This book contains a self-consistent treatment of a geometric averaging technique, induced by the Ricci flow, that allows comparing a given (generalized) Einstein initial data set with another distinct Einstein initial data set, both supported on a given closed n-dimensional manifold.
This is a case study where two vibrant areas of research in geometric analysis, Ricci flow and Einstein constraints theory, interact in a quite remarkable way. The interaction is of great relevance for applications in relativistic cosmology, allowing a mathematically rigorous approach to the initial data set averaging problem, at least when data sets are given on a closed space-like hypersurface.
The book does not assume an a priori knowledge of Ricci flow theory, and considerable space is left for introducing the necessary techniques. These introductory parts gently evolve to a detailed discussion of the more advanced results concerning a Fourier-mode expansion and a sophisticated heat kernel representation of the Ricci flow, both of which are of independent interest in Ricci flow theory.
This work is intended for advanced students in mathematical physics and researchers alike.
Author(s): Mauro Carfora, Annalisa Marzuoli
Series: Mathematical Physics Studies
Publisher: Springer
Year: 2023
Language: English
Pages: 180
City: Singapore
Preface
Acknowledgements
Contents
1 Introduction
1.1 Generalized Einstein Initial Data Sets
1.2 Ricci Flow and Parabolic Conjugation
1.3 Outline of These Lecture Notes
2 Geometric Preliminaries
2.1 Diffusion and Ricci Curvature
2.2 Einstein and Quasi-Einstein Metrics
2.3 Some Properties of the Space of Riemannian Metrics
3 Ricci Flow Background
3.1 Ricci Flow as a Dynamical System on mathcalMet(Σ)
3.2 Ricci Flow on Riemannian Manifolds with Density
3.3 Perelman's Entropy Generating Functional mathcalF
3.4 Non-collapsing Ricci Flow of Bounded Geometry
3.5 The Hodge–DeRham–Lichnerowicz Heat Operator
4 Ricci Flow Conjugation of Initial Data Sets
4.1 The Physical Rationale of Ricci Flow Conjugation
4.2 Motivations from Relativistic Cosmology
4.3 The Existence of an Interpolating Ricci Flow
4.4 Ricci Flow Conjugation Between Einstein Initial Data Sets
4.5 The Ricci Flow Evolution of the Einstein Constraints
4.6 Conjugated Mode Expansion
4.7 Averaging of Matter Initial Data Sets
4.8 The Dominant Energy Condition
4.9 Averaging the Second Fundamental form K
4.10 Heat Kernel Asymptotics of Ricci Flow Conjugated Data
4.11 mathcalF-Energy and Mode Stability of Ricci Flow Conjugation
5 Concluding Remarks
5.1 Projecting the Averaged Data on the Constraint Manifold
Appendix References
Index
