Generalized Ricci Flow

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. Cover Title page Chapter 1. Introduction 1.1. Outline 1.2. On pedagogy 1.3. Acknowledgments Chapter 2. Generalized Riemannian Geometry 2.1. Courant algebroids 2.2. Symmetries of the Dorfman bracket 2.3. Generalized metrics 2.4. Divergence operators Chapter 3. Generalized Connections and Curvature 3.1. Generalized connections 3.2. Metric compatible connections 3.3. The classical Bismut connection 3.4. Curvature and the first Bianchi identity 3.5. Generalized Ricci curvature 3.6. Generalized scalar curvature 3.7. Generalized Einstein-Hilbert functional Chapter 4. Fundamentals of Generalized Ricci Flow 4.1. The equation and its motivation 4.2. Examples 4.3. Maximum principles 4.4. Invariance group and solitons 4.5. Low dimensional structure Chapter 5. Local Existence and Regularity 5.1. Variational formulas 5.2. Short time existence 5.3. Curvature evolution equations 5.4. Smoothing estimates 5.5. Results on maximal existence time 5.6. Compactness results for generalized metrics Chapter 6. Energy and Entropy Functionals 6.1. Generalized Ricci flow as a gradient flow 6.2. Expander entropy and Harnack estimate 6.3. Shrinking Entropy and local collapsing 6.4. Corollaries on nonsingular solutions Chapter 7. Generalized Complex Geometry 7.1. Linear generalized complex structures 7.2. Generalized complex structures on manifolds 7.3. Courant algebroids and pluriclosed metrics 7.4. Generalized Kähler geometry Chapter 8. Canonical Metrics in Generalized Complex Geometry 8.1. Connections, torsion, and curvature 8.2. Canonical metrics in complex geometry 8.3. Examples and rigidity results Chapter 9. Generalized Ricci Flow in Complex Geometry 9.1. Kähler-Ricci flow 9.2. Pluriclosed flow 9.3. Generalized Kähler-Ricci flow 9.4. Reduced flows 9.5. Torsion potential evolution equations 9.6. Higher regularity from uniform parabolicity 9.7. Metric evolution equations 9.8. Sharp existence and convergence results Chapter 10. T-duality 10.1. Topological T-duality 10.2. T-duality and Courant algebroids 10.3. Geometric T-duality 10.4. Buscher rules and the dilaton shift 10.5. Einstein-Hilbert action 10.6. Examples Bibliography Back Cover "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."-- Back cover