Author | Polterovich, Leonid. author |
---|---|
Title | The Geometry of the Group of Symplectic Diffeomorphism [electronic resource] / by Leonid Polterovich |
Imprint | Basel : Birkhรคuser Basel : Imprint: Birkhรคuser, 2001 |
Connect to | http://dx.doi.org/10.1007/978-3-0348-8299-6 |
Descript | XII, 136 p. 2 illus. online resource |
Preface -- 1 Introducing the Group -- 1.1 The origins of Hamiltonian diffeomorphisms -- 1.2 Flows and paths of diffeomorphisms -- 1.3 Classical mechanics -- 1.4 The group of Hamiltonian diffeomorphisms -- 1.5 Algebraic properties of Ham(M, Q) -- 2 Introducing the Geometry -- 2.1 A variational problem -- 2.2 Biinvariant geometries on Ham(M, Q) -- 2.3 The choice of the norm: Lp vs. Loa -- 2.4 The concept of displacement energy -- 3 Lagrangian Submanifolds -- 3.1 Definitions and examples -- 3.2 The Liouville class -- 3.3 Estimating the displacement energy -- 4 The $$ \bar \partial $$-Equation -- 4.1 Introducing the $$ \bar \partial $$-operator -- 4.2 The boundary value problem -- 4.3 An application to the Liouville class -- 4.4 An example -- 5 Linearization -- 5.1 The space of periodic Hamiltonians -- 5.2 Regularization -- 5.3 Paths in a given homotopy class -- 6 Lagrangian Intersections -- 6.1 Exact Lagrangian isotopies -- 6.2 Lagrangian intersections -- 6.3 An application to Hamiltonian loops -- 7 Diameter -- 7.1 The starting estimate -- 7.2 The fundamental group -- 7.3 The length spectrum -- 7.4 Refining the estimate -- 8 Growth and Dynamics -- 8.1 Invariant tori of classical mechanics -- 8.2 Growth of one-parameter subgroups -- 8.3 Curve shortening in Hoferโs geometry -- 8.4 What happens when the asymptotic growth vanishes? -- 9 Length Spectrum -- 9.1 The positive and negative parts of Hoferโs norm -- 9.2 Symplectic fibrations over S2 -- 9.3 Symplectic connections -- 9.4 An application to length spectrum -- 10 Deformations of Symplectic Forms -- 10.1 The deformation problem -- 10.2 The $$ \bar \partial $$-equation revisited -- 10.3 An application to coupling -- 10.4 Pseudo-holomorphic curves -- 10.5 Persistence of exceptional spheres -- 11 Ergodic Theory -- 11.1 Hamiltonian loops as dynamical objects -- 11.2 The asymptotic length spectrum -- 11.3 Geometry via algebra -- 12 Geodesics -- 12.1 What are geodesics? -- 12.2 Description of geodesics -- 12.3 Stability and conjugate points -- 12.4 The second variation formula -- 12.5 Analysis of the second variation formula -- 12.6 Length minimizing geodesics -- 13 Floer Homology -- 13.1 Near the entrance -- 13.2 Morse homology in finite dimensions -- 13.3 Floer homology -- 13.4 An application to geodesics -- 13.5 Towards the exit -- 14 Non-Hamiltonian Diffeomorphisms -- 14.1 The flux homomorphism -- 14.2 The flux conjecture -- 14.3 Links to โhardโ symplectic topology -- 14.4 Isometries in Hoferโs geometry -- List of Symbols