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BRIDGES Lectures: Flows of geometric structures, especially $\mathrm{G}_2$-structures

Spiro Karigiannis

TL;DR

This work surveys flows of geometric structures with a central focus on short-time existence and uniqueness. It develops the DeTurck trick for Ricci-type flows and extends the idea to flows of $G_2$-structures, including a general framework that identifies independent second-order invariants that drive these flows. The notes connect foundational results in Riemannian geometry (Hodge theory, parabolic PDE, and symbol calculus) to modern $G_2$-geometry, emphasizing how torsion and curvature control evolution. Collectively, the text motivates and analyzes a broad family of heat-type flows aimed at reaching torsion-free $G_2$-structures, and highlights recent progress by DGK in classifying invariant flows and establishing short-time existence via DeTurck-type methods.

Abstract

The BRIDGES meeting in gauge theory, extremal structures, and stability was held June 2024 at l'Institut d'Études Scientifiques de Cargèse in Corsica, organized by Daniele Faenzi, Eveline Legendre, Eric Loubeau, and Henrique Sá Earp. The first week was a summer school consisting of four independent but related lecture series by Oscar García Prada, Spiro Karigiannis, Laurent Manivel, and Ruxandra Moraru. The present document consists of notes for the lecture series by Spiro Karigiannis on "Flows of geometric structures, especially $\mathrm{G}_2$-structures". Some assistance in the preparation of these notes by the author was provided by several participants of the summer school. See the Comments field for more information. The main theme is short time existence (STE) and uniqueness for geometric flows. We first introduce geometric structures on manifolds and geometric flows of such structures. We discuss some qualitative features of geometric flows, and consider the notions of strong and weak parabolicity. We focus on the Ricci flow, explaining carefully the DeTurck trick to establish short-time existence and uniqueness, an argument which we then extend to a general class of geometric flows of Riemannian metrics, previewing similar ideas for flows of $\mathrm{G}_2$-structures. Finally, we consider geometric flows of $\mathrm{G}_2$-structures. We review the basics of $\mathrm{G}_2$-geometry and survey several different geometric flows of $\mathrm{G}_2$-structures. In particular, we clarify in what sense STE results for the $\mathrm{G}_2$ Laplacian flow differ from STE results for other geometric flows. We conclude with a summary of some recent results by the author with Dwivedi and Gianniotis, including a classification of all possible heat-type flows of $\mathrm{G}_2$-structures, and a sufficient condition for such a flow to admit STE and uniqueness by a modified DeTurck trick.

BRIDGES Lectures: Flows of geometric structures, especially $\mathrm{G}_2$-structures

TL;DR

This work surveys flows of geometric structures with a central focus on short-time existence and uniqueness. It develops the DeTurck trick for Ricci-type flows and extends the idea to flows of -structures, including a general framework that identifies independent second-order invariants that drive these flows. The notes connect foundational results in Riemannian geometry (Hodge theory, parabolic PDE, and symbol calculus) to modern -geometry, emphasizing how torsion and curvature control evolution. Collectively, the text motivates and analyzes a broad family of heat-type flows aimed at reaching torsion-free -structures, and highlights recent progress by DGK in classifying invariant flows and establishing short-time existence via DeTurck-type methods.

Abstract

The BRIDGES meeting in gauge theory, extremal structures, and stability was held June 2024 at l'Institut d'Études Scientifiques de Cargèse in Corsica, organized by Daniele Faenzi, Eveline Legendre, Eric Loubeau, and Henrique Sá Earp. The first week was a summer school consisting of four independent but related lecture series by Oscar García Prada, Spiro Karigiannis, Laurent Manivel, and Ruxandra Moraru. The present document consists of notes for the lecture series by Spiro Karigiannis on "Flows of geometric structures, especially -structures". Some assistance in the preparation of these notes by the author was provided by several participants of the summer school. See the Comments field for more information. The main theme is short time existence (STE) and uniqueness for geometric flows. We first introduce geometric structures on manifolds and geometric flows of such structures. We discuss some qualitative features of geometric flows, and consider the notions of strong and weak parabolicity. We focus on the Ricci flow, explaining carefully the DeTurck trick to establish short-time existence and uniqueness, an argument which we then extend to a general class of geometric flows of Riemannian metrics, previewing similar ideas for flows of -structures. Finally, we consider geometric flows of -structures. We review the basics of -geometry and survey several different geometric flows of -structures. In particular, we clarify in what sense STE results for the Laplacian flow differ from STE results for other geometric flows. We conclude with a summary of some recent results by the author with Dwivedi and Gianniotis, including a classification of all possible heat-type flows of -structures, and a sufficient condition for such a flow to admit STE and uniqueness by a modified DeTurck trick.

Paper Structure

This paper contains 24 sections, 15 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Geometric structures and geometric flows
  3. Geometric Structures
  4. Geometric Flows
  5. Some qualitative discussion of flows
  6. The "heat equation" on Riemannian manifolds
  7. Principal symbols of linear differential operators
  8. Strongly parabolic flows
  9. The harmonic map heat flow
  10. Exercises
  11. The Ricci flow
  12. Motivation for the Ricci flow
  13. Weak parabolicity and diffeomorphism invariance
  14. The DeTurck trick and short time existence for the Ricci flow
  15. Some known analytic results on the Ricci flow
  16. ...and 9 more sections

Key Result

Theorem 2.4.4

Let $\mathcal{H}^k = \ker (\Delta_d|_{\Omega^k})$ be the vector space of harmonic $k$-forms on $M$. There is an $L^2$-orthogonal decomposition Moreover, we have It follows that every de Rham cohomology class in $H^k (M, \mathbb{R})$ has a unique harmonic representative and that there is an isomorphism $\mathcal{H}^k \cong H^k (M, \mathbb{R})$ given by $\alpha \mapsto [\alpha]$.

Theorems & Definitions (65)

  • Example 2.1.1
  • Example 2.1.2
  • Example 2.1.3
  • Example 2.1.4
  • Example 2.1.5
  • Example 2.1.6
  • Remark 2.1.7
  • Example 2.4.1
  • Example 2.4.2
  • Example 2.4.3
  • ...and 55 more