Einstein Constraints and Ricci Flow

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.

Deals with the application of Ricci flow to the comparison and averaging of Einstein's initial data sets Introduces Ricci flow conjugation between two distinct Einstein initial data sets, providing analysis of its properties Proves Fourier modes expansion and heat kernel estimates for Ricci flow of independent interest in Ricci flow theory

Autorentext

Mauro Carfora is Professor of Mathematical Physics at the University of Pavia, Italy. He is co-author with Annalisa Marzuoli of the Springer Lecture Notes in Physics Quantum Triangulations (LNP 845 and LNP 942), and illustrator of the popular relativity book Flat and Curved Spacetimes by George. F. R. Ellis and Ruth Williams.
Annalisa Marzuoli is Associate Professor of Mathematical Physics at the University of Pavia, Italy. She is co-author with Jan Ambjørn and Mauro Carfora of the Springer Lecture Notes in Physics The Geometry of Dynamical Triangulations (Springer LNP m50).

Zusammenfassung

"The book is well written and well motivated, hence warmly recommended to post-graduate and PhD students as well as researchers (both mathematicians and physicists) interested in modern global analysis and/or mathematical cosmology." (Gábor Etesi, zbMATH 1536.53002, 2024)

Inhalt
Introduction.- Geometric preliminaries.- Ricci ow background.- Ricci ow conjugation of initial data sets.- Concluding remarks.