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March 2019 Stability of torsion-free $\mathrm{G}_2$ structures along the Laplacian flow
Jason D. Lotay, Yong Wei
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J. Differential Geom. 111(3): 495-526 (March 2019). DOI: 10.4310/jdg/1552442608

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.

Funding Statement

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

Citation

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Jason D. Lotay. Yong Wei. "Stability of torsion-free $\mathrm{G}_2$ structures along the Laplacian flow." J. Differential Geom. 111 (3) 495 - 526, March 2019. https://doi.org/10.4310/jdg/1552442608

Information

Received: 22 January 2016; Published: March 2019
First available in Project Euclid: 13 March 2019

zbMATH: 07036514
MathSciNet: MR3934598
Digital Object Identifier: 10.4310/jdg/1552442608

Subjects:
Primary: 53C10, 53C25, 53C44

Rights: Copyright © 2019 Lehigh University

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Vol.111 • No. 3 • March 2019
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