Lectures and Surveys on G2-Manifolds and Related Topics

1. Front Matter

   Pages i-xxii

   PDF

2. Lectures

   1. Front Matter

      Pages 1-1

      PDF

   2. Introduction to \(\mathrm {G}_2\) Geometry

      • Spiro Karigiannis

      Pages 3-50

   3. Constructions of Compact \(G_2\)-Holonomy Manifolds

      • Alexei Kovalev

      Pages 51-67

   4. Calibrated Submanifolds

      • Jason D. Lotay

      Pages 69-101

   5. Calibrated Submanifolds in \(G_{2}\) Geometry

      • Ki Fung Chan, Naichung Conan Leung

      Pages 103-111

   6. Geometric Flows of \({{\,\mathrm{G\!}\,}}_2\) Structures

      • Jason D. Lotay

      Pages 113-140

3. Surveys

   1. Front Matter

      Pages 141-141

      PDF

   2. Distinguishing \(G_2\)-Manifolds

      • Diarmuid Crowley, Sebastian Goette, Johannes Nordström

      Pages 143-172

   3. Gravitational Instantons and Degenerations of Ricci-flat Metrics on the K3 Surface

      • Lorenzo Foscolo

      Pages 173-199

   4. Frölicher–Nijenhuis Bracket on Manifolds with Special Holonomy

      • Kotaro Kawai, Hông Vân Lê, Lorenz Schwachhöfer

      Pages 201-215

   5. Distinguished \(G_2\)-Structures on Solvmanifolds

      • Jorge Lauret

      Pages 217-233

   6. On G\(_{\mathbf 2}\)-Structures, Special Metrics and Related Flows

      • Marisa Fernández, Anna Fino, Alberto Raffero

      Pages 235-251

   7. Laplacian Flow for Closed \(\mathrm{G}_2\) Structures

      • Yong Wei

      Pages 253-269

   8. Flows of Co-closed \(G_{2}\)-Structures

      • Sergey Grigorian

      Pages 271-286

   9. \(\mathrm{G}_2\)-Instantons on Noncompact \(\mathrm{G}_2\)-Manifolds: Results and Open Problems

      • Jason D. Lotay, Goncalo Oliveira

      Pages 287-317

   10. Current Progress on \(\mathrm{{G}}_2\)-Instantons over Twisted Connected Sums

       • Henrique Sá Earp

       Pages 319-348

   11. Complex and Calibrated Geometry

       • Kim Moore

       Pages 349-364

   12. Deformations of Calibrated Submanifolds with Boundary

       • Alexei Kovalev

       Pages 365-382

This book, one of the first on G₂ manifolds in decades, collects introductory lectures and survey articles largely based on talks given at a workshop held at the Fields Institute in August 2017, as part of the major thematic program on geometric analysis. It provides an accessible introduction to various aspects of the geometry of G₂ manifolds, including the construction of examples, as well as the intimate relations with calibrated geometry, Yang-Mills gauge theory, and geometric flows. It also features the inclusion of a survey on the new topological and analytic invariants of G₂ manifolds that have been recently discovered. 

The first half of the book, consisting of several introductory lectures, is aimed at experienced graduate students or early career researchers in geometry and topology who wish to familiarize themselves with this burgeoning field. The second half, consisting of numerous survey articles, is intended to be useful to both beginners and experts in the field.


             
 
                              

• minimal submanifolds
• special Lagrangians
• Cayley submanifolds
• Riemannian holonomy
• Calabi-Yau manifolds
• normed division algebras
• geometric flows
• torsion forms
• hyperkahler metrics
• Dolbeault cohomologies
• Frolicher-Nijenhuis bracket
• solitons
• Einstein metric
• Ricci soliton
• Laplacian flow

• Department of Pure Mathematics, University of Waterloo, Waterloo, Canada

  Spiro Karigiannis

• Institute of Mathematical Sciences, Chinese University of Hong Kong, Shatin, Hong Kong

  Naichung Conan Leung

• Mathematical Institute, University of Oxford, Oxford, UK

  Jason D. Lotay

Policies and ethics