Global analysis on open manifolds

著者

    • Eichhorn, Jürgen

書誌事項

Global analysis on open manifolds

Jürgen Eichhorn

Nova Science Publishers, c2007

大学図書館所蔵 件 / 全6件

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注記

Includes bibliographical reference (p. 625-640) and index

内容説明・目次

内容説明

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 openness and to establish many results and applications on this basis.

目次

  • Preface
  • Introduction
  • A Setting of Linear Analysis
  • Basics of Riemannian Geometry
  • Tools from Hilbert Space Theory
  • Sobolev Spaces on Open Manifolds
  • Uniform Pseudo-differential and Fourier Integral Operators on Open Manifolds
  • Spectral Geometry
  • Generalities of Spectral Geometry
  • Spectral Geometry of the Scalar Laplacian
  • Spectral Geometry of q-forms
  • The Spectral Value Zero
  • The Heat Semigroup and the Heat Kernel
  • Fredholm Properties and Index Theory
  • A setting of Non-linear Analysis
  • Uniform Structures and their Applications to Vector Bundles and Conformal Factors
  • Spaces of Metrics and Connections and their Geometry
  • Characteristic Numbers for Open Manifolds and their Applications
  • Uniform Structures of Clifford Bundles
  • Manifolds of Maps
  • Banach Manifolds of Maps in the Lp-category
  • The Bounded diffeomorphism Group
  • ILH diffeomorphism Groups
  • The Group of Volume Preserving diffeomorphisms
  • The Groups of Contact Transformations and Symplectic diffeomorphisms
  • Lie Groups of Fourier Integral Operators on Open Manifolds
  • Some Non-linear Partial differential Equations on Open Manifolds
  • Gauge Theory on Open Manifolds
  • Fluid Dynamics
  • Teichmuller Theory
  • A Slice Theorem
  • A Classification Approach
  • Uniform Structures of Metric Spaces
  • Functional Algebraic Topology
  • Bordism Theory for Open Manifolds
  • Dirichlet Series for Open Manifolds
  • General Heat Kernel Estimates
  • Trace Class Properties
  • Relative Index Theory
  • Relative Zeta Functions, eta Functions, Determinants and Torsion
  • References
  • Notation
  • Index.

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