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Global Analysis on Open Manifolds
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Authors: Jurgen Eichhorn (Univ. Greifswald, Germany) 
Book Description:
Subject Scope: Mathematics; Mathematical Physics

Global analysis is the analysis on manifolds. Since the middle of the sixties there exists a highly elaborated setting if the underlying manifold is compact, evidence of which can be found in index theory, spectral geometry, the theory of harmonic maps, many applications to mathematical physics on closed manifolds like gauge theory, Seiberg–Witten theory etc. If the underlying manifold is open, i.e. non–compact and without boundary, then most of the foundations and of the great achievements fail. Elliptic operators are no longer Fredholm, the analytical and topological indexes are not defined, the spectrum of self–adjoint elliptic operators is no longer discrete, functional spaces strongly depend on the operators involved and the data from geometry, many embedding and module structure theorems do not hold, manifolds of maps are not defined etc. It is the goal of this new book to provide serious foundations for global analysis on open manifolds, to discuss the difficulties and special features which come from the openess and to establish many results and applications on this basis.

Table of Contents:
Preface pp. i-ix

Introduction

Chapter 1 - A setting of linear analysis; pp. 1-58
1.- Basics of Riemannian geometry
2.- Tools from Hilbert space theory
3.- Sobolev spaces on open manifolds
4.- Uniform pseudo–di_erential and Fourier integral operators on open manifolds

Chapter 2 - Spectral geometry; pp. 59-240
1.- Generalities of spectral geometry
2.- Spectral geometry of the scalar Laplacian
3.- Spectral geometry of q–forms
4.- The spectral value zero
5.- The heat semigroup and the heat kernel
6.- Fredholm properties and index theory

Chapter 3 - A setting of non–linear analysis; pp. 241-437
1.- Uniform structures and their applications to vector bundles and conformal factors
2.- Spaces of metrics and connections and their geometry
3.- Characteristic numbers for open manifolds and their applications
4.- Uniform structures of Cli_ord bundles
5.- Manifolds of maps
6.- Banach manifolds of maps in the Lp–category
7.- The bounded di_eomorphism group
8.- ILH di_eomorphism groups
9.- The group of volume preserving di_eomorphisms
10.- The groups of contact transformations and symplectic di_eomorphisms
11 Lie groups of Fourier integral operators on open manifolds

Chapter 4 - Some non–linear partial di_erential equations on open manifolds; pp. 439-501
1.- Gauge theory on open manifolds
2.- Fluid dynamics
3.- Teichm¨uller theory
4.- A slice theorem

Chapter 5 - A classification approach; pp. 503-583
1.- Uniform structures of metric spaces
2.- Functional algebraic topology
3.- Bordism theory for open manifolds

Chapter 6 - Dirichlet series for open manifolds; pp. 585-623
1.- General heat kernel estimates
2.- Trace class properties .
3.- Relative index theory
4.- Relative zeta functions, eta functions, determinants and torsion

Bibliography
Notation
Index pp. 643-644

   Binding: Hardcover
   Pub. Date: 2007
   Pages: 653, 7 x 10
   ISBN: 978-1-60021-563-6
   Status: AV
  
Status Code Description
AN Announcing
FM Formatting
PP Page Proofs
FP Final Production
EP Editorial Production
PR At Prepress
AP At Press
AV Available
  
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Global Analysis on Open Manifolds