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$S^1$-invariant Laplacian flow

Udhav Fowdar

Abstract

The Laplacian flow is a geometric flow introduced by Bryant as a way for finding torsion free $G_2$-structures. If the flow is $S^1$-invariant then it descends to a flow of $SU(3)$-structures on a $6$-manifold. In this article we derive expressions for these evolution equations. In our search for examples we discover the first inhomogeneous shrinking solitons, which are also gradient. We also show that any compact non-torsion free soliton admits no infinitesimal symmetry.

$S^1$-invariant Laplacian flow

Abstract

The Laplacian flow is a geometric flow introduced by Bryant as a way for finding torsion free -structures. If the flow is -invariant then it descends to a flow of -structures on a -manifold. In this article we derive expressions for these evolution equations. In our search for examples we discover the first inhomogeneous shrinking solitons, which are also gradient. We also show that any compact non-torsion free soliton admits no infinitesimal symmetry.

Paper Structure

This paper contains 17 sections, 19 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Overview.
  3. Motivation.
  4. Outline.
  5. Preliminaries
  6. $SU(3)$-structures
  7. $G_2$-structures
  8. The Laplacian flow
  9. $S^1$ reduction of closed $G_2$-structure
  10. The $S^1$-invariant Laplacian flow
  11. $S^1$-invariant flow equations
  12. The $S^1$-invariant soliton equation
  13. Non-existence of compact solitons with continuous symmetry
  14. The $S^1$-invariant soliton equation
  15. Examples of $S^1$-invariant Laplacian flow
  16. ...and 2 more sections

Key Result

Lemma 2.2

Given a $1$-form $\alpha\in\Lambda^1_6$, let $\beta:=*_\omega(\alpha \wedge \Omega^-)\in \Lambda^2_6$. Then the following hold:

Theorems & Definitions (45)

  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 3.3: Gibbons-Hawking Ansatz for closed $G_2$-structures
  • proof
  • ...and 35 more