Abstract. We develop foundational theory for the Laplacian flow for closed G 2 structures which will be essential for future study. (1). We prove Shi-type derivative estimates for the Riemann curvature tensor Rm and torsion tensor T along the flow, i.e. that a bound onwill imply bounds on all covariant derivatives of Rm and T . (2). We show that Λ(x, t) will blow up at a finite-time singularity, so the flow will exist as long as Λ(x, t) remains bounded. (3). We give a new proof of forward uniqueness and prove backward uniqueness of the flow, and give some applications. (4). We prove a compactness theorem for the flow and use it to strengthen our long time existence result from (2) to show that the flow will exist as long as the velocity of the flow remains bounded. (5). Finally, we study soliton solutions of the Laplacian flow.

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We prove that torsion-free G2 structures are (weakly) dynamically stable along the Laplacian flow for closed G2 structures. More precisely, given a torsion-free G2 structureφ on a compact 7-manifold M , the Laplacian flow with initial value in [φ], sufficiently close toφ, will converge to a point in the Diff 0 (M )-orbit ofφ. We deduce, from fundamental work of Joyce [18], that the Laplacian flow starting at any closed G2 structure with sufficiently small torsion will exist for all time and converge to a torsion-free G2 structure.2010 Mathematics Subject Classification. 53C44, 53C25, 53C10.

This article sets out to serve a dual purpose. On the one hand, we give an explicit description of the Lagrangians in the nearly Kähler 6-sphere S 6 which are ruled by circles of constant radius using 'Weierstrass formulae'. On the other, we recognise all known examples of these Lagrangians as being ruled by such circles. Therefore, we describe all families of Lagrangians in S 6 whose second fundamental form satisfies natural pointwise conditions: so-called 'second order families'.2000 Mathematics Subject Classification: 53B20, 53B25.Remarks The almost complex structure J on S 6 is not integrable. Moreover, the 2-form ω is clearly not closed, but it does satisfy ω ∧ dω = 0.Having defined the nearly Kähler structure on the 6-sphere, we can present the class of submanifolds we wish to study.Here we have generalised the notion of Lagrangian submanifold, usually reserved for symplectic manifolds, to the almost symplectic 6-sphere. However, since S 6 has a nearly Kähler structure, Lagrangians in S 6 have more properties than one would expect from the general almost symplectic case.

Suppose that N is a compact coassociative 4-fold with a conical singularity in a 7-manifold M , with a G 2 structure given by a closed 3-form. We construct a smooth family, {N ′ (t) : t ∈ (0, τ )} for some τ > 0, of compact, nonsingular, coassociative 4-folds in M which converge to N in the sense of currents, in geometric measure theory, as t → 0. This realisation of desingularizations of N is achieved by gluing in an asymptotically conical coassociative 4-fold in R 7 , dilated by t, then deforming the resulting compact 4-dimensional submanifold of M to the required coassociative 4-fold.