## Journal of Differential Geometry

- J. Differential Geom.
- Volume 111, Number 3 (2019), 495-526.

### Stability of torsion-free $\mathrm{G}_2$ structures along the Laplacian flow

#### Abstract

We prove that torsion-free $\mathrm{G}_2$ structures are (weakly) dynamically stable along the Laplacian flow for closed $\mathrm{G}_2$ structures. More precisely, given a torsion-free $\mathrm{G}_2$ structure $\overline{\varphi}$ on a compact $7$-manifold $M$, the Laplacian flow with initial value in $[\overline{\varphi}]$, sufficiently close to $\overline{\varphi}$, will converge to a point in the $\mathrm{Diff}^0 (M)$-orbit of $\overline{\varphi}$. We deduce, from fundamental work of Joyce, that the Laplacian flow starting at any closed $\mathrm{G}_2$ structure with sufficiently small torsion will exist for all time and converge to a torsion-free $\mathrm{G}_2$ structure.

#### Note

This research was supported by EPSRC grant EP/K010980/1.

#### Article information

**Source**

J. Differential Geom., Volume 111, Number 3 (2019), 495-526.

**Dates**

Received: 22 January 2016

First available in Project Euclid: 13 March 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.jdg/1552442608

**Digital Object Identifier**

doi:10.4310/jdg/1552442608

**Mathematical Reviews number (MathSciNet)**

MR3934598

**Subjects**

Primary: 53C10: $G$-structures 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)

**Keywords**

Laplacian flow $\mathrm{G}_2$ structure torsion-free stability

#### Citation

Lotay, Jason D.; Wei, Yong. Stability of torsion-free $\mathrm{G}_2$ structures along the Laplacian flow. J. Differential Geom. 111 (2019), no. 3, 495--526. doi:10.4310/jdg/1552442608. https://projecteuclid.org/euclid.jdg/1552442608