Journal of Differential Geometry

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

Jason D. Lotay and Yong Wei

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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


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