Symplectic Topology and Floer Homology: Volume 1, Symplectic Geometry and Pseudoholomorphic Curves (New Mathematical Monographs)

Published in two volumes, this is the first book to provide a thorough and systematic explanation of symplectic topology, and the analytical details and techniques used in applying the machinery arising from Floer theory as a whole. Volume 1 covers the basic materials of Hamiltonian dynamics and symplectic geometry and the analytic foundations of Gromov's pseudoholomorphic curve theory. One novel aspect of this treatment is the uniform treatment of both closed and open cases and a complete proof of the boundary regularity theorem of weak solutions of pseudo-holomorphic curves with totally real boundary conditions. Volume 2 provides a comprehensive introduction to both Hamiltonian Floer theory and Lagrangian Floer theory. Symplectic Topology and Floer Homology is a comprehensive resource suitable for experts and newcomers alike. 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 V. 1. Symplectic Geometry And Pseudoholomorphic Curves -- V. 2. Floer Homology And Its Applications. Yong-geun Oh, Ibs Center For Geometry And Physics, Pohang University Of Science And Technology, Republic Of Korea. Includes Bibliographical References And Index. This is the first systematic exposition of basic Floer homology theory and its applications to symplectic topology as a whole. In two volumes, it provides a comprehensive resource suitable for experts and newcomers alike.