Symplectic geometry is the geometry underlying Hamiltonian dynamics, and symplectic mappings arise as time-1-maps of Hamiltonian flows. The spectacular rigidity phenomena for symplectic mappings discovered in the last two decades show that certain things cannot be done by a symplectic mapping. For instance, Gromov's famous "non-squeezing" theorem states that one cannot map a ball into a thinner cylinder by a symplectic embedding. The aim of this book is to show that certain other things can be done by symplectic mappings. This is achieved by various elementary and explicit symplectic embedding constructions, such as "folding", "wrapping", and "lifting". These constructions are carried out in detail and are used to solve some specific symplectic embedding problems. The exposition is self-contained and addressed to students and researchers interested in geometry or dynamics.
Author(s): Felix Schlenk
Series: de Gruyter Expositions in Mathematics
Edition: 1
Publisher: Gruyter
Year: 2005
Language: English
Pages: 261
Preface......Page 8
Contents......Page 10
1.1 From classical mechanics to symplectic geometry......Page 12
1.2 Symplectic embedding obstructions......Page 15
1.3 Symplectic embedding constructions......Page 22
2.1 Comparison of the relations ≤......Page 34
2.2 Rigidity for ellipsoids......Page 35
2.3 Rigidity for polydiscs ?......Page 39
3.1 Reformulation of Theorem 2......Page 42
3.2 The folding construction......Page 50
3.3 End of the proof......Page 58
4.1 Modification of the folding construction......Page 63
4.2 Multiple folding......Page 64
4.3 Embeddings into balls......Page 68
4.4 Embeddings into cubes......Page 84
5.1 Four types of folding......Page 93
5.2 Embedding polydiscs into cubes......Page 95
5.3 Embedding ellipsoids into balls......Page 101
6.1 Proof of lim......Page 118
6.2 Proof of lim......Page 134
6.3 Asymptotic embedding invariants......Page 158
7.1 The wrapping construction......Page 160
7.2 Folding versus wrapping......Page 168
8.1 A more general statement......Page 173
8.2 A further motivation for Problem......Page 176
8.3 Proof by symplectic folding......Page 179
8.4 Proof by symplectic lifting......Page 188
Packing symplectic manifolds by hand......Page 199
9.1 Motivations for the symplectic packing problem......Page 200
9.2 The packing numbers of the 4-ball and......Page 205
9.3 Explicit maximal packings in four dimensions......Page 209
9.4 Maximal packings in higher dimensions......Page 224
Appendix......Page 226
Bibliography......Page 252
Index......Page 258
