Symplectic topology and Floer homology. Volume 2, Floer homology and its applications

Cover......Page 1
Half-Title page......Page 3
Series page......Page 4
Title page......Page 5
Copyright page......Page 6
Contents of Volume 1......Page 7
Contents of Volume 2......Page 11
Preface......Page 15
Part 1: Hamiltonian dynamics and symplectic geometry......Page 27
  1.1 The Lagrangian action functional and its first variation......Page 29
  1.2 Hamilton\'s action principle......Page 33
  1.3 The Legendre transform......Page 34
  1.4 Classical Poisson brackets......Page 44
  2.1 The cotangent bundle......Page 47
  2.2 Symplectic forms and Darboux\' theorem......Page 50
  2.3 The Hamiltonian diffeomorphism group......Page 63
  2.4 Banyaga\'s theorem and the flux homomorphism......Page 71
  2.5 Calabi homomorphisms on open manifolds......Page 78
  3.1 The conormal bundles......Page 86
  3.2 Symplectic linear algebra......Page 88
  3.3 The Darboux–Weinstein theorem......Page 97
  3.4 Exact Lagrangian submanifolds......Page 100
  3.5 Classical deformations of Lagrangian submanifolds......Page 103
  3.6 Exact Lagrangian isotopy = Hamiltonian isotopy......Page 108
  3.7 Construction of Lagrangian submanifolds......Page 113
  3.8 The canonical relation and the natural boundary condition......Page 122
  3.9 Generating functions and Viterbo invariants......Page 125
  4.1 Symplectic fibrations and symplectic connections......Page 132
  4.2 Hamiltonian fibration......Page 136
  4.3 Hamiltonian fibrations, connections and holonomies......Page 146
  5.1 Normalization of Hamiltonians......Page 156
  5.2 Invariant norms on C[sup(∞)](M) and the Hofer length......Page 161
  5.3 The Hofer topology of Ham(M,ω)......Page 163
  5.4 Nondegeneracy and symplectic displacement energy......Page 165
  5.5 Hofer\'s geodesics on Ham(M,ω)......Page 169
  6.1 C[sup(0)] symplectic rigidity theorem......Page 172
  6.2 Topological Hamiltonian flows and Hamiltonians......Page 184
  6.3 Uniqueness of the topological Hamiltonian and its flow......Page 189
  6.4 The hameomorphism group......Page 197
Part 2: Rudiments of pseudoholomorphic curves......Page 201
  7.1 Natural connection on almost-Kähler manifolds......Page 203
  7.2 Global properties of J-holomorphic curves......Page 211
  7.3 Calculations of Δe(u) on shell......Page 215
  7.4 Boundary conditions......Page 219
  8.1 Interior a-priori estimates......Page 223
  8.2 Off-shell elliptic estimates......Page 228
  8.3 Removing boundary contributions......Page 235
  8.4 Proof of ε-regularity and density estimates......Page 238
  8.5 Boundary regularity of weakly J-holomorphic maps......Page 247
  8.6 The removable singularity theorem......Page 253
  8.7 Isoperimetric inequality and the monotonicity formula......Page 261
  8.8 The similarity principle and the local structure of the image......Page 265
  9.1 The moduli space of pseudoholomorphic curves......Page 273
  9.2 Sachs–Uhlenbeck rescaling and bubbling......Page 277
  9.3 Definition of stable curves......Page 283
  9.4 Deformations of stable curves......Page 293
  9.5 Stable map and stable map topology......Page 317
  10.1 A quick review of Banach manifolds......Page 349
  10.2 Off-shell description of the moduli space......Page 354
  10.3 Linearizations of [bar(∂)][sub(j,J)]......Page 358
  10.4 Mapping transversality and linearization of [bar(∂)]......Page 361
  10.5 Evaluation transversality......Page 371
  10.6 The problem of negative multiple covers......Page 379
 11 Applications to symplectic topology......Page 381
  11.1 Gromov\'s non-squeezing theorem......Page 382
  11.2 Nondegeneracy of the Hofer norm......Page 390
References......Page 403
Index......Page 418