous works in that direction [ 5–8,15–24,26 ], Bahri was convinced that these topological methods are well adapted to solve the Yamabe problem in the Cauchy Riemann settings. Furthermore, Bahri used this theory to solve non compact variational problems as Yamabe type equations, the prescribed scalar curvature equations, n-body equations in celestial mechanics, fundamental problems in contact and conformal geometries, mean field equations etc. , we refer here to some of his work [ 8–14,25 ]. Given an orientable manifold M of odd dimension 2n + 1, a Cauchy Riemann structure on M is given by a complex n-dimensional subbundle T1,0(M ) of the complexified tangent bundle T(M ) ⊗ C satisfying T1,0(M ) ∩ T1,0(M ) = 0. A Cauchy Riemann in short CR manifold is such a manifold with an integrable Cauchy Riemann structure. The geometry of CR manifolds, the abstract models of real hypersurfaces in complex manifolds, has recently attracted much attention. This is in particular due to the fact that, in the strictly pseudo-convex case, there are many parallels with conformal Riemannian geometry. Indeed, a CR manifold carries a natural Hermitian metric on its holomorphic tangent bundle. The Levi form, which is, like a metric on a conformal manifold, determined only up to multiplication by a smooth function. The multiple is fixed by choosing a contact form θ (a real 1-form) annihilating the holomorphic tangent bundle. A CR manifold together with a choice of a contact form is called a pseudo-Hermitian manifold. The simplest scalar invariant for a pseudo-Hermitian manifold is the pseudo-Hermitian scalar curvature, which we denote by Rθ , defined independently by Webster [70] and Tanaka [68]. If the Levi form is positive definite the pseudo-Hermitian manifold is called a strictly pseudo-convex CR manifold. Consider (M, θ ) a strictly pseudo-convex compact CR manifold of dimension 2n + 1 and K a smooth function. The prescribed Webster scalar curvature problem consists on finding a contact form θ˜ conformal to 2 θ for which the pseudo-Hermitian scalar curvature is equal to K . If we set θ˜ = u n θ , where u is a smooth positive function on M , then the above problem is equivalent to solve the following equation: where −Lθ = − 2(nn+1) θ + Rθ , is the conformal Laplacian, θ is the sub-Laplacian operator on (M, θ ), and Rθ is the Webster scalar curvature of (M, θ ). Problem ( PK ) is the analogue of the prescribed scalar curvature problem on Riemannian manifolds. Comparing to the scalar curvature problem in the Riemannian framework, which was extensively studied (see for example the monograph [ 4 ] and the references therein), only few authors (most of them are students of Bahri) have been interested in solving the problem ( PK ) (see [40,42,45–50,53–55,63]). On the contrary, the Yamabe problem on CR manifolds (the case where K is assumed to be constant) was widely studied by various authors (see among others [43,44,56–58]). The main difficulty one encounters in solving problem ( PK ), appears when we consider it from a variational viewpoint. Indeed, the Euler functional associated with ( PK ) does not satisfy the Palais–Smale condition, that is, there exist noncompact sequences along which the associated functional to ( PK ) is bounded and the gradient of this functional goes to zero. Moreover, as in the Riemannian settings, there are topological obstructions of Kazdan–Warner condition type to solve ( PK ), see [51]. Hence, we do not expect to solve problem ( PK ) for all functions K , and so it is natural to ask the following: under which conditions on K does a positive solution exist for ( PK )? In [63], Malchiodi and Uguzzoni considered the case where M = S2n+1 the unit sphere of Cn+1 and gave a perturbation result for problem ( PK ), that is K is assumed to be a small perturbation of a constant (see also [40]). Their approach is based on a perturbation method due to Ambrosetti [ 1 ]. In [42], Gamara noticed an analogy between the 3-dimensional CR case with the 4-dimensional Riemannian case, see for example [ 18 ] and [ 27 ]: there is a balance phenomenon between the self-interactions and the mutual interactions of the functions failing to satisfy the Palais–Smale condition. In [42], the case where M is locally conformally CR equivalent to the sphere S3 of C2 was considered (thus when n = 1), an Euler-Hopf type criterion for K was provided to find solutions for ( PK ). The existence results of Gamara have been generalized by Chtioui, Ahmedou and Yacoub, see [54], where multiplicity results were also be given. In [40,42,45,53–55,63], to prove the existence or multiplicity results for problem ( PK ), the authors use a non degeneracy condition. In this paper, we focus on the prescribed Webster scalar curvature problem on a Cauchy Riemann manifold M locally conformally equivalent to the unit sphere S3 of C2 endowed with its standard contact form. This manuscript consists of an introduction and three sections organized as follows: • Section 2: We present the problem of prescribing scalar curvature in the Riemannian settings, as an example, we review the Yamabe problem. • Section 3: This chapter is an introduction to Cauchy Riemann geometry: we give some useful definitions and results. In Sect. 3.1, we define the CR structure on a orientable real 2n + 1-dimensional manifold M , which we denoted by T1,0(M ). It is a subbundle of dimension n of the complexified tangent bundle T(M ) ⊗ C satisfying T1,0(M ) ∩ T1,0(M ) = 0. If T1,0(M ) is integrable, then we define the distribution of the CR manifold by H(M ) := T1,0(M ) ⊕ T1,0(M ). The distribution H(M ) is a 1-codimensional subbundle of T(M ), so there is a 1-form θ on M such that H(M ) = K er (θ ). The form θ defines a pseudo-Hermitian structure on M . For a given pseudo-Hermitian structure θ on M , we define in Sect. 3.2, the Levi form lθ . It is a symmetric bilinear form on H(M ). We extend lθ to the hole T(M ) which we denoted also by lθ , so we say that (M, θ ) is strictly pseudo-convex (resp. non degenerate) if lθ is positive definite (resp. non degenerate). In Sect. 3.3, we define the Reeb vector field T (or characteristic direction) for a non degenerate CR manifold (M, θ ) by θ (T) = 1 and T is dθ -orthogonal to H(M ), we obtain the decomposition T(M ) = H(M ) ⊕ RT. In Sect. 3.4, we define the Webster metric gθ for a non degenerate CR manifold (M, θ ) via the Levi form lθ . If (M, θ ) is strictly pseudo-convex then gθ is a Riemannian metric. Sect. 3.5 is devoted to define the Tanaka-Webster connection of a non degenerate CR manifold. In Sect. 3.6, we define the Christofell symbols and the pseudo-Hermitian torsion. In Sect. 3.7, we define the curvature tensors: the global curvature R, the pseudo-Hermitian Ricci curvature and the scalar curvature (called the Webster scalar curvature) of the Tanaka-Webster connection ∇. In Sect. 3.8, we define the divergence and the adjoint of a vector field, we define also the horizontal gradient ∇H f = πH∇ f where πH : T(M ) = H(M ) ⊕ RT −→ H(M ) is the natural projection. We define then the sub-Laplacian operator b by b f = −di v(∇H f ), f ∈ C 2(M ). The Sect. 3.9 is devoted to some examples, we study the Heisenberg group Hn, the Heisenberg group of dimension n = 1 and the unit sphere S2n+1 of C2n . In the last section of this chapter, we introduce normal coordinates for a CR strictly pseudo-convex manifold and give the definition k of the Folland–Stein spaces S p(M ) of M . • Section 4: In this section, we expose the problem of the prescription of a scalar curvature on Cauchy Riemann manifolds and give a quick review on the CR Yamabe problem. Finally, we announce and prove our main result: an existence theorem for problem ( PK ). In Sect. 4.1, we give some preliminaries which are useful for the understanding of the problem. In Sect. 4.2, we formulate the Euler–Lagrange functional J associated with ( PK ). Section 4.3 is devoted to the natural change of the functional. In Sect. 4.4, we review the definitions of the Hessian and the Morse Lemma. In Sect. 4.5, we introduce the notions of a homotopy and homotopy equivalence, while the definition of a retract of a topological space and the notion of deformation retract are the object of Sect. 4.6. In Sect. 4.7, we introduce the Palais–Smale condition and in Sect. 4.8, we give a quick review on the Cauchy Riemann Yamabe Problem. The case of a CR spherical pseudo-Hermitian Manifold of dimension 3 is the object of Sect. 4.9. We define the almost solutions, then we study the properties of the functional related to the associated variational formulation. In Sect. 4.10, we expand the functional near the sets of its critical points at infinity. The Morse Lemma is displayed in Sect. 4.11: we construct a pseudo-gradient for our functional; then, we localize the critical point at infinity of J . Sect. 4.12 is devoted to the proof of our main result an existence theorem for problem ( PK ) using a topological argument. 2 Prescribing the scalar curvature on Riemannian manifolds 2.1 Prescription of a scalar curvature Let (M, g) be a compact Riemannian manifold without boundary of dimension n ≥ 3 and K a positive function of class C 2. The prescribed scalar curvature problem consists to find a metric g conformal to g for which the 4 scalar curvature Rg = K . We write g = u n−2 g, u > 0. If we denote by the Laplace–Beltrami operator of the metric g, we obtain the following transformation law for the scalar curvature of the metrics g and g: Rg = u− nn+−22 (−cn u + Ru) with cn = 4 n−1 . Hence, finding a solution for the prescribed scalar curvature problem is equivalent to solve n−2 the following partial differential equation Let p = n2−n2 and −L = −cn + R be the conformal Laplacian. The last equation can be rewritten as (2.1) (2.2) (2.3) The prescription of the scalar curvature for Riemannian manifolds is known to be the Kazdan–Warner problem; it has been extensively studied by many authors for dimensions 2, 3 and 4 as well as in higher dimensions.There is a big number of papers devoted to this problem as well as for the multiplicity of its solutions, we can mention [ 32–36,39,52,60,62 ]. Here, we will merely refer to the references [ 5,18,27,29 ] and recently [31] which are the most directly related works to our since based on the use of methods related to the theory of critical points at infinity due to Bahri. 2.2 The Yamabe problem The Yamabe problem is the case where the function K to prescribe is constant, K = λ for some constant λ ∈ R. The Yamabe problem goes back to Yamabe himself [71], who claimed in 1960 to have a solution, but in 1968, Trudinger [69] discover an error in his proof and corrected Yamabe’s proof. In [ 2 ], Aubin improved Trudinger’s result, using variational methods and Weyl’s tensor characteristics. In 1984, Schoen [66] solved the remaining cases using variational methods and the positive mass Theorem. We have also to point out the work of Lee and Parker in [61], which is a detailed discussion on the Yamabe problem unifying the work of Aubin [ 2 ] with that of Schoen [66]. Besides the proof by Aubin and Schoen for the Riemannian Yamabe conjecture, another proof by Bahri [ 6 ], Bahri and Brezis [ 17 ] was available by techniques related to the theory of critical points at infinity. Yamabe observed that the Yamabe equation ( Pλ) is the Euler–Lagrange equation of the functional Q0(g) = M Rgdvg˜ 2 M dvg˜ p when restricted to a conformal class [g] = {hg / h ∈ C ∞(M ), h > 0}, where dvg˜ is the volume form of (M, g) and h = u p−2, u > 0. In fact, on [g], we can write Q0(g) = Q0(u p−2g) = J (u), where J (u) = M −Lu udvg 2 = M u pdvg p M (cn |∇u|2 + Ru2)dvg u 2p . We call J (u) the Yamabe quotient of (M, g). So for a given Riemannian manifold (M, g), it is natural to define the following constrained extremal problem: λ(M, g) = inf u∈W 1,2(M,g) Aθ (u) = M −Lu u dvg; M | u | p dvg Therefore λ(M, g) = inf Q0(g) / g ∈ [g] (2.4) λ(M, g) is a conformal invariant, which means that it is determined by the conformal class and is independent of the choice of the initial metric g in the conformal class. It is called the Yamabe invariant of (M, g). Calculus of variation methods has been used to prove that the Yamabe problem can be solved on a general compact Riemannian manifold (M, g) of dimension n, provided that its Yamabe invariant λ(M, g) < λ(Sn, g0), where g0 is the standard metric on the sphere. This is due to Yamabe, Trudinger and Aubin. The modification by Trudinger of Yamabe’s proof works whenever λ(M, g) ≤ 0. In fact, Trudinger showed that there is a positive constant such that the proof works when λ(M, g) < . In 1976, Aubin extended Trudinger’s result by showing that in fact = λ(Sn, g0). We have the following results: Theorem 2.1 ([ 2,3,69,71 ]) Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3 without boundary. The Yamabe problem has a solution if λ(M, g) < λ(Sn, g0). Theorem 2.2 ([ 2 ]) Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3 without boundary. Then λ(M, g) ≤ λ(Sn, g0). Hence, the extremal problem given above is attained by a positive smooth solution of the Yamabe equation 4 ( Pλ). Thus the metric g = u n−2 g has a constant scalar curvature Rg = λ(M, g). Theorem 2.1 is proved by considering a minimizing sequence ui ∈ W 1,2(M, g) satisfying B(ui ) = 1 to minimize functional A. Since the sequence ui is bounded in W 1,2(M, g), there exists a subsequence denoted again by ui which converges weakly to u ∈ W 1,2(M, g). Then, A(u) = λ(M, g), but B(u) could be different from 1 because the embedding of W 1,2(M, g) in L p is continuous but not compact. In particular the limit function u may be identically zero. To overcome this difficulty, a perturbed extremal problem has been considered to lead to a solution of the first one. An alternative proof of Theorem 2.1 has been given by Uhlenbeck, which used Riemannian normal coordinates and blow-up analysis to transplant the minimizing sequences for the perturbed problem. This kind of blow-up analysis was first introduced in 1981 by Sacks and Uhlenbeck in [65] Theorem 2.1 reduces the resolution of Yamabe problem to the estimate of the invariant λ(M, g). In this way, Aubin [ 2 ] proved the following result: Theorem 2.3 If (M, g) is a compact Riemannian manifold of dimension n ≥ 6 not conformally flat, then λ(M, g) < λ(Sn, g0). In [66], Schoen solved all the remaining cases of the Yamabe problem using the positive mass theorem. The proof by Bahri [ 6 ] and Bahri and Brézis [ 17 ] of the Yamabe problem is available using the theory of critical points at infinity. This proof is completely different in spirit as well as in techniques and details from the proof of Aubin and Schoen. It does not require the use of any theory of minimal surfaces neither the use of the positive mass theorem. We remark that for the case (M, g) conformal to Sn, the Yamabe problem clearly has a solution. If : M → Sn is a conformal diffeomorphism then ∗(g0) = f g, where g0 is the standard metric of Sn and f a positive function in C ∞(M ), clearly f g has constant scalar curvature. 3 Cauchy–Riemann manifolds 3.1 CR structures Let M be a real 2n + 1-dimensional C∞ differentiable manifold. Let T(M ) ⊗ C be the complexified tangent bundle over M (T(M ) ⊗ C = {u + i v; u, v ∈ T(M )}, where i = √−1). Definition 3.1 Let us consider a complex subbundle T1,0(M ) of the complexified tangent bundle T(M ) ⊗ C, of complex rank n. We say that T1,0(M ) is a CR structure on M if 1) T1,0(M ) ∩ T0,1(M ) = 0, 2) [ ∞(T1,0(M )), ∞(T1,0(M ))] ⊂ ∞(T1,0(M )) (integrability condition). Where T0,1(M ) = T1,0(M ), over bar denotes complex conjugation, fields X : M −→ T1,0(M ) and [, ] is the Lie bracket. A pair (M, T1,0(M )) is called a CR manifold. Definition 3.2 Let (M, T1,0(M )) and (N , T1,0(N )) be two CR manifolds. A C∞ map f : M −→ N is a CR map if (dx f )T1,0(M )x ⊂ T1,0(N ) f (x), (3.1) for any x ∈ M , where dx f is the C-linear extension to Tx (M ) ⊗ C of the differential of f at x . Let (M, T1,0(M )) be a CR manifold. Its Levi distribution is the real subbundle of rank 2n H(M ) ⊂ T(M ) given by ∞(T1,0(M )) denotes the set of vector It carries the complex structure J : H(M ) −→ H(M ) given by H(M ) = Re{T1,0(M ) ⊕ T0,1(M )}. J (V + V ) = i (V − V ), ∀ V ∈ T1,0(M ). Definition 3.3 A function f : M diffeomorphism and a CR map. −→ N is a CR isomorphism (or a CR equivalence) if f is both a C∞ Standard examples of CR manifolds are those of real hypersurfaces of complex manifolds. For example if M is a hypersurface of Cn+1), the CR structure is the one induced by the ambient space, for any x ∈ M T1,0(M )x = T(M ) ⊗ C T1,0(Cn+1) x ; here, T1,0(Cn+1) is the tangent holomorphic subbundle having local generator system ∂∂z j , 1 ≤ j ≤ n + 1 where, (z1, z2 . . . zn+1) are the complex cartesian coordinates of Cn+1. Recall that the Heisenberg group Hn, (n ≥ 1), is the homogeneous Lie group whose underlying manifold is Cn × R = R2n+1 and whose group law is given by τ(z ,t )(z, t ) = (z , t ) · (z, t ) = (x + x , y + y , t + t + 2(< x , y > − < x , y >)), where < ., . > denotes the inner product in the Euclidian space Rn, (z, t ) = (x1, . . . , xn, y1, . . . , yn, t ) and (z , t ) = (x1, . . . , xn, y1, . . . , yn, t ). Following the geometrical interpretation due to I. Piatetski-Shapiro [59], one can introduce the Heisenberg group Hn using its identification with the boundary Mn of the Siegel Domain: Dn+1 = (ξ0, ξ1, . . . , ξn) = (ξ0, ξ ) ∈ C × Cn; |ξ j |2 − I mξ0 < 0 Mn = ∂ Dn+1 = (ξ0, ξ ) ∈ C × Cn ; |ξ j |2 = I mξ0 . n 1 n 1 The Siegel domain Dn+1 is holomorphically equivalent to the unit ball in Cn+1. The Heisenberg group Hn acts on Cn+1 by holomorphic affine transformation which preserves Dn+1 and Mn as follows: if (z, t ) ∈ Hn and ξ ∈ Cn+1, (z, t ) • ξ = ξ where ξ0 = ξ0 + t + i |z|2 + 2i ξ j z¯ j ξ j = ξ j + z j , 1 ≤ j ≤ n. n 1 Since this action is transitive on Mn, the group Hn is identified with Mn via the correspondence: (z, t ) ↔ (z, t ) • 0 = (t + i |z|2, z1, . . . , zn). Under this identification the CR structure on Hn described above coincides with the CR structure on Mn induced from Cn+1. 3.2 The Levi form Let M be an orientable connected CR manifold. Let E(M ) = H(M )⊥ := {α ∈ T∗(M ); ∀x ∈ M, H(M )x ⊂ ker(αx )}. E(M ) is a real line subbundle of the cotangent bundle T∗(M ) and (a vector bundle isomorphism). Since M is orientable and H(M ) is oriented by its complex structure J , it follows that E(M ) is orientable. Since E(M ) is an orientable real line bundle over a connected manifold, then E(M ) has a nowhere vanishing C∞ section θ : M −→ E(M ). The section θ is a 1-form and we have Any such section θ is referred to as a pseudo-Hermitian structure on M . Definition 3.4 Given a pseudo-Hermitian structure θ on M , the Levi form lθ is the symmetric bilinear form defined by lθ (V , W ) = dθ (V , J (W )) ∀ V , W ∈ H(M ). The C-linear extension to CH(M ) gives an Hermitian form on T1,0(M ) defined by lθ (V , W ) = −i dθ (V , W ) ∀ V , W ∈ T1,0(M ). Since E(M ) is a real line bundle, then for any two pseudo-Hermitian structures θ and θ˜ there exists a nowherezero C∞ function λ : M −→ R such that Let us apply the exterior differentiation operator d to (3.4); we get Since ker(θ ) = H(M ), the C-linear extension of θ vanishes on T1,0(M ) and T0,1(M ) as well. Consequently, the Levi form changes according to E(M ) T(M )/H(M ) H(M ) = ker(θ ). dθ˜ = dλ ∧ θ + λdθ . θ˜ = λθ . lθ˜ = λlθ . (3.2) (3.3) (3.4) (3.5) (3.6) (3.7) Definition 3.5 Let (M, T1,0(M )) be a CR manifold and θ a pseudo-Hermitian structure on M . 1) We say that (M, T1,0(M )) (or (M, θ )) is non degenerate if the Levi form lθ is nondegenerate. 2) We say that (M, θ ) is strictly pseudo-convex if lθ is positive definite. 3.3 The Reeb field Proposition 3.6 (See [ 37 ]) Let (M, T1,0(M )) be a nondegenerate CR manifold, θ a pseudo-Hermitian structure on M . Then, there exists a unique globally defined nowhere zero tangent vector field T on M such that T is transverse to the Levi distribution H(M ). Moreover, the tangent bundle decomposes as θ (T) = 1, and dθ (T, .) = 0. T(M ) = H(M ) ⊕ RT. Definition 3.7 This vector field T is called the characteristic direction or Reeb field of (M, θ ). 3.4 Webster metric Definition 3.8 Let (M, T1,0(M)) be a non degenerate CR manifold and θ a pseudo-Hermitian structure on M. Let gθ be the semi-Riemannian metric given by gθ is called the Webster metric of (M, θ ). Remark 3.9 where denotes the symmetric tensor product defined by ⎧ gθ (X, Y ) = lθ (X, Y ) ⎨ gθ (X, T) = 0 ⎩ gθ (T, T) = 1 ∀X, Y ∈ H(M). gθ = lθ + θ θ, θ θ (X, Y ) = θ (X )θ ( J Y ), X, Y ∈ T(M) Proposition 3.10 Let (M, T1,0(M)) be a non degenerate CR manifold and θ a pseudo-Hermitian structure on M. If lθ is positive definite ((M, θ ) is strictly pseudo-convex), then gθ is a Riemannian metric on M. 3.5 The Tanaka-Webster connection Let (M, T1,0(M)) be a non degenerate CR manifold and θ a fixed pseudo-Hermitian structure on M. Let T be the Reeb field of (M, θ ). If ∇ is a linear connection on M, we denote T∇ the associated torsion tensor field. Definition 3.11 ([ 37 ]) We say that T∇ is pure if for any Z , W ∈ T1,0(M). Here T∇ (Z , W ) = 0, T∇ (Z , W ) = 2ilθ (Z , W )T, τ ◦ J + J ◦ τ = 0, τ : T(M) −→ T(M) X −→ T∇ (T , X ). (3.8) (3.9) (3.10) On each non degenerate CR manifold on which a pseudo-Hermitian structure has been fixed, there is a canonical linear connection compatible with both the complex structure of the Levi distribution and the Levi form. Precisely, we have the following result: Theorem 3.12 Let (M, T1,0(M)) be a non degenerate CR manifold and θ a pseudo-Hermitian structure on M. Let T be the Reeb field of (M, θ ) and J the complex structure in H(M) (extended to an endomorphism of T(M) by requiring that J T = 0). Let gθ be the Webster metric of (M, θ ). There is a unique linear connection ∇ on M satisfying the following axioms: (i) H(M) is parallel with respect to ∇, that is ∇X ∞(H(M)) ⊂ ∞(H(M)), for any X ∈ X (M). (ii) ∇ J = 0, ∇gθ = 0. (iii) The torsion T∇ of ∇ is pure. Proof Since we have the following direct sum decomposition, we can define the natural projections and T(M ) ⊗ C = T1,0(M ) ⊕ T0,1(M ) ⊕ T ⊗ C, π+ : T(M ) ⊗ C −→ T1,0(M ) π− : T(M ) ⊗ C −→ T0,1(M ). Then, for any Z ∈ T1,0(M ), π−(Z ) = π+(Z ). We establish first the uniqueness of a linear connection ∇ on M obeying the axioms (i ),(i i ) and (i i i ). Since the torsion T∇ is pure then for any Y, Z ∈ T1,0(M ), we have where (as ∇Y Z ∈ ∞(T1,0(M )) and ∇Z Y ∈ ∞(T0,1(M ))), we obtain [Y , Z ] = ∇Y Z − ∇Z Y + 2ilθ (Z , Y )T, π+[Y , Z ] = ∇Y Z Let be the 2-form defined by (X, Y ) = gθ (X, J Y ) = −dθ (X, Y ), Since ∇ gθ = 0, we obtain X, Y ∈ T(M ). It satisfies (T, .) = 0. for any X, Y, Z ∈ T(M ). In particular, for Y = T , it yields We distinguish two cases: Z ∈ H(M ) and Z = T. X (gθ (Y, Z )) = gθ (∇X Y, Z ) + gθ (Y, ∇X Z ) X (θ (Z )) = gθ (∇X T , Z ) + θ (∇X Z ) • If Z ∈ H(M ), then (3.12) yields gθ (∇X T , Z ) = 0 or πH(∇X T) = 0, where πH : T(M ) −→ H is the natural projection associated with the direct sum decomposition (3.7). • Let Z = T; we use (3.12), we obtain (3.11) (3.12) (3.13) (3.14) (3.15) = 0, where L denotes the Lie derivative. On the other hand, (by ∇T = 0) ∇T X = τ X + LT X, X ∈ T(M ). since ∇X T is parallel to T , we deduce that ∇X T = 0. By using (3.12) with X = T , we obtain that ∇T T = 0; hence we conclude that Note that as a consequence of axiom (ii) in Theorem 3.12. Therefore, for any X, Y, Z ∈ T1,0(M ). Using (3.11) we may rewrite this identity as X ( (Y, Z )) = (∇X Y, Z ) + (Y, ∇X Z ) (∇X Y, Z ) = X ( (Y, Z )) − (Y, π−[X, Z ]), which, in view of the non degeneracy of the bundle endomorphism KT given by on H(M ), determines ∇X Y for any X, Y ∈ T1,0(M ). We shall need Note that as a consequence of property (3.10), τ is H(M )-valued. We may use ∇ J = 0 and (3.15) to perform the following calculation: 0 = (∇T J )X = ∇T J X − J ∇T X = τ ( J X ) + LT( J X ) − J (τ X + LT X ) = − J τ X + LT( J X ) − J (τ X + LT X ) = −2 J τ X + (LT J )X. Let us apply J in both members of this identity and use the fact that τ is H(M )-valued to obtain τ = KT. Therefore, (3.15) may be rewritten as ∇T X = KT X + LT X, We use the identities (3.11), (3.13), (3.14) and (3.16) to prove uniqueness statement in Theorem 3.12. To prove existence, we consider ∇ : ∞(T(M ) ⊗ C) × ∞(T(M ) ⊗ C) −→ ∞(T(M ) ⊗ C) be the differential operator defined by ∇X Y = π+[X , Y ] ∇X Y = UXY , ∇X Y = ∇X Y , ∇X Y = ∇X Y , ∇T X = LT X + KT X, ∇T X = ∇T X , ∇T = 0, U : ∞(T(M )) × ∞(T(M )) −→ ∞(T(M )) for any X, Y ∈ T1,0(M ). Here Then we can verify that ∇ satisfies axioms (i), (ii) and (iii) in Theorem 3.12. Definition 3.13 The connection ∇ given by Theorem 3.12 is the Tanaka-Webster connection of (M, T1,0(M ), θ ). The vector-valued 1-form τ on M is the pseudo-Hermitian torsion of ∇. 3.6 Expressions in local coordinates 3.6.1 Christoffel symbols ([ 37 ]) Let Tα : α ∈ {1, 2 . . . , n} be a local frame of T1,0(M ) defined on a given open set U ⊂ M . Since the Tanaka-Webster connection parallelizes the eigenbundles of J, there exist uniquely defined complex 1-forms ωβα ∈ ∞(T∗(M ) ⊗ C) (locally defined on U ) such that These are the connection 1-forms. Let us set Tα = Tα. Then, ∇Tβ = ωαβ ⊗ Tα. T1, . . . , Tn, T1, . . . , Tn is a frame of T(M) ⊗ C on U . Let us set ωβα = ωβα. Then (since ∇ is a real operator) we have with the convention T0 = T. Therefore, We denote by ∇Tγ Tβ = γαβ Tα, ∇Tγ Tβ = γαβ Tα, ∇TTβ = 0αβ Tα. For any α, β ∈ {1, . . . , n} and A ∈ {0, 1, . . . , n, 1, . . . , n}, we define the Christoffel symbols α Aβ : U −→ C the components of the Levi form. Recall that Let us set for simplicity Then on the one hand, On the other hand, taking into account that where hαβ = lθ (Tα, Tβ ). We have and contraction by hσ α leads to Using the equality [hαβ ] = [hαβ ]−1, we obtain hαβ = lθ (Tα, Tβ ) ∇X Y = UXY , X, Y ∈ T1,0(M). Uαβ = UTαTβ . γ Uαβ = αβ Tγ . (Tα, Tβ ) = −i hαβ , −i γαβ hασ = −i Tγ (hβσ ) − (Tβ , [Tγ , Tσ ])) γβ = hσ α Tγ (hβσ ) − gθ (Tβ , [Tγ , Tσ ]) α (3.17) Next, we will give the computation of the Christoffel symbols α . Since γ β hαβ hβσ = δσ . α π+[Tγ , Tβ ] = ∇Tγ Tβ = γαβ Tα, γ β = hμα gθ ([Tγ , Tβ ], Tμ) α (3.18) 3.6.2 Pseudo-Hermitian torsion Lemma 3.14 τ (T1,0(M)) ⊂ T0,1(M). The proof follows from (3.10). By Lemma 3.14, there exist uniquely defined C∞ functions Aβα : U −→ C, such that If {T1, . . . , Tn} is a local frame of T1,0(M) and {θ 1, . . . , θ n} the dual coframe in T1∗,0(M), that is, where denotes the symmetric tensor product defined by T∇ = 2 (θ ∧ τ − ⊗ T) . θ ∇ = ∇ + ( − A) ⊗ T + τ ⊗ θ + 2θ J, 2θ J (X, Y ) = θ (X )J Y + θ (Y )J X, X, Y ∈ T(M). Lemma 3.15 Let (M, T1,0(M)) be a nondegenerate CR manifold and θ a fixed pseudo-Hermitian structure on M. Let ∇ be the Tanaka-Webster connection of (M, θ ). Then the torsion tensor field T∇ of ∇ is given by Moreover, the Levi-Civita connection ∇θ of the semi-Riemannian manifold (M, gθ ) is related to ∇ by Since then Set and let Then We define τ α by Then, where τ α = τ α. For any X = xαTα, we have We have also τ Tβ = Aβα Tα. τ Tα = T∇(T, Tα) = ∇TTα − [T, Tα], Aβ T α β = −π−[T, Tα]. A(X, Y ) = gθ (τ X, Y ), X, Y ∈ T(M) Aαβ = A(Tα, Tα). Aαβ = Aγα hγ β . θ α(Tβ ) = δβ , θ α(Tβ ) = 0, θ α(T ) = 0. α τ α = Aβα θ . τ = τ α ⊗ Tα + τ α ⊗ Tα, T∇ (X, X ) = i Aαβ xα xβ − Aαβ xα xβ . dθ = 2i hαβ θ α ∧ θ β , dθ α = θ β ∧ ωβα + θ ∧ ξ α. Conformal transformation: Let θ˜ = euθ . Then (3.19) (3.20) 3.7 Curvature tensors 3.7.1 The curvature tensor field We refer here to [70]: Definition 3.16 Let (M, T1,0(M)) be a non degenerate CR manifold and θ a fixed pseudo-Hermitian structure on M. The curvature tensor field R of the Tanaka-Webster connection ∇ of (M, θ) is defined by R(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z , X, Y, Z ∈ X (M). Notation: We denote by R(X, Y, Z , W ) = gθ (R(X, Y )Z , W ), X, Y, Z , W ∈ T(M). Proposition 3.17 1) For any X, Y, Z , W ∈ T(M), we have a) R(X, Y, Z , W ) = R(Y, X, Z , W ) b) R(X, Y, Z , W ) = R(Y, X, W, Z ) c) R(X, Y, Z , W ) = R(Z , W, X, Y ) 2) R satisfies the Bianchi identity: R(X, Y )Z = (T∇(T∇(X, Y ), Z ) + (∇X T∇)(Y, Z )) , XY Z XY Z for any X, Y, Z ∈ T(M). Here XY Z denotes the cyclic sum over X, Y, Z . Remark 3.18 For any X, Y, Z ∈ H(M), we have Then, Denote by Then, Let {T1, . . . , Tn} be a local frame of T1,0(M). Then T∇(T∇(X, Y ), Z ) = −2 (X, Y )ξ(Z ), (∇X T∇) (Y, Z ) = −2 (∇X ) (Y, Z )T = 0. R(X, Y )Z = −2 (X, Y )τ (Z ), X, Y, Z ∈ H(M). XY Z XY Z R(X, Y )Tα = ∇X ∇Y Tα − ∇Y ∇X Tα − ∇[X,Y ]Tα = ∇X ωαβ(Y )Tβ − ∇Y ωαβ(X)Tβ + ωαβ([X, Y ])Tβ = (dωαβ)(X, Y )Tβ − ωαβ(Y )ωαγ (X) − ωαβ(X)ωαγ (Y ) Tγ . Rβαρσ = R(Tβ, Tα, Tρ, Tσ ). α γ α γ Rβαρσ = Tρ( βασ ) − Tσ ( βαρ) + βγ ρσ − βγ σρ + iδσρ βα0. 3.7.2 Pseudo-Hermitian Ricci and scalar curvatures Let {T1, . . . , Tn} be a local frame of T1,0(M). Definition 3.19 The Ricci tensor of the Webster connection is defined by The pseudo-Hermitian Ricci tensor is then given by Ricαβ = Ric(Tα, Tβ ).. Definition 3.20 The Webster scalar curvature is defined by Ric(Y, Z ) = traceX −→ R(X, Z )Y , Y, Z ∈ T(M). Rθ = hαβ Ricαβ where uα = Tα(u), uα = Tα(u), uγ = hγ β uβ , and uγ = uγ . 3.8 The sub-Laplacian operator ([ 37 ]) 3.8.1 Divergence of a vector field Definition 3.22 Let (M, T1,0(M)) be a nondegenerate CR manifold, θ a fixed pseudo-Hermitian structure on M and ∇ be the Tanaka-Webster connection of (M, θ ). The divergence of a vector field X , div(X ), is defined by Let {T1, . . . , Tn} be a local frame of T1,0(M) on an open set U ⊂ M. Then for any Z = Z αTα, we have div(X ) = trace {Y ∈ T(M) −→ ∇Y X } . div(Z ) = Tα(Z α) + Z β ααβ . 3.8.2 The adjoint of a vector field Let M be a 2n + 1-dimensional non degenerate CR manifold and θ a pseudo-Hermitian structure on M. Proposition 3.23 The 2n + 1-form θ ∧ (dθ )n is a volume form on M. In other words, θ is a contact form on M. We define the L2(M) inner product (u, v) = M uv θ ∧ (dθ )n, u, v ∈ L2(M). (X u, v) = (u, X ∗v), u, v ∈ L2(M). Definition 3.24 The adjoint X ∗ of a vector field X is defined by Let (M, T1,0(M)) be a strictly pseudo-convex CR manifold, of real dimension 2n + 1 and θ be a pseudoHermitian structure. Let {Xa, 1 ≤ a ≤ 2n} be a local gθ -orthonormal frame of H(M) (i.e., lθ (Xa, Xb) = δab) defined on an open set U ⊂ M and (U, x1, . . . , x2n+1) be a local coordinate system on U . Then Xa = bai ∂∂xi , where bai ∈ C∞(U ). Proposition 3.25 For any 1 ≤ a ≤ n, the adjoint of the vector field Xa is given by where ijk are the local coefficients of ∇ with respect to the local frame { ∂∂xi , 1 ≤ i ≤ 2n + 1}. Proposition 3.26 Let {Tα, 1 ≤ α ≤ n} be a local frame of T1,0(M) on an open set U . Then ∂ Xa∗u = − ∂ xi (baiu) − baj iij u, u ∈ C0∞(U ), Tα∗ = −Tα + n α βαβ . 3.8.3 The sub-Laplacian operator Let (M, T1,0(M)) be a strictly pseudo-convex CR manifold, of real dimension 2n + 1 and θ be a pseudoHermitian structure. Let πH : T(M) −→ H(M) the natural projection associated with the direct sum decomposition T(M) = H(M) ⊕ RT (T be the characteristic direction of (M, θ )). Definition 3.27 Let f ∈ C2(M). We define the Hessian of f , ∇2 f , by (∇2 f )(X, Y ) = (∇X d f )Y = X (Y ( f )) − (∇X Y )( f ) − (∇Y X )( f ), X, Y ∈ X (M). Let {Tα, 1 ≤ α ≤ n} be a local frame of T1,0(M) on an open set U . We denote by fα = Tα( f ), fα = Tα( f ), f0 = T( f ), f AB = (∇2 f )(TA, TB ), A, B ∈ {0, 1, . . . , n, 1¯, . . . , n} Then, fαβ = Tα( fβ ) − αγβ fγ , fαβ = Tα( fβ ) − αγβ fγ , f0β = T( fβ ) − 0γβ fγ , fα0 = Tα( f ). ∇H f = πH∇ f, Definition 3.28 The horizontal gradient ∇ H is defined by where ∇ f is the ordinary gradient of f with respect to the Webster metric i.e., gθ (∇ f, X ) = X ( f ), for any X ∈ X (M). Definition 3.29 The sub-Laplacian operator of M is the operator b defined by b f = −div(∇H f ), f ∈ C2(M). Proposition 3.30 Proposition 3.31 For any u, v ∈ C2(U ) we have b f = fαα + fαα n = fα α + = − n α=1 α=1 fαα Tα∗Tα( f ) + Tα∗Tα( f ) . U ( bu)v θ ∧ (dθ )n = − gθ ∇Hu, ∇Hv . U (3.21) where Since and z j = x1j + i x2j , 1 ≤ j ≤ n. Define T1,0(Hn) as the space spanned by the Tj s, i.e., it follows that (Hn, T1,0(Hn)) is a CR manifold. Next, we consider the real 1-form θ0 on Hn defined by it is a pseudoHermitian structure on (Hn, T1,0(Hn)). By differentiating (3.24) we obtain by taking into account (3.3), it follows that 3.9 Examples 3.9.1 The Heisenberg group The Heisenberg group Hn is the homogeneous Lie group whose underlying manifold is Cn × R = R2n+1 with coordinates (z1, . . . , zn, t) = (x11, x21, . . . , x1n, x2n, t), where for all 1 ≤ j ≤ n, z j = x1j + i x2j , and whose group law is defined by ∀(z1, . . . , zn, t), (w1, . . . , wn, s) ∈ Hn ⎛ (z1, . . . , zn, t)(w1, . . . , wn, s) = ⎝ z1 + w1, . . . , zn + wn, t + s + 2Im n j=1 z j w j ⎞ ⎠ We consider the complex vector fields on Hn: ∂ ∂ Tj = ∂ z j + i z j ∂t , Tj = ∂ z j − i z j ∂ , ∂ ∂t ∂ 1 ∂ z j = 2 ∂ ∂ ∂ x1j − i ∂ x2j , ∂ ∂ z j = 2 1 ∂ ∂ ∂ x1j + i ∂ x2j , (3.22) (3.23) (3.24) T1,0(Hn) = CTj . n j=1 [Tj , Tk ] = 0, ∀ 1 ≤ j, k ≤ n, θ0 = dt + i z j dz j − z j dz j , n j=1 n j=1 where Tj = Tj , 1 ≤ j ≤ n. ∂ The Reeb field of (Hn, θ0) is T = ∂t . The Horizontal gradient of (Hn, θ0) is given by and the sub-Laplacian operator of (Hn, θ0) is given by ∇ H = (X1, . . . , Xn, Xn+1, . . . , X2n) n α=1 b = − Xα2 + Xα2 . Definition 3.32 The map δλ : Hn −→ Hn given by δλ(z, t) = (λz, λ2t), for any (z, t) ∈ Hn, is called the dilation by the factor λ > 0. Proposition 3.33 Each dilation is a group homomorphism and a CR isomorphism. Definition 3.34 The Heisenberg norm is ρ : Hn −→ R+ (z1, . . . , zn, t) −→ ⎝ ⎛ n j=1 The Heisenberg group H1 is the Lie group R3 equipped with the law defined for all ξ = (x; y; t), ξ0 = (x0; y0; t0) ∈ R3 by ξ0ξ = (x0 + x; y0 + y; t0 + t + 2(x y0 − yx0)) ρ(ξ ) = ((x2 + y2)2 + t2) 41 and δλ(ξ ) = (λx; λy; λ2t), for ξ = (x; y; t) ∈ H1. Proposition 3.35 A basis of the corresponding Lie algebra: the Heisenberg algebra h = T0H1 is given by the following vector fields: Proof Let f be a differentiable function defined on H1; we have, respectively, ∂ ∂ X (x, y, t) = ∂ x + 2y ∂t ; ∂ ∂ Y (x, y, t) = ∂ y − 2 ∂t ; f (x + , y, t + 2y ) − f (x, y, t) T ( f ) = lim →0 Definition 3.36 A tangent vector is left invariant if for all f ∈ C∞(H1), we have where fh is the left translation of f in H1 given by V ( fh) = (V f )h, fh(g) = f (hg), ∀h, g, ∈ H1. Proposition 3.37 The vector fields X, Y and T are left invariant. Proof Let f be a left translation on H1, for example f = L(s,t,u) , (s, t, u) ∈ H1; then ,we have f (x, y, z) = L(s,t,u)(x, y, z) = (s, t, u) ◦ (x, y, z) = (x + s, y + t, z + u + 2(t x − sy)), ∀(x, y, z) ∈ H1 The derivative of f is hence On the other hand, we have therefore, X f = X ◦ f ; hence X is left invariant. It is the case for the vector fields Y and Z , since we have: The Lie brackets are given by Lemma 3.38 The left invariant metric g on H1 is [X, Y ] = 4Z , [X, Z ] = [Y, Z ] = 0. gθ = 4(dx2 + dy2) + (dz + 2xdy − 2ydx)2. Proof Let gθ denote the Riemannian metric associated with the Levi form of H1 gθ (u, v) = gθ,i j dxi ⊗ dy j where gθ,i j = gθ ∂∂xi , ∂∂x j . where from (3.22) hence, Z = 21 (X + iY ) and Z¯ = 21 (X − iY ) Z = ∂z j + i z j ∂ , ∂ ∂t Z = ∂z j − i z j ∂ ∂ ∂t ; gθ (X, T ) = Lθ (X, T ) = dθ(T, −Y ) = 0. gθ (Y, T ) = Lθ (Y, T ) = dθ(T, X) = 0. gθ (X, Y ) = Lθ (X, Y ) = dθ(X, −X) = 0. Finally, a simple computation gives The result follows since gθ (T, T ) = Lθ (T, T ) = 1. 4 = gθ (X, X) = gθ ∂∂x − 2y ∂∂t , ∂∂x − 2y ∂∂t = gθ ∂∂x , ∂∂x + 4ygθ ∂∂x , ∂∂t + 4y2gθ ∂∂t , ∂∂t − 2xgθ ∂∂x , ∂∂t + 2ygθ ∂∂t , ∂∂y − 4x y; gθ,12 = −4x y gθ,23 = 2x. gθ,11 = 4(1 + y2) gθ,22 = 4(1 + x2) ⇒ gθ,13 = gθ ∂∂x , ∂∂t = −2y, g = g(u, v) = gθ,i j dxi ⊗ dy j 1≤i, j≤3 = gθ,11dx2 + gθ,12dxdy + gθ,21dydx + gθ,13dxdt + gθ,31dtdx +gθ,22dy2 + gθ,23dydt + gθ,32dtdy + gθ,33dt2. gθ (X, X) = dθ(X, J X) = dθ(X, J (Z + Z¯ ) = dθ(X, i(Z − Z¯ ) = dθ(X, −Y ) == θ(4T ) = 4. gθ (Y, Y ) = dθ(Y, J Y ) = dθ(X, J Z −i Z¯ = −dθ(Y, Z + Z¯ ) = dθ(X, Y ) = θ[X, Y ] = θ(4T ) = 4, (3.25) (3.26) (3.27) (3.28) (3.29) The dual basis associated with ε = (e1 = X, e2 = Y, e3 = T ) is the triplet of 1-forms (θ 1, θ 2, θ 3) satisfying the following conditions: θ i (e j ) = δi j δi j is the Kronecker symbol. This base is given by θ 1 = dx θ 2 = d y θ 3 = dt + 2(x d y − ydx ) θ 3 = dt + 2(x d y − ydx ) is the contact form θ0 (see (3.24) of the CR structure of H1. The Tanaka Webster connection The Tanaka Webster connection ∇ associated with the contact form θ0 expressed on the basis (X, Y, T ) and using (3.11)is given by ∇X Y = πH([X, Y ]) = π+(4T ) = 0 ∇X T = πH([X, T ]) = 0 ∇Y T = πH([Y, T ]) = 0, where πH is the projection on the distribution H for the decomposition of the tangent space, see (3.7). Hence, ∇ is identically zero since T is parallel to the connection ((3.13)). The torsion tensor The torsion tensor associated with ∇ is given by T∇ (X, Y ) = ∇X Y − ∇Y X − [X, Y ]) = 4T T∇ (X, T ) = ∇X T − ∇T X − [X, T ]) = 0 T∇ (Y, T ) = ∇Y T − ∇T Y − [Y, T ]) = 0 (3.30) (3.31) (3.32) (3.33) (3.34) (3.35) (3.36) (3.37) (3.38) The pseudo-Hermitian torsion We have τ (W ) = T∇ (T , W ), W ∈ T M . So τ (X ) = T∇ (T , X ) = 0 τ (Y ) = T∇ (T , Y ) = 0 τ (T ) = T∇ (T , T ) = 0 Hence, the pseudo-Hermitian torsion is identically zero. Curvature tensors Using the expression of the curvature tensor R given in (3.18) where Hence, the tensor curvature R is identically zero; therefore, both the Ricci Tensor Ric and the Webster scalar curvature Rθ are zero. 3.9.3 The CR manifold S2n+1 Let S2n+1 be the unit sphere of Cn+1. The standard CR structure of S2n+1 is given by T1,0(S2n+1) = T1,0(Cn+1) ∩ T(S2n+1) ⊗ C , T1,0(Cn+1) = span 1) If s < 1 + n2 , then there exist q < 2 + n2 , and p1 < 1 + n2 such that L(2+ n2 ) is only continuous but not compact. 2) Let s = 1 + n2 , to ensure the embedding of L(2+ n2 ) p11 (M) in S12(M); we suppose the existence of such real p1 ≤ 1 + n2 . It yields that equality (4.5) gives q = 2 + n2 . Hence, in this case, the inclusion of S12(M) in apply to solve the scalar curvature equation (PK ). Therefore, the function F1+ n2 is not compact and the methods based on compactness arguments do not 4.3 Natural change of the CR functional We write u ∈ S12(M) as u = (|u|−L . |u|u−L ) = (λ.v). Hence, 1 2 I (u) = I (λ.v) = 2 λ − 2 + n M 1 ⎟ ⎟ K v2+ n2 θ ∧ dθ n ⎠ The second derivative in λ(v) is M . 1 K v2+ n2 θ ∧ dθ n Iλ (λ.v) = 1 − (4.5) (4.6) (4.7) (4.8) Hence, the second derivative at the critical point λ(v) = ( maximum for I (λ.v). A critical point of I has to satisfy M K v2+ n2 θ ∧ dθ n n ) 2 is negative, so λ(v) is a since (1) is realized, if λ.v = λ(v).v, (2) is equivalent to Iv(λ(v).v) = 0. We consider the following function: Where S is the unit sphere of the space S12(M) for the norm ||−L . Denote by Tv S the tangent space of S at v; we have : S ⊂ S12(M) −→ 1 S12(M) −→ 2 R v −→ λ(v).v −→ I (λ(v), v) (v) : Tv S −→ R h −→ (v)(h) = (I (λ(v).v)) (h) since for Hence, we have a new choice for the functional; we set The maximum is attained in a unique point λ = λ(v) and and This equation is reduced to (I (λ(v).v)) (h) = I (λ(v).v))[λ (v).v + λ(v)](h). (I (λ(v).v)) (h) = λ(v) I (λ(v).v))(h) d I (λ(v).v))(v) = dλ I (λv) |λ(v)= 0. J (v) = I (λ(v).v)) v ∈ S = {v ∈ S12(M), |v|−L = 1} J (v) = mλ>a0x I (λ(v).v)) J (v) = (2n + 2) M 1 n K v2+ n2 θ ∧ dθ n 2 The functionals I and J are both of class C2 and there is a one-to-one correspondence between the non zero critical points of I and J. 4.4 Hessian and the Morse Lemma [64] Definition 4.1 Let M be a compact C∞ manifold; a point x0 is said to be a non degenerate critical point of a smooth function f : M → R if 1) the derivative of f at x0, d fx0 ≡ 0 2) the Hessian of f at x0, H ess f (x0) is a non degenerate quadratic form. Definition 4.2 If M is of finite dimension, the index of the non degenerate critical point x0 of f , denoted by i nd(x0), is the number of negative eigenvalues of H ess f (x0). Theorem 4.3 Morse Lemma Let M be a C∞ manifold of dimension n and f : M → R a smooth function having a non degenerate critical point x0 of index λ. There exists an open neighborhood U of and a local chart y = φ(x), φ : U −→ Rn such that φ(x0) = 0 and f ◦ φ−1(y) = f ◦ φ−1(0) − 2 yi + i=λ i=1 i=n i=λ+1 yi2 (4.9) 4.5 Homotopy and homotopy type • F(x, 0) = f0(x), ∀x ∈ X. • F(x, 1) = f1(x) ∈ Y, ∀x ∈ X. It is an equivalence relation. Definition 4.4 Let X and Y be two topological spaces, f0 : X → Y and f1 : X → Y two continuous maps from X to Y. We say that f0 is homotopic to f1 if there exists a continuous map F : X × [ 0, 1 ] −→ Y such that Definition 4.5 Two topological spaces X and Y are said to have the same homotopy type or homotopy equivalent, if there exist f : X −→ Y and g : Y −→ X two continuous maps such that g ◦ f is homotopic to the identity of X and f ◦ g is homotopic to the identity of Y. Definition 4.6 A topological space X is said to be contractible if it has the same homotopy type of "a point". Remark 4.7 A contractible space is simply connected. The results and definitions given in this section are extracted from [64] 4.6 Deformation retract Definition 4.8 Let Y ⊂ X be two topological spaces and r : X → Y an onto continuous map from X to Y ; r is called a retraction by deformation of X onto Y, if • r ◦ iY = I dY , (iY : Y → X is the inclusion map.) • iY ◦ r is homotopic to I dX . Y is said to be a deformation retract of X. Let M be a compact manifold and f : M −→ R a smooth function defined on M. Let a, b ∈ R, a < b. Define Ma := f −1((−∞, a]) = {x ∈ M : f (x) ≤ a} Mab := f −1([a, b]) = Mb\int(Ma) Theorem 4.9 We suppose that Mab is compact and contains no critical point of f . Then Ma is diffeomorphic to Mb. Furthermore, Ma is a deformation retract of Mb so that the inclusion map i : Ma → Mb is a homotopy equivalence. Theorem 4.10 Let f : M −→ R be a smooth function and let x0 be a non degenerate critical point with index λ. Setting f (x0) = c, suppose that f −1([c − , c + ]) is compact, and contains no critical point of f other than x0, for some > 0. Then, for all sufficiently small , the set Mc+ has the homotopy type of Mc− with a λ-cell attached. We give here some elements of the proof of the second theorem and for detailed proofs of the two theorems above one can see [64]. From the Morse Lemma, we have Denote by Br−( ) = {φ−1(X, 0), |X |2 ≤ } and suppose that the scalar product around x0 is defined in the coordinates (X, Y ) by |X |2 + |Y |2. The gradient of f is then given by hence gr ad f(X,Y ) = 2 ⇔ ⎩⎪ ⎧ ∂ f ◦ φ−1 ⎪⎨ ∂ X ∂ f ◦ φ−1 ∂Y (X, Y ) = −2X (X, Y ) = 2Y (4.10) The differential equation, ("X#,˙$Y%) = grad f(X,Y ) ⇔ is easy to integrate and gives ⎧ ∂ X ⎨⎪ ∂s = −2X ∂Y ⎩⎪ ∂s = 2Y, X (s) = e−2s X (0) Y (s) = e2s Y (0) (4.11) (4.12) The trajectories of this differential equation "crash" on the Y axis when s → +∞ and " crash" on the X axis when s → −∞. So, the differential equation is defined on the hole manifold M and admits a solution η(s, z), ∀s and ∀z ∈ M since M is compact. Definition 4.11 The unstable manifold of x0 for the vector field grad f or ∇ f denoted by Wu (x0) is the set of points z ∈ M for which the solution η(s, z) converges to x0 as s → +∞. Definition 4.12 The stable manifold of x0 for the vector field grad f or ∇ f denoted by Ws (x0) is the set of points z ∈ M for which the solution η(s, z) converges to x0 as s → −∞. The dimension of the unstable manifold Wu (x0) is equal to the index of the critical point x0; in fact we have Wu (x0) = Br−( ). If > 0 is sufficiently small then M c+ retracts by deformation on M c− & Br−( ). 4.7 The Palais–Smale condition The Palais–Smale condition, which relates both to the function and to the metric, imposes conditions at infinity which allow a control of the dynamics of the gradient on a manifold without boundary. Suppose that I is a differentiable functional on a Hilbert space H ; we say that I satisfies the Palais–Smale condition, (PS) in short, if for any sequence (uk ) of H satisfying 1) I (uk ) is bounded. 2) ∂ I (uk ) −→ 0. We can extract from (uk ) a convergent subsequence. Proposition 4.13 If the Palais–Smale condition is not satisfied by a functional I in an interval [a, b] and if I b does not retract by deformation onto I a , then I admits a critical value in [a, b]. Since the injection S2(M ) in L2+ n2 (M ) is continuous but not compact, the functional J given by (4.9) 1 fails to satisfy the (PS) condition. One can see that the standard solutions of the Yamabe problem on Hn after superposition are the good candidate sequences which violate the Palais–Smale condition. 4.8 The CR Yamabe problem In [56], Jerison and Lee have extensively studied the CR Yamabe problem and showed that there is a deep analogy between the CR Yamabe problem and the Riemannian one. Their results can be formally compared to the partial completion of the proof of the Riemannian Yamabe conjecture by Aubin. As it was the case in the Riemannian settings, the CR Yamabe equation is the Euler–Lagrange equation for the constrained variational problem λ(M ) = inf u∈S12(M) M M In 1986, Jerison and Lee [56] gave a necessary condition on the conformal constant (4.13) to have existence of solutions for the CR Yamabe problem: Theorem 4.14 Let M be a compact, orientable strictly pseudo-convex and integrable CR manifold of dimension 2n + 1, and θ any contact form on M . 1) λ(M ) depends on the CR structure on M , not on the choice of θ . 2) λ(M ) ≤ λ(S2n+1), where S2n+1 ⊂ Cn+1 is the unit sphere with its standard CR structure. 3) If λ(M ) < λ(S2n+1), then the infimum in (4.13) is attained by a positive C ∞ solution of ( Pλ). If we denote 2 this solution u, the contact form θ˜ = u n θ has a constant Webster scalar curvature Rθ = λ(M ). Theorem 4.14 (1) is an analogue of Aubin’s Theorem 2.2 for the Riemannian Yamabe problem and its proof is also similar to the one given for Aubin’s Theorem 2.2. Since the asymptotic expansion of the Yamabe functional Y (M, θ ) = M R θ ∧ dθ n M θ ∧ dθ n 2p on M is expressed in terms of pseudoHermitian curvature and torsion invariants. In order to make the calculation as easy as possible Jerison and Lee refined their notion of normal coordinates, defined in [56], by constructing in [58] new intrinsic CR normal coordinates for an abstract CR manifold. These coordinates are called pseudoHermitian normal coordinates. Using these coordinates, Jerison and Lee have simplified the pseudo-Hermitian curvature and torsion invariants at a base point q and showed that the contact form can be chosen in a neighborhood of q so that the pseudo-Hermitian Ricci and torsion tensors and certain combination of their covariant derivatives vanish at this base point. The notions and results introduced and proved by Jerison and Lee are parallel, with drastically different techniques, to the one introduced by Lee and Parker for the Riemannian Yamabe conjecture [61]. For a family of contact forms θ which concentrate more and more around the base point q. The asymptotic expression of the functional for θ is given by Y (M, θ ) = λ(S2n+1)(1 − c(n)|S(q)|2 4) + O( 5) for n ≥ 3 λ(S2n+1) 1 − c(2)|S(q)|2 4 log 1 + O( 4) for n = 2. (4.14) Here, S(q) is the Chern curvature tensor [ 30 ] of M evaluated at q and c(n) > 0 S is identically zero precisely when M is locally CR equivalent to the sphere. In 1987, Jerison and Lee established the following result: Theorem 4.15 [58] Let M be a compact strictly pseudo-convex 2n + 1 dimensional CR manifold. If n ≥ 2 and M is not locally CR equivalent to S2n+1, then λ(M ) < λ(S2n+1). Hence, the CR Yamabe problem can be solved on M. H = u ∈ S12(M )/ M |du|θ2 θ ∧ dθ n < ∞, M |u|2+ n2 θ ∧ dθ n < ∞ . The remaining cases left open by Jerison and Lee should by analogy be solved by using some CR positive mass theorem. Unfortunately, such a CR version of the positive mass theorem did not exist at that time. Besides the proof of Aubin and Schoen of the Riemannian Yamabe conjecture another proof by Bahri [ 6 ], Bahri and Brézis [ 17 ] of the same conjecture was available by techniques related to the theory of critical points at infinity. This proof is completely different in spirit as well as in techniques and details from the proof of Aubin and Schoen. It does not require the use of any theory of minimal surfaces neither the use of a CR positive mass theorem. It turns out that this proof can be carried to the CR framework. We consider the subspace of S12(M ), defined by Let = {u ∈ H, st u H = 1} , u H = 1 M ((2 + n ) |du|θ2 + Rθ u2)θ ∧ dθ n 2 , and let 2 For u ∈ H, we define the CR Yamabe functional: , /u > 0} . If u is a critical point of J on n +, then J (u) 2 u is a solution of the Yamabe equation. Let us recall that the standard solutions of the CR Yamabe equation on the Heisenberg group , Hn, are obtained by left translations and dilations (z, t ) → (λz, λ2t ), (λ ∈ R) of the functions u(z, t ) = K |w + i |−n , w = t + i |z|2 (z, t ) ∈ H n, K ∈ C. Since the injection S2(H n) → L2+ n2 (H n ) is continuous but not compact, the functional J does not satisfy the 1 Palais–Smale condition denoted by (PS). More precisely, one can see that the standard solutions on H n after superposition are the good candidate sequences which violate (PS). Therefore, the classical variational theory, based on compactness arguments, does not apply in this case. The cases left open by Jerison and Lee of the CR Yamabe problem have been the purpose of two papers [43,44] published in 2001, in the first paper Yacoub and Gamara solved the CR Yamabe problem for spherical CR manifolds. The main result of [44] is Theorem 4.16 Let (M, θ ) be an orientable compact (2n +1)-dimensional CR manifold, locally CR equivalent to S2n+1; then the CR Yamabe problem has a solution. In the second paper [43] Gamara completed the resolution of the CR Yamabe conjecture for all dimensions by solving the 3-dimensional non spherical case: Theorem 4.17 Let (M, θ ) be a compact 3-dimensional CR manifold, not locally CR equivalent to the sphere S3; then the CR Yamabe problem has a solution. The proofs of Theorems 4.16 and 4.17 are both based on a contradiction argument. Before giving a sketch of the proofs of these theorems, we will introduce the general settings. 4.8.1 Spherical CR manifold: general settings 2 Let (M, θ ) be a compact spherical CR manifold; we show the existence of a conformal factor u˜an depending dif2 2 ferentiably on a ∈ M, such that if θ is replaced by u˜an θ in a ball B(a, ρ), then (M, u˜an θ ) is locally (H n, θ0). We may use in B(a, ρ) the usual multiplication of H n and the standard solutions of the CR Yamabe problem, which we denote by δ(b, λ), where λ ∈ R. The function δ(b, λ) satisfies Lθ0 δ(b, λ) = δ(b, λ)1+ n2 on B(a, ρ) b ∈ B(a, ρ) We then define on M a family of "almost solutions", which we denote by δ(a, λ). These functions are the solutions of ' where Lδ(a, λ) = δ (a, λ)1+ n2 , ' δ (a, λ) = ωa u˜a δ(a, λ) on B(a, ρ) δ (a, λ) = 0 on Bc(a, ρ) ((δˆ(a, λ) − δ (a, λ)(( = O ( ( ((δˆ(a, λ) − δ (a, λ)(((C2 = O ( 1 λn 1 λn , , when λ → ∞. Here ωa is a cut-off function used to localize our function near the base point a; as λ goes to infinity, we show that these "almost solution" closely approximate at infinity the Yamabe solutions of the Heisenberg group Neighborhoods of critical points at infinity Following Bahri [ 6,15 ], we set the following definitions and notations: Definition 4.18 A critical point at infinity of J , on + is a limit of a flow line u(s) of the following equation: ∂u ∂s = −∂ J (u) u(0) = u0 We define neighborhoods of critical points at infinity of J as follows: + s.t, there exist p concentration ⎫ p cpooninctesnatr1a,tiii.==o.1np.s'aδλ(pa1ini,,.Mλ. i.)aλ)))npds.<t ε ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬ )H with λi > 1ε , and for i = j ⎪⎪⎪⎪⎪⎪ λ j ⎪⎩⎪⎪ εi j = λλij + λi + λi λ j d˜2(ai , a j ) −n L2+ n2 (H n)). where d˜(x, y), if x and y are in a small ball of M of radius ρ, is ))exp−x1(y)))Hn expx is the CR exponential map for the point x and d˜(x, y) is equal to ρ2 otherwise. (S is the Sobolev constant for the inclusion S12(H n) → Because of the estimates of the "almost solution" given above, we can replace in the analysis of the (PS) condition, the functions δ or δ by the functions δ. Hence, we will be able to characterize the sequences of functions which violate the Palais–Smale conditio'n. In fact, we prove that, if (uk ) is a sequence of H satisfying ∂ J (uk ) → 0 and J (uk ) bounded, then (uk ) has a weak limit u in H . Hence, If u is non-zero, we prove that u is a critical point of J. Since we will prove the CR Yamabe problem using a contradiction argument, we suppose that (P) has no solutions. Then we have the following characterization of the sequences failing (PS) condition Proposition 4.19 Let {uk } be a sequence such that ∂ J (uk ) → 0 and J (uk ) is bounded. Then there exists an integer p ∈ N+, a sequence εk → 0 (εk > 0) and an extracted subsequence of (uk ), such that uukk ∈ V ( p, εk ). This proposition was first introduced in the Riemannian settings in [ 15,17 ]; the proof follows from iterated blow-up around the concentration points. For the CR settings, a complete proof is given in [44]. 4.8.2 CR Yamabe problem: ideas of the Proof of theorem 4.16 Considering for p ∈ N, the formal barycentric sets where δxi is the Dirac mass at the point xi , and the following level sets of the functional J We define a map f p(λ) from Bp(M) to B0(M) = ∅ Bp(M) = αi δxi , αi = 1, αi > 0, xi ∈ M , p 1 u ∈ + by p i=1 Wp = 1 + / J (u) < ( p + 1) n S . f p(λ) ⎝ ⎛ i=p αi δxi ⎠ = i=p i=1 αi 'δ(xi , λi ) i=p i=1 αi 'δ(xi , λi ) . It is proved in [44] that Theorem 4.20 1) For any integer p ≥ 1, there exists a real λp > 0, such that f p(λ) sends Bp(M) in Wp, for any λ > λp. 2) There exists an integer p0 ≥ 1, such that for any integer p ≥ p0 and for any λ > λp0 , the map of pairs f p(λ) : (Bp(M), Bp−1(M)) → (Wp, Wp−1), is homologically trivial i.e., where and H∗(•) is the homology group with Z /2Z coefficients of •. f p∗ (λ) = 0, f p∗ (λ) : H∗(Bp(M), Bp−1(M)) → H∗(Wp, Wp−1). On the other hand, arguing by contradiction, we will assume that the weak limit u of any ( P S) sequences (uk ) of H satisfying ∂ J (uk ) → 0 and J (uk ) bounded, is zero; otherwise, our problem would be solved, since we would have found a solution. Then assuming that (uk ) is non- negative, we prove that we can extract from (uk ) uk a subsequence denoted again by (uk ), such that uk H ∈ V ( p, εk ) with εk > 0 and limk→∞ εk = 0. In this case, we proved that the pair (W p, W p−1) retracts by deformation on the pair (W p−1 ∪ A p, W p−1), where A p ⊂ V ( p, ε). More precisely, we prove that the elements of A p are of the form ii==1p αi 'δxi ,λi + v, v small in the norm H . Therefore, the model …⊂ Bp−1(M ) ⊂ Bp(M ) ⊂ · · · can be compared via f p to.... ⊂ W p−1 ⊂ W p, and we proved that f p∗ (λ) = 0 , for every p ∈ N∗, which is a contradiction with the result of Theorem 4.20 and, therefore, achieves the proof of the CR Yamabe problem in this case. 4.8.3 CR Yamabe problem: ideas of the proof of theorem 4.17 In the paper [43], it is shown how the techniques of critical points at infinity can settle the case of a strictly pseudo-convex CR manifold (M, θ ) of dimension 2n + 1, without assuming that M is locally conformally flat. In fact in [43], we focus on the case n = 1, but the techniques apply to higher CR dimensions with no more assumptions. We have just to follow the sketch of the proof given for the case n = 1, with introducing where it is needed some required modifications due to the dimension of the CR manifold. Here, we will give some ideas about the proof for the CR Yamabe problem in the case of a generic 3-dimensional non spherical CR manifold. The proof of the result in this case is similar to the one given for the CR spherical case; it is obtained by using a contradiction argument. We use the same techniques given by Bahri and Brézis in [ 17 ]. However, in this case the study of “the almost solutions” δˆa is not straightforward as in the CR spherical case, where we have locally a relation between the conformal Laplacians of M and H n. Here, we have to use the Green’s function associated with L to derive a good asymptotic expansion of the Yamabe funtional J near the sets of its critical points at infinity. Finally, to compute the numerator and the denominator of J, we used the approach of Jerison and Lee who refined in [58] the notion of normal coordinates by constructing the so-called pseudo-Hermitian normal coordinates. In pseudo-Hermitian normal coordinates, Jerison and Lee gave the Taylor series of θ and {θ α} to high order at a base point q ∈ M, in terms of the pseudo-Hermitian curvature and torsion. Since the problem is CR invariant they had to choose θ so as to simplify the curvature and the torsion at a base point q as much as possible. Using the results of [58], we proved the following estimates in pseudo-Hermitian normal coordinates near a base point q ∈ M Gq (z, t ) = C (ρ−2(z, t )) + A + O(ρ(z, t )), Gq > 0, R = O(2), W = Z + O(3), W = Z + O(3), L = −2(Z Z + Z Z ) + O(2) where O(m) is a homogenous polynomial in ρ of degree a least m. Using these estimates and the topological method based on the theory of critical points at infinity explained earlier, we derive the result in this case. 4.9 The case of a CR spherical pseudo-Hermitian manifold of dimension 3 4.9.1 Introduction and main results Let (M, θ ) be a compact spherical pseudo-Hermitian 3-dimensional manifold and K a C2 positive function defined on M. Our aim was to find suitable conditions on K such that we can find a contact form θ˜ conformal to θ having K as Webster scalar curvature. The new contact form reads θ˜ = u2θ , where u is a positive function on M . The problem of prescribing the Webster scalar curvature is equivalent to the resolution of following partial differential equation: where −L is the conformal Laplacian of M , −L = −4 θ + Rθ where (M, θ ) and Rθ is the Webster scalar curvature of (M, θ ). θ is the sub-Laplacian operator on (4.15) Problem ( PK ) has a variational structure, with associated Euler functional: u ∈ S12(M ); u S12(M) = M −Lu u θ ∧ dθ = 1 and As for the Yamabe problem, the functional J fails to satisfy the Palais–Smale condition on +, which means that there exist noncompact sequences along which the functional J is bounded and its gradient goes to zero. The failure of the (PS) condition has been analyzed for the Riemannian case throughout the works of [ 5,15,18,27,28,60,62,65,67 ]. For the CR case, a complete description of sequences failing to satisfy (PS) is given in [44]. Since this problem has been formulated, obstructions have to be pointed out. The main encountered difficulty, when one tries to solve equation of type ( PK ) consists of the failure of the Palais–Smale condition, which leads to the failure of classical existence mechanisms. We will use a gradient flow to overcome the noncompactness. Thinking of the sequences failing to satisfy the Palais–Smale condition as "critical points", our objective was to try to find suitable parameters, in order to complete a Morse Lemma at infinity analogous to the one given for the Riemannian case. The Morse Lemma is crucial to prove the existence of solution for equation ( PK ); more precisely, the method, we used to prove the existence of solutions for problem ( PK ) is based on the work of Bahri [ 5,18,24 ]. This method involves a Morse lemma at infinity, which establishes near the set of critical points at infinity of the functional J a change of variables in the space (ai , αi , λi , v), 1 ≤ i ≤ p to (ai , αi , λi , V ), (αi = αi ), where V is a variable completely independent of ai and λi such that J ( αi δˆai ,λi ) behaves like J ( αi δˆai ,λi ) + V 2 . The Morse lemma relies on the construction of a suitable pseudo-gradient for the associated variational prob−lLem, which is based on the expansion of J and its gradient ∂ J near infinity. we define also a pseudo-gradient for the V -variable with the aim to make this variable disappear by setting ∂∂Vs = −ν V , where ν is taken to be a very large constant. Then, at s = 1, V (s) = exp(−νs)V (0) will be as small as we wish. This shows that in order to define our deformation, we can work as if V was zero. The deformation will be extended immediately with the same properties to a neighborhood of zero in the V -variable. We prove that the Palais–Smale condition is satisfied along the decreasing flow lines of this pseudo-gradient, as long as these flow lines do not enter the neighborhood of a finite number of critical points of K . This method allows to study the critical points at infinity of the variational problem, by computing their total index and comparing this total index to the Euler-Poincaré characteristic of the space of variations. This procedure was extensively used in earlier Riemannian works and has displayed the role of the Green’s function in equation of type ( PK ). It is important to recall that for the case we review, we have a balance phenomenon between the self interactions and interactions between the functions failing to satisfy the Palais–Smale condition. To state our results we set up the following conditions and notations: Let G(a, ) be a Green’s function for L at a ∈ M and Aa the value of the regular part of G evaluated at a. We assume that K has only nondegenerate critical points ξ1, ξ2 . . . ξr such that θ K (ξi ) − 3K (ξi ) − 2 Aξi = 0, i = 1, . . . , r Assume that ξi , i = 1, . . . , r1 are the critical points of K with − 3θKK(ξ(ξi)i) − 2 Aξi > 0. Let τl = (i1, . . . , il ) denote any l-tuple of (1, . . . , r1), 1 ≤ l ≤ r1, we define the following matrix M (τl ) = (Mst ) with θ K (ξs ) Aξs Mss = − 3K 2(ξs ) − 2 K (ξs ) G(ξs , ξt ) Mst = −2 √K (ξs )K (ξt ) , for 1 ≤ s = t ≤ l ki j the index of the critical point ξi j with respect to K , then i (τl ) = 4l − 1 − critical point at infinity τl . We obtain the following result: We say that K satisfies condition (C ) if for any τl , 1 ≤ l ≤ r1, M (τl ) is nondegenerate. If we denote by lj=1 ki j is the index of the Theorem 4.21 Suppose the function K satisfies (4.16) and condition (C ). If r1 l=1 τl , M(τl )>0 This result means that if the total contribution of the critical points at infinity to the topology of the level sets of the associated functional J is not trivial, then we have a solution for ( PK ). Let (M, θ ) be a compact spherical pseudo-Hermitian manifold of dimension 3. Any point a in M has a neighborhood Va ⊃ B(a, ρ), ρ independent of a, such that the contact form of M is conformal to the contact form θ0 of the Heisenberg group H1, so if there exists a conformal factor v˜a depending smoothly on a such that 2 θ0 = v˜a θ in the ball B(a, ρ), then (M, v˜a2 θ ) is locally (H1, θ0). Therefore, we may use the usual multiplication of H1 in B(a, ρ); we also may use the standard solutions of the CR Yamabe equation which we denote by δ(a, λ), where λ is a large positive parameter; we have where (z, t ) = expa−1(ξ ) and the constant c0 is such that the following equation is satisfied: Let va (ξ ) = ωa (ξ )v˜a (ξ ), where ωa (ξ ) = χ ( ξ ), χ is a cut-off function which is used to localize the function δ(a, λ) near the base point a when λ → ∞, λ δ(a, λ)(ξ ) = c0 |1 + λ2(|z|2 − i t )| , −Lθ0 δ(a, λ) = δ3(a, λ) on B(a, ρ). χ : R −→ [ 0, 1 ] t −→ χ (t ) = 1 0 ρ if 0 ≤ t ≤ 2 if t ≥ ρ . −Lδˆ(a, λ)(ξ ) = δ 3(a, λ)(ξ ) δ (a, λ)(ξ ) = va δ(a, λ)(ξ ) 0 on B(a, ρ) on Bc(a, ρ). We define a family of "almost solutions" δˆ(a, λ) to be the unique solutions on M of with The following result of Jerison and Lee will be very useful later: Lemma 4.22 Let be in C 2(B(a, ρ), R); we have the following relation between the conformal Laplacian of M and the one of H1: 3 L(v˜a ) = v˜a Lθ0 ( ). For a proof one can see [56]. Let Ha,λ(x ) = λ(δˆa,λ − δa,λ)(x ); we have Lemma 4.23 [42] For λ large enough, there exists a constant C = C(ρ) such that |Ha,λ(x)|L∞ ≤ C, |λ ∂ Ha,λ ∂λ |L∞ ≤ C, |λ−1 ∂ Ha,λ ∂a |L∞ ≤ C. Moreover, for ρ small and λ large, we obtain and outside B(a, 2ρ) lim Ha,λ(a) = Aa, λ−→∞ lim Ha,λ(ξ ) = G(a, ξ ). λ−→∞ Now, we define the set of potential critical points at infinity of the functional J. For any ε > 0 and p ∈ N+, let +; ∃ (a1, . . . , ap) ∈ M, α1, . . . , αp > 0 and (λ1, . . . , λp) ∈ (ε−1, ∞)p s.t : p Box where εi j = λi λ j + λi λ j αi δˆai ,λi )) < ε, i=1 K (ai ) 21 )))S12(M) ⎪⎪ αi2 K (ai ) ⎪⎩⎪⎪⎪ εi j < ε, | α2j K (a j ) − 1| < ε, ∀ 1 ≤ i = j ≤ p. + λi λ j (d(ai , a j )2 −1 and d(x, y) = exp−x1(y) H1 if x and y are in a small ball r of M of radius r , and d(x, y) is equal to otherwise. 2 Let (uk ) be a sequence of + satisfying J (uk ) bounded and ∂ J (uk ) → 0; then (uk ) is a bounded sequence in S12(M); hence (uk ) has a weak limit u¯ in S12(M). If u¯ = 0, we prove that u¯ > 0, and it is a critical point of J. The proof is similar to the one given for the Yamabe case (Proposition 5 of [44]). Since we are going to prove the existence of solution for problem PK by contradiction, we assume that the weak limit u¯ of any sequence (uk ) of + satisfying J (uk ) bounded and ∂ J (uk ) → 0 is zero. Using the estimates of Lemma 4.22, one can replace in the analysis of the Palais–Smale condition the functions δ or δ by the function δˆ; we then proceed as in [44], Proposition 8 to characterize the sequences which violate the (PS) condition as follows: Proposition 4.24 Let {uk } be a sequence such that ∂ J (uk ) → 0 and J (uk ) is bounded. Then there exist an integer p ∈ N∗, a sequence εk → 0 (εk > 0) and an extracted subsequence of {uk }, again denoted by {uk }, such that uk ∈ V ( p, εk ). While the final characterization of the sequences which violate the Palais–Smale condition is basically identical to the Riemannian case, the proof is different here from the one given by Struwe in [67]. M. Struwe used the H 1- spaces and the projections on them in order to give a characterization of the sequences violating 0 the Palais–Smale condition in the Riemannian framework. We consider the following minimization problem for a function u ∈ V ( p, ε), with ε small ) min ))u − αi >0,λi >0,ai ∈M ) ) i=1 ) ) αi δˆai ,λi ) ))S12(M) . (4.16) We obtain as showed in [ 5,44 ] the following parametrization of V ( p, ε): Proposition 4.25 For any p ∈ N∗, there exists εp > 0 such that, for any 0 < ε < εp, u ∈ V ( p, ε), the minimization problem (4.16) has a unique solution (α¯ 1, . . . , α¯ p, λ¯1, . . . , λ¯p, a¯1, . . . , a¯ p) (up to permutation on the set of indices {1, . . . , p}). In particular, we can write u ∈ V ( p, ε) as follows: u = ip=1 αi δˆai ,λi + v, where v ∈ S12(M) satisfies (V0) < v, ψ >−L = 0 for all ψ ∈ δˆai ,λi , ∂ai ∂δˆai ,λi , ∂δˆai ,λi ∂λi 1≤i≤p Here <, >−L denotes the −L-scalar product defined on S12(M) by < u, v >−L = −Luv θ ∧ dθ. M One of the basic phenomena that it displays is the behavior of the functional J with respect to v. We will prove the existence of a unique v¯ which minimizes J ( ip=1 αi δˆai ,λi + v) with respect to v ∈ Hεp(a, λ), where Hεp(a, λ) = Hεp(δˆa1,λ1 , . . . , δˆap,λp ) = v ∈ S12(M); v satisfies (V0) and v −L < ε p . 4.10 Expansion of the functional near the sets of potential critical points at infinity For any u = αi δˆai,λi + v ∈ V ( p, ε), ε > 0, we have αi δˆai,λi + v ∈ V ( p, ε), ε < ε0, v satisfying where M αi2 M M −Lv vθ ∧ dθ. −Lu uθ ∧ dθ = M αi δˆai,λi + v αi δˆai,λi + v θ ∧ dθ −Lδˆai,λi δˆai,λi θ ∧ dθ + 2 αi α j −Lδˆai,λi δˆa j,λ j θ ∧ dθ i< j M All the other terms are zero since u satisfies conditions (V0). We obtain the following expansion of the functional J : Proposition 4.26 There exists ε0 > 0 such that, for any u = (V0), we have where f is a linear form in v and Q is a bilinear form in v given by J (u) = i=1 + ci j εi j + c i = j c Hi (ai ) 1 S2 λi2 H j (ai ) 0 M K (x) Q(v, v) = S p i=1 v 2−L p 2 − S i=1 αi i = j 3 (αi δˆai ,λi )3v θ ∧ dθ p i=1 αi4 K (ai ) M i=1 K (αi δˆai ,λi )2 v2. (4.17) (4.18) and Furthermore, 4 S = c0 H1 ((1 + |z|2 − i t(( ( ( i = j ⎞ εi j (Logεi−j1) 21 ⎠ Proof Let u = p given in Appendix iA=1oαfi[δˆ4a2i ,]λtih+efvo,llvowsaintigs:fies conditions (V0). We derive from the expansions of N and D p i=1 αi2 S ⎛ p =1 1 + i = j i / c × 1 + 2 i = j +4 +O +O c M M αi α j p k=1 αk S αi2 p k=1 αk S p i=1 c Hi (ai ) λi2 αi4 p k=1 αk4 K (ak )S + o αi3α j K (ai ) p k=1 αk4 K (ak )S |∇v|2) 21 |∇v|2) 23 ⎛ p i=1 + 6 i=1 ci j εi j + c + o(εi j ) ci j εi j + o(εi j ) + 1 + λ2 i αi2 K (ai ) p k=1 αk4 K (ak )S B(ai ,ρ) + v 2 p −L 0 k=1 αk S + i = j + f (v) 1 ⎞ εi j Logεi−j1 2 (δ(ai ,λi ))2v20 2 −1 Lemma 4.27 ([42]) For ε > 0 very small, there is α0 > 0 such that, for all v ∈ Hεp(a, λ), Q(v, v) ≥ α0 v 2−L . Lemma 4.28 There exists a C1 map which to each (α1, . . . , αp, a1, . . . , ap, λ1, . . . , λp) such that p i=1 αi δˆai ,λi ∈ V ( p, ε), with small enough ε, associates v¯ = v¯(αi , ai , λi ) satisfying Moreover, there exists c > 0 such that the following holds: Proof We expand ∂ J along a variation of h in the v-space Hεp(a, λ) (that is h is a variation with respect to v with (α, a, λ) fixed). Since Q is positive definite, we write Q(v, v) = Av, then A is invertible and there exists p a unique v¯, which minimizes J αi δˆai ,λi + v i.e i=1 v¯ −L ≤ c ⎝ ⎛ p i=1 i=1 αi δˆai ,λi + v¯ = v satimsfiiens (V0) J αi δˆai ,λi + v . i=1 |∇ K (ai )| 1 + λ2 i i = j ⎞ εi j (Log(εi j )−1) 21 ⎠ . f + Av¯ + o( v¯ −L ) = 0. v¯ −L ≤ c A−1 f ≤ c f v¯ −L = O( f , ⎛ p =1 1 + λ2 i i = j ⎞ εi j (Logεi−j1) 21 ⎠ . since v¯ is a minimizer, it yields f (v¯) + Q(v¯, v¯) + o( v¯ 2−L ) = 0; f (v) + Q(v, v) + o v 2−L = Q(v − v¯, v − v¯) + o( v¯ 2−L ). Proposition 4.29 There exists ε0 > 0 (ε0 < ε) such that, for any Set and where We have We derive we have i=1 J αi δˆai ,λi + v p i=1 u = αi δˆai ,λi + v, v ∈ Hεp(a, λ), 1 + S + ci j εi j + c i = j p c Hi (ai ) i S λi2 αi α j p 2 − k=1 αk i = j εi j 0 4.11 Morse Lemma at infinity We begin by characterizing the critical points at infinity of J in the sets V ( p, ε). This characterization is obtained through the construction of a suitable pseudo-gradient at infinity for the functional J for which the Palais–Smale condition is satisfied along its decreasing flow lines as long as these flow lines do not enter in the neighborhood of a finite number of critical points ξi ; 1 ≤ i ≤ p satisfying condition (C ). Notice that the deformation lemmas in Morse theory are realized by using the gradient flow lines or the flow lines of any decreasing pseudo-gradient vector field. We first introduce some definitions and notations due to Bahri [ 5,15 ]. Let ∂ J denote the gradient of the functional J. Definition 4.30 A critical point at infinity of J on + is a limit of a flow line u(s) of the equation: ∂u ∂s u(0) = u0 = −∂ J (u) ξ∞ or (a1, . . . , a p)∞ or αi δˆ(ai ,∞) such that u(s) remains in V ( p, ε(s) for s ≥ s0, and ε(s) satisfies lims−→∞ε(s) = 0. One can write u(s) = ip=1 αi (s)δ(ai (s),λi (s)) + v(s); let ai := lims−→∞ai (s) and αi := lims−→∞αi (s); we denote such a critical point at infinity by To a critical point at infinity ξ∞ are associated stable and unstable manifolds Ws (ξ∞) and Wu (ξ∞); those manifolds allow to compare critical points at infinity by what we call a "domination property", one can see [ 5,42 ], where a detailed description of theses manifolds is given. Definition 4.31 A critical point at infinity ξ∞ is said to be dominated by another critical point at infinity ξ , ∞ if Ws (ξ∞) ∩ Wu (ξ∞) = ∅ and we write ξ If we assume that the intersection Ws (ξ∞) ∩ Wu (ξ ) is transverse, then we obtain ∞ 4.11.1 Construction of the pseudo-gradient In the set V ( p, ε), we obtain Proposition 4.32 Assume that K satisfies (4.16) and condition (C ) For any p, there exists a pseudo-gradient W so that the following hold: there is a positive constant c independent of u = ip=1 αi δai ,λi ∈ V ( p, ε), ε small enough such that, if we denote u = u + v, we have 1) 2) − J (u)(W ) ≥ c ⎝ ⎛ i=1 − J (u¯) ∂v¯ W + ∂(α, a, λ) (W ) ≥ c ⎝ |∇ K (ai )| p i=1 p 1 i=1 λi2 + i = j εi j ⎠ |∇ K (ai )| λi p 1 i=1 λi2 + i = j 3) |W | is bounded and dλi0 ≤ cλi0 where λi0 is the highest of the concentration λ1, λ2, . . . , λ p. We will now give the following result, which establishes our Morse Lemma at infinity: Proposition 4.33 [42] For any u = ip=1 αi δˆai ,λi ∈ V ( p, ε1), (ε1 < 2ε ), we find a change of variables in the space (ai , αi , λi , v), 1 ≤ i ≤ p to (ai , αi , λi , V ), (αi = αi ), such that (4.19) (4.20) with and with initial condition Proof Here, we give only the proof key idea. For the complete proof, see [42], (Lemma 4.4). Since the vector field W constructed in the next section is lipschitz, there is a one parameter group ηs generated by W solution of the equation i = j J εi j + αi δˆai ,λi + v(α, a, λ) = J αi δˆai ,λi , i = j εi j + ai − ai −→ 0 as εi j + i = j i=1 η0 αi δˆai ,λi = W αi δˆai ,λi αi δˆai ,λi αi δˆai ,λi , i=1 p i=1 p i=1 ∂ ∂s ηs p i=1 i=1 where J (ηs ( p minimizer, we hi =av1eαi δˆai ,λi )), and J (ηs ( p i=1 αi δˆai ,λi )) + v(s) are decreasing functions of s. As v(s) is a J αi δˆai ,λi + v(s) ≤ J αi δˆai ,λi . Regarding the construction of the vector field W the flow line ηs ( condition if it does not approach the critical points at infinity. Since the im=a1xαiimδˆaui m,λio)fstahteisλfiie(ss)thseisPaaldaeisc–reSamsianlge function, and the flow line started far away from these critical points at infinity, it will take an infinite time to this later to go to infinity. During this trip, we would be down the level J ( ip=1 αi δˆai ,λi ) + v(s). In any case, as long as we do not cut the lower bound level, the speed of decay is at least −c. Hence, we are forced to cut the level J ( p of the equationi =:1 αi δˆai ,λi ) + v(s) unless the flow line exits V ( p, ) which means there is at most one solution p during the trip between the boundaries of V ( p, 1) and V ( p, ), which we suppose of length l(ε). If we denote s the corresponding time to travel on this portion of the flow trajectory, we have l(ε) ≤ c s. Let αi δˆai ,λi = J αi δˆai ,λi + v(s) (4.21) Does the flow line ηs exit from V ( p, )? We assume that p i=1 αi δˆai ,λi ∈ V ( p, 1), 1 < 2 ; then we have i=1 −∂ J αi δˆai ,λi W αi δˆai ,λi ≥ C ⎝ p i=1 1 + λ2 + i i = j εi j ⎠ ⎞ ≥ c(ε) > 0 (4.22) p i=1 J ηs W αi δˆai ,λi + v¯ > J ηs − δ(ε) (4.23) δ(ε) = − c(ε)cl(ε) ; then J (ηs p V ( p, 1) to the boundary of V ( pi=,1 )α.i Tδˆaoi ,pλrio)vdeetchreearseessulat,t wleeashtav−eδt(oε)shdouwrinthgatthe trip from the boundary of which establishes (4.19). Now, we will prove (4.20): we have |a˙i (s)| ≤ λi (s) ≤ C eλxip(0c)s ; thus C To this end, we know from [42] that J (ηs ip=1 αi δˆai ,λi ) − J (ηs p to 0. Hence, by choosing ε1 small enough, we have (4.23) and, thereforei,=E1qα.i(δˆ4a.i 2,λ1i)+hav¯s)aguoneisqtuoe0soalsutεi1ontetnhdast wNeexdte,nwoetearbeygηosi0n(g toip=p1rαoivδeai(,4λ.i1).9), set ip=1 αi (s)δˆai (s),λi (s)) = ηs ( ip=1 αi δˆai ,λi ). Since the vector field W has no action on the variables αi , we have αi λi (s) αi λi (s) ∂δˆai (s),λi (s) ∂λi (s) λ˙i (s) λi (s) i=1 where a˙i (s) and λ˙i (s) denote the actions of W on the variables ai and λi . Since W is bounded, ∂εi j and we obtain ∂s ≤ C εi j . Therefore, and λi (s) ∂δˆai (s),λi (s) are nearly orthogonal and bounded (both are O(δai (s),λi (s)), it yields that ∂λi (s) |λi (s)a˙i (s)| + | λi (s) | ≤ C , i = 1, . . . , p. Regarding λ1i ∂∂εaiij and λi ∂∂ελiij both are O(εi j ) since εi j = o(1) λ˙i (s) λi (s) ∂δˆai (s),λi (s) ∂ai (s) W = 2) and since s0 satisfies Eq. (4.21), |ai (s) − ai | is bounded; hence we have (4.20). Here and for the sake of simplicity, we will use the notation δˆ j instead of δˆa j ,λ j . To construct a vector field W satisfying Proposition 4.32, we have first to find the expansions of < ∂δˆ j 3 ∂ J (u), λ j ∂λ j We have the following result: >, and < ∂ J (u), λ j ∂a j > . ip=1 αi δˆi ∈ V ( p, ε), we obtain εi j (s) exp −cs ≤ εi j (0) exp cs, |ai (s) − ai | ≤ C s exp cs λi (0) 1 ω3 i = j εi j ⎦ i = j 1 ⎞ ⎤ εi j + λ2 ⎠ ⎦ j −∂ J (u), λ j = 2 J (u) ∂δˆ j 4 ∂λ j i αi Hi (a j ) (1 + o(1)) − 24 α j K'(Ka(ja)λj )2j (1 + o(1)) ω3 + ci j αi λ j ∂εi j (1 + o(1)) + o ∂λ j 3 −∂ J (u), 1 ∂δˆ j 4 λ j ∂a j = 2 J (u) ⎣ K (a j ) 48 λ j α j ω3 ∇ K (a j ) (1 + o(1)) + O ⎝ For the proof, one can see the Appendix of [42]. The second step for the construction of the vector field W is to divide the domains V ( p, ε) in subdomains and to construct partial vector field on such domains satisfying Proposition 4.32. Thus, the final vector field will be defined as a convex combination of the vector fields constructed in the subdomains of V ( p, ε). For details of this construction, we refer to [42]. For technical reasons and for ε0 > 0 small enough, we introduce the following neighborhood of + : Vε0 ( +) = {u ∈ u− L4 ≤ ε0}, where u− = max(0, −u) is the negative part of u and u− L4 = M Once the vector field W is constructed in the new variables, we build a global vector field Z on Vε0 ( +) such that Proposition 4.32 is satisfied. For the V -part, we construct a pseudo-gradient T by setting ∂∂Vs = −ν V , locally on the base space of the bundle V ( p, ε), where ν is taken to be a very large constant. Define Z on Vε0 ( +) to be Z = W + T . Thus, the defined vector field Z is a pseudo-gradient vector field for the functional − J on Vε0 ( +) which is invariant under the flow generated by Z (the proof of this claim is similar to the one given in [ 27 ]). |u−|4θ ∧ dθ 1 4 4.11.2 Critical points at infinity In the sequel, we have to check the critical points at infinity of the functional J, which lead us to the study of the concentration phenomenon of J. First we claim that if u0 ∈ Vε0 ( +) there is p ∈ N∗ and s0 ≥ 0 such that if η(s, u0) denotes the flow line of the vector field Z with initial condition u0, that is η(s, u0) satisfies ∂∂s η(s, u0) = Z (η(s, u0)) η(0, u0) = u0 (4.24) η(s, u0) is in V ( p, 34ε ) for s ≥ s0. Indeed outside &rp1=1 V ( p, 34ε ), −∂ J (Z (u)) ≥ c > 0. and which contains u = We come back to the subdivision of the neighborhood V ( p, ε). We denote by V¯ ( p, , ε) the subset of V ( p, ε) containing u = ip=1 αi δˆai ,λi + v, v satisfying conditions (V0) and for which there is a subcollection of the critical points ξ1, . . . , ξr of K such that any point is very close to all the concentration points a1 . . . a p and where < 13 mini = j d(ξi , ξ j ). Then, we consider the subset of V¯ ( p, , ε) which we denote by V¯2( p, , ε) p i=1 αi δˆai ,λi + v such that 1) two different concentration points are close to different critical points of K . 2) for which the matrix M (τ ) for τ = (ξη1 , . . . , ξηp ) defined in (4.16) is positive definite. 3) θ K (ai ) 3K (ai ) + 2 Aai < 0, i = 1, . . . , p. We have the following result: Lemma 4.35 The critical points at infinity of the functional J lie in &rp1=1 V¯2( p, , ε) for any ε, > 0 small. Proof By using the argument above η(s, u0) = new variables ip=1 αi (s)δˆai (s),λi (s) is outside V¯2( p, , 2ε ), then we derive from the construction of Z that the ip=1 αi δˆai (s),λi (s) + v(s) is in V ( p, 34ε ). Suppose that in the maximum of the λi (s) and the λi (s) are bounded by a positive constant c (we refer to [42] for all the details). Since − J (u)Z (u) > 0 and ip=1 αi δˆai ,λi is in the compact set {αi ≤ 1, λi ≤ c, ai ∈ M }, the minimum is achieved; hence − J (u)Z (u) > C > 0. Therefore, J (η(s, u0)) = J (η(0, u0)) + 0 ≤ J (η(0, u0)) − C (s − s0), s J (u)(t )Z (u)(t )dt which gives that J is not bounded; hence a contradiction. Lemma 4.36 For any u = ip=1 αi δˆai ,λi in V¯2( p, , ε) (ε, > 0 small) close to a critical point at infinity of J, we obtain the following expansion of J in the new variables: p K (ξi ) 1 2 p (|ais |2 − |aiu |2) + c where (ais , aiu ) are the coordinates of ai near ξi along the manifolds Ws (ξi ) and Wu (ξi ) and Q ∈ R p−1 is the coordinate of (α1, . . . , α p). Proof Using Proposition 4.29, we obtain the following expansion of the functional J in V¯2( p, , ε) in the new variables (v = 0): p αi δˆai ,λi p α2 S i=1 i p [ i=1αi4 K (ai )]1/2 p l=1αl4 K (al ) 'K (ai ) λi2 c + S2 i = j p Aai 2 i=1 λi G(ai , a j ) αi α j lp=1αl2 − αi2 lp=1αl4 − 2 αi4 K (ai ) p l=1αl4 K (al ) + o εi j + o i = j 1 λi2 0 . p i=1 αi δˆai ,λi belongs to V¯2( p, , ε), the expansion of the functional can Under the assumption that u = be rewritten as follows: J (u) = i = j p α2 S i=1 i ip=1αi4 K (ai ).1/2 p / 'K (ai ) 2 Aai 0 1 3K 2(ai ) + K (ai ) λi2 1 λi2 We can refine the expansion of J, since in this set, we have αi2 K (ai ) (λi λ j d2(ai , ai ))−1. Hence, we obtain α2j K (a j ) and εi j = p αi δˆai ,λi ip=1αi2 / ip=1αi4 K (ai ).1/2 S 1 + 8S2 ω3 p 1 k=1 K (ak ) M + o(1), t 0 where = ( λ11 , . . . , λ1p ). Let us turn now to the term G(α1, . . . , α p) = p i=1αi4 K (ai ) 1/2 , where G is a homogeneous function and ( K (1a1) , . . . , K (ap) ) is a critical point (a maximum) with critical value 1 p j=1 K (1ai ) . By performing a Morse lemma for G, we obtain in the new variables J (u) = S p i=1 K (ξi ) 1 − |Q|2 + p i=2 |ais |2 − |aiu |2 + c p 1 2 i=1 λi2 since M t ≥ c| |2 = p j=1 λci2 , and the lemma follows. 4.12 Topological argument l j=1 K (ξi j ) For any l-tuple τl = (i1, . . . , il ), 1 ≤ i j ≤ r1, j = 1, . . . , l such that M (τl ) is positive definite, let c(τl ) = S denote the associated critical value. Here, we choose to consider a simplified situation, where denotes the level set for the functional, J a = {u ∈ the critical point at infinity (ξl )∞. for τ = τ we have c(τ ) = c(τ ) and thus order the c(τ )’s as c(τ1) < c(τ2) < · · · < c(τk ). By using a deformation lemma (one can see [ 15 ]), we derive the existence of a positive constant σ0(ε, ) such that for any 0 < σ < σ0, the set J c(τl )−σ ∪ Wu∞(ξl )∞ is a retract by deformation of J c(τl )+σ , where J a + / J (u) ≤ a} and Wu∞(ξl )∞ is the unstable manifold of Lemma 4.37 If c(τl−1) < a < c(τl ) < b < c(τl+1), then for any coefficient group G, we have Hq ( J b, J a ) = q = i (τl ), q = i (τl ). l j=1 ki j with ki j = i nd (K , ξ j ). We are now ready to state the proof of our result. 4.12.1 Proof of Theorem 4.21 Let b1 < c(τ1) = minu∈ + J (u) < b2 < c(τ2) < · · · < bk < c(τk ) < bk+1. Since we assume that problem ( PK ) has no solution, J bk+1 is a retract by deformation of the set +, which is a retract by deformation of Vε0 ( +) and hence they have the same Euler-Poincaré characteristic, We derive after recalling that χ ( J b1 ) = χ (∅) = 0, that χ (Vε0 ( +)) = χ ( J bk+1 ). χ ( J bk+1 ) = χ ( J bk ) + (−1)i(τk ). r1 l=1 τl =(i1,...,il ),M(τl )>0 (−1)i(τl ) = 1. using the fact that u0 is in Vε0 ( +), we obtain Therefore, ( PK ) has a solution u0 in Vε0 ( +) if the equality above is not true. We claim that u0 > 0, when ε0 is small enough. Otherwise, by multiplying ( PK ) by u0− and integrating, u0− 2 ≤ C u0− 4L4 ≤ C u0− 2. Hence, either u− 0 0 = 0 or u− Therefore, u− 0 = 0 and u0 > 0. ≥ C0, where C0 > 0. Thus, we have a contradiction if ε0 is small enough. Acknowledgements The author would like to express posthumously her deep gratitude to her Professor, the great mathematician Abbas Bahri, without whom this work would not have been accomplished. The valuable research supervision of Professor Bahri, during the planning and development of the resolution of the CR Yamabe Conjecture and his willingness to give his knowledge and time so generously has been very much appreciated. The author also would like to thank the staff of Rutgers University, New Jersey, USA, where most of this work was realized. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 39. Hebey, E.: Changements de métriques conformes sur la sphère, le problème de Nirenberg. Bull. Sci. Math. 114, 215–242 (1990) 40. Felli, V.; Uguzzoni, F.: Some existence results for the Webster scalar curvature problem in presence of symmetry. Ann. Math. 183, 469–493 (2004) 41. Folland, G.B.; Stein, E.: Estimates for ∂b complex and analysis on the Heisenberg group. Commun. Pure Appl. Math. 27, 429–522 (1974) 42. Gamara, N.: The prescribed scalar curvature on a 3-dimensional CR manifold. Adv. Nonlinear Stud. 2, 193–235 (2002) 43. Gamara, N.: The CR Yamabe conjecture the case n=1. J. Eur. Math. Soc. 3, 105–137 (2001) 44. Gamara, N.; Yacoub, R.: CR Yamabe conjecture, the conformally flat case. Pacif. J. Math. 201, 121–175 (2001) 45. Gamara, N.; El Jazi, S.: The Webster scalar curvature revisited - the case of the three dimensional CR sphere. Calc. Var. Partial Differ. Equ. 42(1–2), 107–136 (2011) 46. Gamara, N.; Riahi, M.: Multiplicity results for the prescribed Webster scalar curvature on the three CR sphere under flatness condition. Bull. Sci. Math. 136, 72–95 (2012) 47. Gamara, N.; Riahi, M.: The interplay between the CR “flatness condition” and existence results for the prescribed Webster scalar curvature. Adv. Pure. Appl. Math. 3(3), 281–291 (2012) 48. Gamara, N.; Riahi, M.: The impact of the flatness condition on the prescribed Webster scalar curvature. Arab. J. Math. 2, 381–392 (2013) 49. Gamara, N.; Guemri, H.; Amri, A.: Optimal control in prescribing Webster scalar curvature on 3-dimensional pseudoHermitian manifold. Non Linear Anal. TMA 127, 235–262 (2015) 50. Gamara, N.; Hafassa, B.: The β-flatness condition in CR spheres. Adv. Nonlinear Stud. 17(1), 193–213 (2017) 51. Garofalo, N.; Lanconelli, E.: Existence and nonexistence results for semilinear equations on the Heisenberg group. Indiana Univ. Math. J. 41, 71–98 (1992) 52. Han, Z.: Prescribing Gaussian curvature on S2. Duke. Math. J. 61, 679–703 (1990) 53. Chtioui, H.; Elmehdi, K.; Gamara, N.: The Webster scalar curvature problem on the three dimentional CR manifolds. Bull. Sci. Math. 131, 361–374 (2007) 54. Chtioui, H., Ould Ahmedou, M., Yacoub, R.: Existence and multiplicity results for the prescribed webster scalar curvature problem on three CR manifolds. J. Geom. Anal. doi:10.1007/s12220-011-9267-z 55. Chtioui, H.; Ould Ahmedou, M.; Yacoub, R.: Topological methods for the prescribed webster scalar curvature problem on CR manifolds. Differ. Geom. Appl. 28, 264–281 (2010) 56. Jerison, D.; Lee, J.M.: The Yamabe problem on CR manifolds. J. Differ. Geom. 25, 167–197 (1987) 57. Jerison, D.; Lee, J.M.: Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem. J. Am. Math. Soc. 1, 1–13 (1988) 58. Jerison, D.; Lee, J.M.: Intrinsic CR normal coordinates and the CR Yamabe problem. J. Differ. Geom. 29, 303–343 (1989) 59. Kaneyuki, S.: Homogenous Bounded Domains and Siegel Domains Lecture Notes in Math n0241. Springer, Berlin (1971) 60. Kazdan, J.; Warner, F.: Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvature. Ann. Math. (2) 101, 317–331 (1975) 61. Lee, J.M.; Parker, T.H.: The Yamabe problem. Bull. Am. Math. Soc. (NS) 17(1), 37–91 (1987) 62. Li, Y.Y.: Prescribing scalar curvature on Sn and related problems, Part I. J. Differ. Equ. 120, 319–410 (1995) 63. Malchiodi, A.; Uguzzoni, F.: A perturbation result for the Webster scalar curvature problem on the CR sphere. J. Math. Pures Appl. 81, 983–997 (2002) 64. Milnor, J.: Lectures on h-Cobordism. Princeton University Press, Princeton (1965) 65. Sacks, J.; Uhlenbeck, K.: The existence of minimal immersions of 2-spheres. Ann. Math. (2) 113(1), 1–24 (1981) 66. Schoen, R.: Conformal deformation of a metric to constant scalar curvature. J. Differ. Geom. 20, 479–495 (1984) 67. Struwe, M.: A Global Compactness Result for Elliptic Boundary Value Problems Involving Limiting Non Linearities, Mathematische Zeitschrift, vol. 187, pp. 511–517. Springer, Berlin (1984) 68. Tanaka, N.: A Differential Geometric Study on Strongly Pseudo-convex Manifolds. Kinokuniya, Tokyo (1975) 69. Trudinger, N.S.: Remarks concerning the conformal deformation of riemannian structures on comapact manifolds. Ann. scuola Norm. Sup. Pisa (3) 22, 265–274 (1968) 70. Webster, S.: Pseudo-Hermitian structures on a real hypersurface. J. Differ. Geom. 13, 25–41 (1975) 71. Yamabe, H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12, 21–37 (1960) 1. Ambrosetti , A. ; Badiale , M. : Homoclinics: Poincaré-Melnikov type results via a variational approach . Ann. Inst. H. Poincaré Anal. Non Linéaire 15 , 233 - 252 ( 1998 ) 2. Aubin , T. : Equations differentielles non lineaires et probleme de Yamabe concernant la courbure scalaire . J. Math. Pures Appl . 55 , 269 - 296 ( 1976 ) 3. Aubin , T. : Problème de Yamabe concernant la courbure scalaire . C. R. Acad. Sci. t. 280 ( série A ), 721 ( 1975 ) 4. Aubin , T. : Some Nonlinear Problem in Differential Geometry . Springer, New York ( 1997 ) 5. Bahri , A. : An invariant for Yamabe-type flows with application to scalar curvature problems in high dimensions . Duke. Math. J . 281 , 323 - 466 ( 1996 ) 6. Bahri , A. : Proof of the Yamabe conjecture for locally conformally flat manifolds . Non Linear Anal. T. M. A 20 ( 10 ), 1261 - 1278 ( 1993 ). MR 94e:53033, Zbl 782 . 53027 7. Bahri , A. : Variations at infinity in contact form geometry . J. Fixed Point Theor. Appl . 5 ( 2 ), 265 - 289 ( 2009 ) 8. Bahri , A. : Pseudo-orbits of contact forms . In: Pitman Research Notes in Mathematics Series , vol. 173 . Longman Scientific and Technical, Harlow ( 1988 ) 9. Bahri , A. : An invariant for Yamabe-type flow with applications to scalar curvature problems in high dimensions. A celebration of John F . Nach, Jr. Duke Math. J. 81 ( 2 ), 323 - 466 ( 1996 ) 10. Bahri , A. : Classical and Quantic Periodic Motions of Multiply Polarized Spin Particles . Pitman Research Notes in Mathematics Series , vol. 378 . Longman , Harlow ( 1998 ) 11. Bahri , A. : Morse relations and Fredholm deformation of v-contact forms . Arab. J. Math. 3 ( 2 ), 93 - 187 ( 2014 ) 12. Bahri , A. : Fredholm pseudo-gradient for the action functional on a sub-manifold of dual Legendrian curves of a three dimensional manifold (M3, α) contact of v-contact forms . Arab. J. Math. 3 ( 2 ), 189 - 198 ( 2014 ) 13. Bahri , A. : Linking numbers in contact form geometry, with an applicationto the computation of the intersection operator for the first contact form of J. Gonzalo and F. Varela . Arab. J. Math. 3 ( 2 ), 199 - 210 ( 2014 ) 14. Bahri , A. : On the contact homology of the first exotic contact form of J. Gonzalo and F. Varela . Arab. J. Math. 3 ( 2 ), 211 - 289 ( 2014 ) 15. Bahri , A. : Critical Points at Infinity in Some Variational Problems, Pitman Res. Notes Math, Ser 182 , Longman Sci. Tech. , Harlow ( 1989 ) 16. Bahri , A. ; Berestycki , H.: A perturbation method in critical point theory and applications . Trans. Am. Math. Soc . 267 ( 1 ), 1 - 32 ( 1981 ) 17. Bahri , A. ; Brezis , H. : Nonlinear elliptic equations . In: Gindiken, S. (Ed) Topics in Geometry in memory of Joseph d' Atri , pp. 1 - 100 . Birkhauser, Boston ( 1996 ) 18. Bahri , A. ; Coron , J.M.: The scalar curvature problem on the standard three-dimensional spheres . J. Funct. Anal . 95 , 106 - 172 ( 1991 ) 19. Bahri , A. ; Coron , J.M. : On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of topology of the domain . Commun. Pure Appl . Math. 41 , 253 - 294 ( 1988 ) 20. Bahri , A. ; Coron , J.M.: Une theorie des points critiques à l'infini pour l'equation de Yamabe et le probleme de Kazdan-Warner. C. R. Acad . Sci, Paris 300 , 513 - 516 ( 1985 ) 21. Bahri , A. ; Lions , P.L. : Remarques sur la theorie variationnelle des points critiques et Applications, (French) [ Remarks on variational critical point theory and applications]. C. R. Acad . Sci Paris. Ser. I Math . 301 ( 5 ), 145 - 147 ( 1985 ) 22. Bahri , A. ; Lions , P.L. : Morse index of some min-max critical points I, Application to multiplicity results . Commun. Pure Appl . Math. 41 , 1027 - 1037 ( 1988 ) 23. Bahri , A. ; Lions , P.L.: Remarks on the variational theory of critical points . Manuscripta Math . 66 , 129 - 152 ( 1989 ) 24. Bahri , A. ; Rabinowitz , P.H. : Periodic solutions of 3-body problems . Ann. Inst. H. Poincaré Ana. Non linéaire 8 , 561 - 649 ( 1991 ) 25. Bahri , A. ; Xu , Y. : Recent Progress in Conformal Geometry , ICP Advanced Texts in Mathematics, 1 . Imperial College Press, London ( 2007 ) 26. Bahri , A. ; Lee , Yanyan; Rey, O. : On a variational problem with lack of compactness: the topological effect of the critical points at infinity . Calc. Var . 3 ( 1 ) 95 , 67 - 93 ( 1995 ) 27. Ben Ayed , M. ; Chen , Y. ; Chtioui , H. ; Hammami, M. : On the prescribed scalar curvature problem on 4-manifolds . Duke Math. J. 84 , 633 - 677 ( 1996 ) 28. Ben Ayed , M. ; Chtioui , H. ; Hammami, M.: The scalar curvature problem on higher dimensional spheres . Duke Math. J . 93 , 379 - 424 ( 1998 ) 29. Ben Ayed , M. ; Ould Ahmedou , M. : Multiplicity results for the prescribed scalar curvature on low spheres . Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) VII , 1 - 26 ( 2008 ) 30. Chern , S.S. ; Moser , J.K. : Real hypersurfaces in complex manifolds . Acta Math . 133 , 219 - 271 ( 1974 ) 31. Chtioui , H.; Abdelhedi , W. ; Hajaiej, H.: A Complete Study of the Lack of Compactness and Existence Results of a Fractional Nirenberg Equation Via a Flatness Hypothesis: Part I . arXiv:1409.5884v1 [math.AP] (20 Sep 2014 ) 32. Chen , W. ; Ding, W. : Scalar curvatures on S2 . Trans. Am. Math. Soc . 303 , 365 - 382 ( 1987 ) 33. Chang , K.C. ; Liu , J.Q. : On Nirenberg's problems . Int. J. Math. 4 , 35 - 58 ( 1993 ) 34. Chang , S.Y. ; Yang , P.: A perturbation result in prescribing scalar curvature on Sn . Duke. Math. J. 64 , 27 - 69 ( 1991 ) 35. Chang , S.Y. ; Yang , P. : Conformal deformation of metrics on S2 . J. Differ. Geom . 27 , 259 - 296 ( 1988 ) 36. Chang , S.Y. ; Yang , P. : Prescribing Gaussian curvature on S2 . Acta Math . 159 , 215 - 259 ( 1987 ) 37. Dragomir , S. ; Tomassini, G. : Differential Geometry and Analysis on CR Manifolds, Progress in Mathematics , vol. 246 . Birkhäuser, Basel, Boston 38. Daomin , C. ; Shuangjie , P. ; Shusen , Y. : On the webster scalar curvature problem on the CR sphere with a cylindrical-type symmetry . J. Geom. Anal . ( 2012 ). doi:10.1007/s12220-012-9301-9


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Najoua Gamara. Methods of the theory of critical points at infinity on Cauchy Riemann manifolds, Arabian Journal of Mathematics, 2017, 1-47, DOI: 10.1007/s40065-017-0180-6