We study the behaviour of the Laplacian flow evolving closed G2-structures on warped products of the form M 6 ×S 1 , where the base M 6 is a compact 6-manifold endowed with an SU(3)-structure. In the general case, we reinterpret the flow as a set of evolution equations on M 6 for the differential forms defining the SU(3)-structure and the warping function. When the latter is constant, we find sufficient conditions for the existence of solutions of the corresponding coupled flow. This provides a method to construct immortal solutions of the Laplacian flow on the product manifolds M 6 × S 1 . The application of our results to explicit cases allows us to obtain new examples of expanding Laplacian solitons.2010 Mathematics Subject Classification. 53C44, 53C10, 53C30. Key words and phrases. Laplacian flow, G2-structure, SU(3)-structure, warped product, soliton. The authors were supported by GNSAGA of INdAM. The first author was also supported by PRIN 2015 "Real and complex manifolds: geometry, topology and harmonic analysis" of MIUR. 1 2 ANNA FINO AND ALBERTO RAFFEROconditions. Then, there are various methods to define a G 2 -structure on the product M 6 ×S 1 by means of the SU(3)-structure on M 6 , and in each case this 7-manifold turns out to be a Riemannian product or a warped product (see e.g. [6,8,25]).Recall that a G 2 -structure on a 7-manifold M 7 is defined by a stable 3-form ϕ giving rise to a Riemannian metric g ϕ and to a volume form dV ϕ . The intrinsic torsion of a G 2structure ϕ is completely determined by dϕ and d * ϕ ϕ, * ϕ being the Hodge operator defined by g ϕ and dV ϕ , and it vanishes identically if and only if both ϕ and * ϕ ϕ are closed [3,17]. When this happens, the Riemannian holonomy group Hol(g ϕ ) is a subgroup of G 2 , and g ϕ is Ricci-flat. G 2 -structures whose defining 3-form is both closed and co-closed are called torsion-free, and they play a central role in the construction of metrics with holonomy G 2 . The first complete examples of such metrics were obtained by Bryant and Salamon in [4], while compact examples of Riemannian manifolds with holonomy G 2 were constructed first by Joyce [23], and then by Kovalev [26], and by Corti, Haskins, Nordström, Pacini [12]. A potential method to obtain new results in this direction is represented by geometric flows evolving G 2 -structures.Let M 7 be a 7-manifold endowed with a closed G 2 -structure ϕ, i.e., satisfying dϕ = 0. The Laplacian flow starting from ϕ is the initial value problemwhere ∆ ϕ(t) denotes the Hodge Laplacian of the Riemannian metric g ϕ(t) induced by ϕ(t). This geometric flow was introduced by Bryant in [3] as a tool to find torsion-free G 2structures on compact manifolds. Short-time existence and uniqueness of the solution when M 7 is compact were proved by Bryant and Xu in the unpublished paper [5]. Recently, Lotay and Wei investigated the properties of the Laplacian flow in the series of papers [32,33,34].In [25], a flow evolving the 4-form * ϕ ϕ in the direction of minus its Hodge Laplacian was introduced, and its beha...

We study locally conformal calibrated G 2 -structures whose underlying Riemannian metric is Einstein, showing that in the compact case the scalar curvature cannot be positive. As a consequence, a compact homogeneous 7-manifold cannot admit an invariant Einstein locally conformal calibrated G 2 -structure unless the underlying metric is flat. In contrast to the compact case, we provide a non-compact example of homogeneous manifold endowed with a locally conformal calibrated G 2 -structure whose associated Riemannian metric is Einstein and non Ricci-flat. The homogeneous Einstein metric is a rank-one extension of a Ricci soliton on the 3-dimensional complex Heisenberg group endowed with a left-invariant coupled SU(3)-structure (ω, Ψ), i.e., such that dω = cRe(Ψ), with c ∈ R − {0}. Nilpotent Lie algebras admitting a coupled SU(3)-structure are also classified.2010 Mathematics Subject Classification. 53C10, 53C30, 53C25. Research partially supported by the project PRIN Varietà reali e complesse: geometria, topologia e analisi armonica, the project FIRB Differential Geometry and Geometric functions theory and by GNSAGA (Indam) of Italy.

We study the automorphism group of a compact 7-manifold M endowed with a closed non-parallel G2-structure, showing that its identity component is abelian with dimension bounded by min{6, b2(M )}. This implies the non-existence of compact homogeneous manifolds endowed with an invariant closed non-parallel G2-structure. We also discuss some relevant examples.2010 Mathematics Subject Classification. 53C10.

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We consider 6-manifolds endowed with a symplectic half-flat SU(3)-structure and acted on by a transitive Lie group G of automorphisms. We review a classical result allowing to show the non-existence of compact non-flat examples. In the noncompact setting, we classify such manifolds under the assumption that G is semisimple. Moreover, in each case we describe all invariant symplectic half-flat SU(3)-structures up to isomorphism, showing that the Ricci tensor is always Hermitian with respect to the induced almost complex structure. This last condition is characterized in the general case.2010 Mathematics Subject Classification. 53C30, 53C15, 53D05.