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Einstein Constraints and Ricci Flow

A Geometrical Averaging of Initial Data Sets

  • Book
  • © 2023

Overview

Authors:
  1. Mauro Carfora

    1. Dipartimento di Fisica, University of Pavia, Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Pavia and Gruppo Nazionale di Fisica Matematica (GNFM), Pavia, Italy

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  2. Annalisa Marzuoli

    1. Dipartimento Matematica F Casorati, University of Pavia, Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Pavia and Gruppo Nazionale di Fisica Matematica (GNFM), Pavia, Italy

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  • 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

Part of the book series: Mathematical Physics Studies (MPST)

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Table of contents (5 chapters)

  1. Front Matter

    Pages i-xii
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  2. Introduction

    • Mauro Carfora, Annalisa Marzuoli
    Pages 1-10

  3. Geometric Preliminaries

    • Mauro Carfora, Annalisa Marzuoli
    Pages 11-43

  4. Ricci Flow Background

    • Mauro Carfora, Annalisa Marzuoli
    Pages 45-79

  5. Ricci Flow Conjugation of Initial Data Sets

    • Mauro Carfora, Annalisa Marzuoli
    Pages 81-156

  6. Concluding Remarks

    • Mauro Carfora, Annalisa Marzuoli
    Pages 157-162

  7. Back Matter

    Pages 163-173
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Keywords

About this book

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. 


Authors and Affiliations

  • Dipartimento di Fisica, University of Pavia, Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Pavia and Gruppo Nazionale di Fisica Matematica (GNFM), Pavia, Italy

    Mauro Carfora

  • Dipartimento Matematica F Casorati, University of Pavia, Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Pavia and Gruppo Nazionale di Fisica Matematica (GNFM), Pavia, Italy

    Annalisa Marzuoli

About the authors

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). 

Bibliographic Information

  • Book Title: Einstein Constraints and Ricci Flow

  • Book Subtitle: A Geometrical Averaging of Initial Data Sets

  • Authors: Mauro Carfora, Annalisa Marzuoli

  • Series Title: Mathematical Physics Studies

  • DOI: https://doi.org/10.1007/978-981-19-8540-9

  • Publisher: Springer Singapore

  • eBook Packages: Physics and Astronomy, Physics and Astronomy (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023

  • Hardcover ISBN: 978-981-19-8539-3Published: 11 January 2023

  • Softcover ISBN: 978-981-19-8542-3Published: 11 January 2024

  • eBook ISBN: 978-981-19-8540-9Published: 10 January 2023

  • Series ISSN: 0921-3767

  • Series E-ISSN: 2352-3905

  • Edition Number: 1

  • Number of Pages: XII, 173

  • Number of Illustrations: 1 b/w illustrations, 32 illustrations in colour

  • Topics: Mathematical Physics, Differential Geometry

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