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El. knyga: Symplectic Topology and Floer Homology: Volume 1, Symplectic Geometry and Pseudoholomorphic Curves

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  • Formatas: PDF+DRM
  • Serija: New Mathematical Monographs
  • Išleidimo metai: 27-Aug-2015
  • Leidėjas: Cambridge University Press
  • Kalba: eng
  • ISBN-13: 9781316382943
Kitos knygos pagal šią temą:
Symplectic Topology and Floer Homology: Volume 1, Symplectic Geometry and Pseudoholomorphic CurvesDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Serija: New Mathematical Monographs
  • Išleidimo metai: 27-Aug-2015
  • Leidėjas: Cambridge University Press
  • Kalba: eng
  • ISBN-13: 9781316382943
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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.

Published in two volumes, this is the first book to provide a 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.

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The first part of a two-volume set offering a systematic explanation of symplectic topology. This volume covers the basic materials of Hamiltonian dynamics and symplectic geometry.
Contents of Volume 2 ix
Preface xiii
PART 1 HAMILTONIAN DYNAMICS AND SYMPLECTIC GEOMETRY
1 The least action principle and Hamiltonian mechanics
3(18)
1.1 The Lagrangian action functional and its first variation
3(4)
1.2 Hamilton's action principle
7(1)
1.3 The Legendre transform
8(10)
1.4 Classical Poisson brackets
18(3)
2 Symplectic manifolds and Hamilton's equation
21(39)
2.1 The cotangent bundle
21(3)
2.2 Symplectic forms and Darboux' theorem
24(13)
2.3 The Hamiltonian diffeomorphism group
37(8)
2.4 Banyaga's theorem and the flux homomorphism
45(7)
2.5 Calabi homomorphisms on open manifolds
52(8)
3 Lagrangian submanifolds
60(46)
3.1 The conormal bundles
60(2)
3.2 Symplectic linear algebra
62(9)
3.3 The Darboux--Weinstein theorem
71(3)
3.4 Exact Lagrangian submanifolds
74(3)
3.5 Classical deformations of Lagrangian submanifolds
77(5)
3.6 Exact Lagrangian isotopy = Hamiltonian isotopy
82(5)
3.7 Construction of Lagrangian submanifolds
87(9)
3.8 The canonical relation and the natural boundary condition
96(3)
3.9 Generating functions and Viterbo invariants
99(7)
4 Symplectic fibrations
106(24)
4.1 Symplectic librations and symplectic connections
106(4)
4.2 Hamiltonian libration
110(10)
4.3 Hamiltonian librations, connections and holonomies
120(10)
5 Hofer's geometry of Ham(M, Ω)
130(16)
5.1 Normalization of Hamiltonians
130(5)
5.2 Invariant norms on C∞ (M) and the Hofer length
135(2)
5.3 The Hofer topology of Ham (M, Ω)
137(2)
5.4 Nondegeneracy and symplectic displacement energy
139(4)
5.5 Hofer's geodesies on Ham (M, Ω)
143(3)
6 C°-Symplectic topology and Hamiltonian dynamics
146(31)
6.1 C° symplectic rigidity theorem
146(12)
6.2 Topological Hamiltonian flows and Hamiltonians
158(5)
6.3 Uniqueness of the topological Hamiltonian and its How
163(8)
6.4 The hameomorphism group
171(6)
PART 2 RUDIMENTS OF PSEUDOHOLOMORPHIC CURVES
7 Geometric calculations
177(20)
7.1 Natural connection on almost-Kahler manifolds
177(8)
7.2 Global properties of J-holomorphic curves
185(4)
7.3 Calculations of Δe(u) on shell
189(4)
7.4 Boundary conditions
193(4)
8 Local study of J-holomorphic curves
197(50)
8.1 Interior a-priori estimates
197(5)
8.2 Off-shell elliptic estimates
202(7)
8.3 Removing boundary contributions
209(3)
8.4 Proof of e-regularity and density estimates
212(9)
8.5 Boundary regularity of weakly J-holomorphic maps
221(6)
8.6 The removable singularity theorem
227(8)
8.7 Isoperimetric inequality and the monotonicity formula
235(4)
8.8 The similarity principle and the local structure of the image
239(8)
9 Gromov compactification and stable maps
247(76)
9.1 The moduli space of pseudoholomorphic curves
247(4)
9.2 Sachs--Uhlenbeck rescaling and bubbling
251(6)
9.3 Definition of stable curves
257(10)
9.4 Deformations of stable curves
267(24)
9.5 Stable map and stable map topology
291(32)
10 Fredholm theory
323(32)
10.1 A quick review of Banach manifolds
323(5)
10.2 Off-shell description of the moduli space
328(4)
10.3 Linearizations of ∂(j,J)
332(3)
10.4 Mapping transversality and linearization of ∂
335(10)
10.5 Evaluation transversality
345(8)
10.6 The problem of negative multiple covers
353(2)
11 Applications to symplectic topology
355(22)
11.1 Gromov's non-squeezing theorem
356(8)
11.2 Nondegeneracy of the Hofer norm
364(13)
References 377(15)
Index 392
Yong-Geun Oh is Director of the IBS Center for Geometry and Physics and is Professor in the Department of Mathematics at POSTECH (Pohang University of Science and Technology) in Korea. He was also Professor in the Department of Mathematics at the University of Wisconsin, Madison. He is a member of the KMS, the AMS, the Korean National Academy of Sciences, and the inaugural class of AMS Fellows. In 2012 he received the Kyung-Ahm Prize for Science in Korea.
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