**University Lecture Series**

Volume: 76;
2021;
248 pp;
Softcover

MSC: Primary 53;

**Print ISBN: 978-1-4704-6258-1
Product Code: ULECT/76**

List Price: $55.00

AMS Member Price: $44.00

MAA Member Price: $49.50

**Electronic ISBN: 978-1-4704-6411-0
Product Code: ULECT/76.E**

List Price: $55.00

AMS Member Price: $44.00

MAA Member Price: $49.50

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#### Supplemental Materials

# Generalized Ricci Flow

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*Mario Garcia-Fernandez; Jeffrey Streets*

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.

#### Readership

Graduate students and researchers interested in the generalized Ricci Flow and mathematical physics.

#### Reviews & Endorsements

Generalized geometry is still a young field, and even expository accounts are generally found in papers. In the authors' words, "The primary purpose of this book is to provide an introduction to the fundamental geometric, algebraic, topological, and analytic aspects of the generalized Ricci flow equation." Many results about generalized geometry and generalized Ricci flow are currently open. The authors note, "The secondary purpose of this book is to formulate questions and conjectures about the generalized Ricci flow as an invitation to the reader."

-- Andrew D. Hwang, College of the Holy Cross

#### Table of Contents

# Table of Contents

## Generalized Ricci Flow

- Cover Cover11
- Title page iii4
- Chapter 1. Introduction 18
- Chapter 2. Generalized Riemannian Geometry 714
- Chapter 3. Generalized Connections and Curvature 3138
- Chapter 4. Fundamentals of Generalized Ricci Flow 6572
- Chapter 5. Local Existence and Regularity 8592
- Chapter 6. Energy and Entropy Functionals 109116
- Chapter 7. Generalized Complex Geometry 127134
- Chapter 8. Canonical Metrics in Generalized Complex Geometry 165172
- Chapter 9. Generalized Ricci Flow in Complex Geometry 183190
- 9.1. Kähler-Ricci flow 183190
- 9.2. Pluriclosed flow 185192
- 9.3. Generalized Kähler-Ricci flow 190197
- 9.4. Reduced flows 196203
- 9.5. Torsion potential evolution equations 200207
- 9.6. Higher regularity from uniform parabolicity 202209
- 9.7. Metric evolution equations 207214
- 9.8. Sharp existence and convergence results 210217

- Chapter 10. T-duality 219226
- Bibliography 241248
- Back Cover Back Cover1257