Carfora M., Marzuoli A. Einstein Constraints and Ricci Flow: A Geometrical Averaging of Initial Data Sets

Singapore: Springer, 2023. — 181 p.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.Preface
Acknowledgements
Introduction
Generalized Einstein Initial Data Sets
Ricci Flow and Parabolic Conjugation
Outline of These Lecture Notes
Geometric Preliminaries
Diffusion and Ricci Curvature
Einstein and Quasi-Einstein Metrics
Some Properties of the Space of Riemannian Metrics
Ricci Flow Background
Ricci Flow as a Dynamical System on mathcalMet(Σ)
Ricci Flow on Riemannian Manifolds with Density
Perelman's Entropy Generating Functional mathcalF
Non-collapsing Ricci Flow of Bounded Geometry
The Hodge–DeRham–Lichnerowicz Heat Operator
Ricci Flow Conjugation of Initial Data Sets
The Physical Rationale of Ricci Flow Conjugation
Motivations from Relativistic Cosmology
The Existence of an Interpolating Ricci Flow
Ricci Flow Conjugation Between Einstein Initial Data Sets
The Ricci Flow Evolution of the Einstein Constraints
Conjugated Mode Expansion
Averaging of Matter Initial Data Sets
The Dominant Energy Condition
Averaging the Second Fundamental form K
Heat Kernel Asymptotics of Ricci Flow Conjugated Data
mathcalF-Energy and Mode Stability of Ricci Flow Conjugation
Concluding Remarks
Projecting the Averaged Data on the Constraint Manifold
Appendix
References
Index