Book contents
- Frontmatter
- Contents
- List of contributors
- Introduction
- 1 A brief introduction to Dirac manifolds
- 2 Differential geometry of holomorphic vector bundles on a curve
- 3 Paths towards an extension of Chern–Weil calculus to a class of infinite dimensional vector bundles
- 4 Introduction to Feynman integrals
- 5 Iterated integrals in quantum field theory
- 6 Geometric issues in quantum field theory and string theory
- 7 Geometric aspects of the Standard Model and the mysteries of matter
- 8 Absence of singular continuous spectrum for some geometric Laplacians
- 9 Models for formal groupoids
- 10 Elliptic PDEs and smoothness of weakly Einstein metrics of Hölder regularity
- 11 Regularized traces and the index formula for manifolds with boundary
- Index
- References
10 - Elliptic PDEs and smoothness of weakly Einstein metrics of Hölder regularity
Published online by Cambridge University Press: 05 May 2013
- Frontmatter
- Contents
- List of contributors
- Introduction
- 1 A brief introduction to Dirac manifolds
- 2 Differential geometry of holomorphic vector bundles on a curve
- 3 Paths towards an extension of Chern–Weil calculus to a class of infinite dimensional vector bundles
- 4 Introduction to Feynman integrals
- 5 Iterated integrals in quantum field theory
- 6 Geometric issues in quantum field theory and string theory
- 7 Geometric aspects of the Standard Model and the mysteries of matter
- 8 Absence of singular continuous spectrum for some geometric Laplacians
- 9 Models for formal groupoids
- 10 Elliptic PDEs and smoothness of weakly Einstein metrics of Hölder regularity
- 11 Regularized traces and the index formula for manifolds with boundary
- Index
- References
Summary
Abstract
This chapter is broadly divided into two parts. In the first, a brief but self-contained review of the interior regularity theory of elliptic PDEs is presented, including relevant preliminaries on function spaces. In the second, as an application of the tools introduced in the first part, a detailed study of the Einstein condition on Riemannian manifolds with metrics of Holder regularity is undertaken, introducing important techniques such as the use of harmonic coordinates and giving some consideration to the smoothness of the differentiable structure of the underlying manifold.
Introduction
Partial differential equations (PDEs) play a fundamental role in many areas of pure and applied sciences. Particularly known is their ubiquitous presence in the description of physical systems, but it is less obvious to students how their application to geometry is not only useful but often decisive. As an introduction to the world of PDEs in geometry, the aim of this chapter is two-fold. First, to give a brief (and rather condensed) overview of the minimum number of basic preliminaries that allow us to write down precise and complete statements of several important theorems from the regularity theory of elliptic PDEs, which is one of the most needed aspects for applications. Second, to present a detailed treatment of some useful techniques required to study geometric equations involving the Ricci tensor and, in particular, to put into use the definitions and tools introduced in the first part to study the smoothness of weak solutions to the Einstein condition on Riemannian manifolds with metrics of Hölder regularity C1,α.
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- Chapter
- Information
- Geometric and Topological Methods for Quantum Field TheoryProceedings of the 2009 Villa de Leyva Summer School, pp. 340 - 365Publisher: Cambridge University PressPrint publication year: 2013