Contents -- Preface A. Bahri and Y. Xu -- 1. Sign-Changing Yamabe-Type Problems -- 1.1 General Introduction -- 1.2 Results and Conditions -- 1.3 Conjecture 2 and Sketch of the Proof of Theorem 1 -- Outline -- 1.4 The Difference of Topology -- 1.5 Open Problems -- 1.5.1 Understand the difference of topology -- 1.5.2 Non critical asymptots -- 1.5.3 The exit set from infinity -- 1.5.4 Establishing Conjecture 2 and continuous forms of the discrete inequality -- 1.5.5 The Morse Lemma at in.nity, Part I, II, III -- 1.5.6 Notations v, vi, -- 1.6 Preliminary Estimates and Expansions, the Principal Terms -- 1.7 Preliminary Estimates -- 1.7.1 The equation satis.ed by -- 1.7.2 First estimates on vi and -- 1.7.3 The matrix A -- 1.7.4 Towards an H1 H0 -estimate on vi and an L-estimate on ht. -- 1.7.5 The formal estimate on hi -- 1.7.6 Remarks about the basic estimates -- 1.7.7 Estimating the right hand side of Lemma 12 . -- 1.7.8 Ri and the estimate on

Bibliography -- 2. Contact Form Geometry -- 2.1 General Introduction -- 2.2 On the Dynamics of a Contact Structure along a Vector Field of its Kernel -- 2.2.1 Introduction -- 2.2.2 Introducing a large rotation -- 2.2.3 How γ is built -- 2.2.4 Modification of a into -- 2.2.5 Computation of N -- 2.2.6 Conformal deformation -- 2.2.7 Choice of λ -- 2.2.8 First step in the construction of -- 2.3 Appendix 1 -- 2.3.1 The normal form for (α, v) when α does not turn well -- 2.4 The Normal Form of (α, v) Near an Attractive Periodic Orbit of v -- 2.5 Compactness -- 2.5.1 Some basic facts -- 2.5.2 A model for Wu(xm), the unstable manifold in Cβ of a periodic orbit of index m -- 2.5.3 Hypothesis (A), Hypothesis (B), Statement of the result . -- 2.5.4 The hole flow -- 2.5.4.1 Combinatorics -- 2.5.4.2 Normals -- 2.5.4.3 Hole flow and Normal (II)-flow on curves of G4k near x8 -- 2.5.4.4 Forced repetition -- 2.5.4.5 The Global picture, the degree is zero -- 2.5.5 Companions -- 2.5.5.1 Their definition, births and deaths -- 2.5.5.2 Families and nodes -- 2.5.6 Flow-lines for x2k+1 to x -- 2.5.7 The S1-classifying map -- 2.5.8 Small and high oscillation, consecutive characteristic pieces -- 2.5.9 Iterates of critical points at infinity -- 2.5.10 The Fredholm aspect -- 2.5.11 Transversality and the compactness argument -- 2.6 Transmutations -- 2.6.1 Study of the Poincare-returnmaps -- 2.6.2 Definition of a basis of Tx8 G2s for the reduction of d2J( x8) -- 2.6.3 Compatibility -- 2.7 On the Morse Index of a Functional Arising in Contact Form Geometry -- 2.7.1 Introduction -- 2.7.2 The Case of Γ2 -- 2.7.3 Darboux Coordinates -- 2.7.4 The v-transportmaps -- 2.7.5 The equations of the characteristic manifold near x -- the equations of a critical point -- 2.7.5.1 The characteristic manifold for the unperturbed problem -- 2.7.6 Critical points, vanishing of the determinant.

2.7.7 Introducing the perturbation -- 2.7.8 The characteristic manifold for the perturbed problem -- the determinant equations -- 2.7.9 Reduction to the Case k=1 -- 2.7.10 Modification of d2Jt8 (x8)