Recent progress in conformal geometry

Abbas Bahri; Yongzhong Xu1860947727, 9781860947728, 9781860948602

This book presents a new front of research in conformal geometry, on sign-changing Yamabe-type problems and contact form geometry in particular. New ground is broken with the establishment of a Morse lemma at infinity for sign-changing Yamabe-type problems. This family of problems, thought to be out of reach a few years ago, becomes a family of problems which can be studied: the book lays the foundation for a program of research in this direction. In contact form geometry, a cousin of symplectic geometry, the authors prove a fundamental result of compactness in a variational problem on Legrendrian curves, which allows one to define a homology associated to a contact structure and a vector field of its kernel on a three-dimensional manifold. The homology is invariant under deformation of the contact form, and can be read on a sub-Morse complex of the Morse complex of the variational problem built with the periodic orbits of the Reeb vector-field. This book introduces, therefore, a practical tool in the field, and this homology becomes computable.

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Table of contents :
Contents……Page 10
Preface A. Bahri and Y. Xu……Page 6
1.1 General Introduction……Page 14
1.2 Results and Conditions……Page 15
1.3 Conjecture 2 and Sketch of the Proof of Theorem 1; Outline……Page 20
1.4 The Difference of Topology……Page 24
1.5.1 Understand the difference of topology……Page 27
1.5.3 The exit set from infinity……Page 28
1.5.6 Notations v, vi,……Page 29
1.6 Preliminary Estimates and Expansions, the Principal Terms……Page 30
1.7 Preliminary Estimates……Page 31
1.7.1 The equation satis.ed by……Page 32
1.7.2 First estimates on vi and……Page 36
1.7.3 The matrix A……Page 38
1.7.4 Towards an H1 H0 -estimate on vi and an L-estimate on ht…….Page 39
1.7.5 The formal estimate on hi……Page 44
1.7.7 Estimating the right hand side of Lemma 12 …….Page 48
1.7.8 Ri and the estimate on |vi|H1……Page 58
1.8 Proof of the Morse Lemma at Infinity When the Concentrations are Comparable……Page 67
1.9.1 Content of Part II……Page 79
1.9.2 Redirecting the estimates……Page 81
1.10.2 Content of Part III……Page 121
1.10.3 Basic conformally invariant estimates……Page 122
1.10.4 Estimates on v – (vI + vII)……Page 134
1.10.5 The expansion……Page 142
1.10.6 The coeficient in front of ek d……Page 167
1.10.7 The -equation, the estimate on……Page 172
1.10.8 The system of equations corresponding to the variations of the points……Page 187
1.10.9 Rule about the variation of the points of concentrations of the various groups……Page 195
1.10.11 Remarks on the basic parameters……Page 198
1.10.12 The end of the expansion and the concluding remarks……Page 202
Bibliography……Page 212
2.1 General Introduction……Page 214
2.2.1 Introduction……Page 218
2.2.2 Introducing a large rotation……Page 223
2.2.3 How γ is built……Page 227
2.2.4 Modification of a into……Page 239
2.2.5 Computation of N……Page 240
2.2.6 Conformal deformation……Page 248
2.2.7 Choice of λ……Page 253
2.2.8 First step in the construction of……Page 254
2.3.1 The normal form for (α, v) when α does not turn well……Page 288
2.4 The Normal Form of (α, v) Near an Attractive Periodic Orbit of v……Page 289
2.5 Compactness……Page 292
2.5.1 Some basic facts……Page 293
2.5.2 A model for Wu(xm), the unstable manifold in Cβ of
a periodic orbit of index m……Page 295
2.5.3 Hypothesis (A), Hypothesis (B), Statement of the result …….Page 301
2.5.4.1 Combinatorics……Page 304
2.5.4.2 Normals……Page 307
2.5.4.3 Hole flow and Normal (II)-flow on curves of G4k near x8……Page 309
2.5.4.4 Forced repetition……Page 312
2.5.4.5 The Global picture, the degree is zero……Page 314
2.5.5.1 Their definition, births and deaths……Page 317
2.5.5.2 Families and nodes……Page 318
2.5.6 Flow-lines for x2k+1 to x……Page 336
2.5.7 The S1-classifying map……Page 345
2.5.8 Small and high oscillation, consecutive characteristic pieces……Page 347
2.5.9 Iterates of critical points at infinity……Page 367
2.5.10 The Fredholm aspect……Page 372
2.5.11 Transversality and the compactness argument……Page 377
2.6 Transmutations……Page 397
2.6.1 Study of the Poincare-returnmaps……Page 415
2.6.2 Definition of a basis of Tx8 G2s for the reduction of d2J( x8)……Page 426
2.6.3 Compatibility……Page 430
2.7.1 Introduction……Page 433
2.7.2 The Case of Γ2……Page 437
2.7.3 Darboux Coordinates……Page 438
2.7.4 The v-transportmaps……Page 443
2.7.5.1 The characteristic manifold for the unperturbed problem……Page 447
2.7.6 Critical points, vanishing of the determinant……Page 449
2.7.7 Introducing the perturbation……Page 450
2.7.8 The characteristic manifold for the perturbed problem; the determinant equations……Page 454
2.7.9 Reduction to the Case k=1……Page 460
2.7.10 Modification of d2Jt8 (x8) |span{u2,···,uk-1}……Page 467
2.7.11 Calculation of ﾝ2J (x ).u2.u3……Page 472
2.8 Calculation of ﾝ2J (x ).u2.u2……Page 478
2.9 Calculation of ﾝ2J (x ).u2.u4……Page 484
2.10 Other Second Order Derivatives……Page 487
2.11.1 The Proof of Lemma 42……Page 489
2.11.2 The proof of Lemma 47……Page 493
2.11.4 Proof of Lemma 48……Page 496
2.11.5 The Proof of Lemma 49 …….Page 497
2.11.6 The proof of Lemma 50……Page 498
2.11.7 Proof of Claim 1……Page 500
2.11.8 Proof of Claim 3……Page 502
2.11.9 The Final Details of the Calculation of ﾝ2J(x ).u2.u3……Page 505
2.11.11 Proof of Lemma 52……Page 507
2.11.12 Proof of Lemma 53……Page 508
2.11.13 Proof of Claim 3……Page 509
2.11.14 Proof of Claim 4……Page 510
2.11.15 Details of the Calculation of ﾝ2J.u2…….Page 516
Bibliography……Page 522