A Course on Holomorphic Discs (Birkhäuser Advanced Texts Basler Lehrbücher)

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