Generalized Ricci Flow
About this Title
Mario Garcia-Fernandez, Instituto de Ciencias Matemáticas, Madrid, Spain and Jeffrey Streets, University of California, Irvine, Irvine, CA
Publication: University Lecture Series
Publication Year: 2021; Volume 76
ISBNs: 978-1-4704-6258-1 (print); 978-1-4704-6411-0 (online)
The generalized Ricci flow is a geometric evolution equation which has recently emerged from investigations into mathematical physics, Hitchin's generalized geometry program, and complex geometry. This book gives an introduction to this new area, discusses recent developments, and formulates open questions and conjectures for future study.
The text begins with an introduction to fundamental aspects of generalized Riemannian, complex, and Kähler geometry. This leads to an extension of the classical Einstein-Hilbert action, which yields natural extensions of Einstein and Calabi-Yau structures as ‘canonical metrics’ in generalized Riemannian and complex geometry. The book then introduces generalized Ricci flow as a tool for constructing such metrics and proves extensions of the fundamental Hamilton/Perelman regularity theory of Ricci flow. These results are refined in the setting of generalized complex geometry, where the generalized Ricci flow is shown to preserve various integrability conditions, taking the form of pluriclosed flow and generalized Kähler-Ricci flow, leading to global convergence results and applications to complex geometry. Finally, the book gives a purely mathematical introduction to the physical idea of T-duality and discusses its relationship to generalized Ricci flow.
The book is suitable for graduate students and researchers with a background in Riemannian and complex geometry who are interested in the theory of geometric evolution equations.
Graduate students and researchers interested in the generalized Ricci Flow and mathematical physics.
Table of Contents
- Generalized Riemannian Geometry
- Generalized Connections and Curvature
- Fundamentals of Generalized Ricci Flow
- Local Existence and Regularity
- Energy and Entropy Functionals
- Generalized Complex Geometry
- Canonical Metrics in Generalized Complex Geometry
- Generalized Ricci Flow in Complex Geometry