Closed warped G_2-structures evolving under the Laplacian flow

Abstract

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...

Laplacian coflow for warped G2-structures

“…Remark 2.1. If we consider the pair pα, βq " p1, 0q, this definition of warped G 2 -structure is exactly the one already given in [11].…”

“…The metrics g ω,ψ˘a nd g ϕ on the base manifold M 6 and the warped product M 6ˆf S 1 respectively define two star operators˚6 and˚7 related by the following: [11]). Let η P Ω k pM 6 q be a differential k-form on M 6 , and let˚6 and˚7 be the Hodge star operator determined by the SUp3q-structure and the warped G 2 -structure, respectively.…”

“…Recent papers by Lotay and Wei [17,18,19] derived important properties of the Laplacian flow as long time existence or convergence results. Even more recently Fino and Raffero on [11] obtained sufficient conditions for the existence of solution of this flow on warped products of the form M 6ˆf S 1 with M 6 a 6-dimensional manifold endowed with an SUp3q-structure. Recall that, if pB, g B q and pF, g F q are Riemannian manifolds and f is a non-vanishing differentiable function on B, then the warped product W " Bˆf F consists on the product manifold BˆF endowed with the metric g " π1 pg B q`f 2 π2 pg F q where π 1 and π 2 are the projections of W onto B and F respectively.…”

“…In Section 3 we reinterpret the Laplacian flow and coflow of a G 2 -structure as a set of evolution equations of the SUp3q-structure and we describe the Laplacian coflow operator of the warped G 2 -structure by means of the torsion forms of the SUp3qstructure and the warping function. In particular for the Laplacian flow we reobtain the equations due to Fino and Raffero in [11]. Finally the goal of Section 4 is to obtain new examples of long time solutions of the Laplacian coflow constructed as warped products where the base manifolds are 6-dimensional and they are endowed with nearly Kähler, symplectic half-flat or balanced SU(3)-structures.…”

Cohomogeneity‐one G2‐Laplacian flow on the 7‐torus

“…For the hypersymplectic flow on a compact 4‐manifold, the long‐time existence assuming bounded torsion was obtained in . There is also an interesting reduction of a warped ‐Laplacian flow on to a coupled flow of the ‐structure and the warped function on in .…”