An introduction to compactness results in symplectic field theory [electronic resource] /
Abbas, Casim.author.
This book provides an introduction to symplectic field theory, a new and important subject which is currently being developed. The starting point of this theory are compactness results for holomorphic curves established in the last decade. The author presents a systematic introduction providing a lot of background material, much of which is scattered throughout the literature. Since the content grew out of lectures given by the author, the main aim is to provide an entry point into symplectic field theory for non-specialists and for graduate students. Extensions of certain compactness results, which are believed to be true by the specialists but have not yet been published in the literature in detail, top off the scope of this monograph.
Description based upon print version of record.
""An Introduction to Compactness Results in Symplectic Field Theory""; ""Preface""; ""Contents""; ""Chapter 1: Riemann Surfaces""; ""1.1 Smooth and Noded Riemann Surfaces""; ""1.2 Riemann Surfaces and Hyperbolic Geometry""; ""1.2.1 Stable Surfaces""; ""1.2.2 The Hyperbolic Plane""; ""1.2.3 Gluing Hyperbolic Surfaces Along Their Boundaries""; ""1.2.4 Annuli""; ""1.2.5 Hexagons in the Upper Half Plane and Pairs of Pants""; ""1.2.6 Pairs of Pants Decompositions""; ""1.2.7 Thick-Thin Decomposition and Collar Lemma""; ""1.3 The Deligne-Mumford Compactness Result""
""1.3.1 The Notion of Convergence""""1.3.2 The Proof of the Compactness Result for Surfaces Without Boundary""; ""1.3.3 Surfaces with Boundary""; ""Chapter 2: Pseudoholomorphic Curves""; ""2.1 Basic De nitions""; ""2.2 Asymptotic Behavior Near a Puncture""; ""2.2.1 Introduction""; ""2.2.2 Estimates for the Linear Cauchy Riemann Operator""; ""2.2.3 Regularity: Gradient Bounds Imply Cinfty-Bounds""; ""2.2.4 Behavior Near an Interior Puncture""; ""2.2.5 Behavior Near a Boundary Puncture""; ""2.3 Isoperimetric Inequality, Monotonicity Lemma, Removal of Singularities""
""2.4 Finite-Energy Strips and Cylinders of Small Area""""Chapter 3: The SFT Compactness Results""; ""3.1 Holomorphic Buildings for Curves Without Boundary""; ""3.1.1 Holomorphic Buildings of Height 1""; ""3.1.2 Holomorphic Buildings of Height N""; ""3.2 Adding Additional Marked Points""; ""3.3 The Compactness Result for the Case Without Boundary""; ""3.3.1 Statement of the Result""; ""3.3.2 Gradient Bounds""; ""3.3.3 Convergence in the Thick Part""; ""3.3.4 Convergence in the Thin Part and Level Structure""; ""3.4 More General Holomorphic Buildings and Compactness Results""
""3.4.1 Holomorphic Buildings of Height 1""""3.4.2 Holomorphic Buildings of Height N""; ""3.4.3 Holomorphic Buildings in Manifolds with Cylindrical Ends""; ""3.4.4 A More General Compactness Result""; ""References""; ""Index""
Includes bibliographical references and index.
This book provides an introduction to symplectic field theory, a new and important subject which is currently being developed. The starting point of this theory are compactness results for holomorphic curves established in the last decade. The author presents a systematic introduction providing a lot of background material, much of which is scattered throughout the literature. Since the content grew out of lectures given by the author, the main aim is to provide an entry point into symplectic field theory for non-specialists and for graduate students. Extensions of certain compactness results, which are believed to be true by the specialists but have not yet been published in the literature in detail, top off the scope of this monograph.
Differential geometry.
Manifolds (Mathematics).
Complex manifolds.
Differential Geometry.
Manifolds and Cell Complexes (incl. Diff.Topology).