This textbook, based on a one-semester course taught several times by the authors, provides a self-contained, comprehensive yet concise introduction to the theory of pseudoholomorphic curves. Gromov’s nonsqueezing theorem in symplectic topology is taken as a motivating example, and a complete proof using pseudoholomorphic discs is presented. A sketch of the proof is discussed in the first chapter, with succeeding chapters guiding the reader through the details of the mathematical methods required to establish compactness, regularity, and transversality results. Concrete examples illustrate many of the more complicated concepts, and well over 100 exercises are distributed throughout the text. This approach helps the reader to gain a thorough understanding of the powerful analytical tools needed for the study of more advanced topics in symplectic topology.
This text can be used as the basis for a graduate course, and it is also immensely suitable for independent study. Prerequisites include complex analysis, differential topology, and basic linear functional analysis; no prior knowledge of symplectic geometry is assumed.
This book is also part of the Virtual Series on Symplectic Geometry.
Author(s): Hansjörg Geiges, Kai Zehmisch
Series: Birkhäuser Advanced Texts Basler Lehrbücher
Publisher: Birkhäuser
Year: 2023
Language: English
Pages: 202
City: Basel
Contents
Preface
Outline of the text
The rationale for this text
Discs versus spheres
Mathematical prerequisites
A guide for the reader and the lecturer (and a little rant)
The sources we consulted
Acknowledgements
Chapter I Gromov's Nonsqueezing Theorem
I.1 Liouville's theorem in Hamiltonian mechanics
I.2 Symplectomorphisms and symplectic embeddings
I.3 Gromov's nonsqueezing theorem
I.3.1 Statement of the theorem
I.3.2 Almost complex structures
I.3.3 Idea of the proof
I.4 The monotonicity lemma
I.4.1 The strong maximum principle
I.4.2 Symplectic energy of holomorphic curves
I.4.3 Proof of the monotonicity lemma
I.5 Isoperimetric inequalities
I.5.1 Proof of Lemma I.4.25
I.5.2 Symplectic action and energy of loops
I.6 Holomorphic curves are minimal surfaces
I.7 The space of almost complex structures
I.7.1 Linear algebra of complex structures
I.7.2 The Cayley transformation as a Möbius transformation
I.7.3 The Cayley transformation on matrices
I.7.4 Interpolating almost complex structures
I.8 A moduli space of J-holomorphic discs
I.8.1 The Schwarz lemma
I.8.2 Automorphisms of the unit disc
I.8.3 The moduli space M
I.8.4 Symplectic energy of J-holomorphic discs
I.8.5 Flat discs
Chapter II Compactness
II.1 Geometric a priori bounds
II.1.1 Schwarz re ection in the unit circle
II.1.2 A bound on boundaries
II.1.3 A bound on nonstandard discs
II.1.4 The boundary lemma of E. Hopf
II.1.5 Boundaries are embedded
II.1.6 The evaluation map
II.2 Uniform gradient bounds
II.2.1 The Fréchet metric
II.2.2 The Arzelà–Ascoli theorem
II.2.3 Examples of bubbling
II.2.4 The bubbling-o argument
II.3 An asymptotic isoperimetric inequality
II.4 Boundary singularities
Chapter III Bounds of Higher Order
III.1 The a priori estimate
III.1.1 Sobolev norms
III.1.2 The Poincaré inequality
III.1.3 The inhomogeneous Cauchy–Riemann equation
III.1.4 The Calderón–Zygmund inequality
III.1.5 The a priori estimate
III.1.6 Why do we need Sobolev norms?
III.2 Various Sobolev estimates
III.3 The Ck-norm of nonstandard discs is bounded
Chapter IV Elliptic Regularity
IV.1 The linearisation
IV.1.1 Notions of di erentiability
IV.1.2 The linearisation of Ck
IV.1.3 Spaces of continuous maps are smooth Banach manifolds
IV.1.4 The linearisation of ∂J
IV.2 The Sobolev completion
IV.2.1 The de nition of Sobolev spaces
IV.2.2 The Sobolev embedding theorem
IV.2.3 Rules of differentiation in Sobolev spaces
IV.2.4 Sobolev estimates
IV.2.5 The completion B of C is a Banach manifold
IV.2.6 Differentiability of ∂J
IV.3 Elliptic regularity
IV.3.1 The topology on M
IV.3.2 A local estimate
IV.3.3 The shift operator
IV.3.4 Proof of Theorem IV.3.1
IV.3.5 Proof of the deceptively simple lemma
IV.3.6 The need for the Sobolev completion
Chapter V Transversality
V.1 Fredholm theory
V.1.1 Fredholm operators
V.1.2 The Fredholm index of
V.1.3 Transformation to a perturbed ∂-operator
V.1.4 Fredholm plus compact is Fredholm
V.1.5 The components of ∂
V.2 Regular values and the Sard{Smale theorem
V.2.1 The implicit function theorem
V.2.2 Regular values
V.2.3 The Sard{Smale theorem
V.2.4 How to prove that M is a manifold
V.3 The Carleman similarity principle
V.3.1 Behaviour near the boundary
V.3.2 From a nonlinear to a linear equation
V.3.3 The similarity principle and its corollaries
V.3.4 Proof of the Carleman similarity principle
V.4 Injective points
V.4.1 Inj(u) is open and dense
V.4.2 (Self-)Intersections of holomorphic discs
V.5 The Floer space of almost complex structures
V.5.1 The Floer norm
V.5.2 A separable Banach space
V.5.3 A Banach manifold of almost complex structures
V.6 The universal moduli space
V.6.1 The tangent space TJJ0
V.6.2 The linearisation of ∂
V.6.3 The Weyl lemma
V.6.4 The universal moduli space is a Banach manifold
V.7 The moduli space M and the evaluation map
V.7.1 The moduli space MJ is a manifold
V.7.2 The mod 2 degree of a smooth map
V.7.3 The degree of the evaluation map
Bibliography
Index
