Abstract
We consider the Laplacian flow of locally conformal calibrated G 2 -structures as a natural extension to these structures of the well-known Laplacian flow of calibrated G 2 -structures. We study the Laplacian flow for two explicit examples of locally conformal calibrated G 2 manifolds and, in both cases, we obtain a flow of locally conformal calibrated G 2 -structures, which are ancient solutions, that is they are defined on a time interval of the form ( − ∞ , T ) , where T > 0 is a real number. Moreover, for each of these examples, we prove that the underlying metrics g ( t ) of the solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric as t goes to − ∞ , and they blow-up at a finite-time singularity.
Highlights
A G2 -structure on a 7-manifold M can be characterized by the existence of a globally defined3-form φ on M, which can be written at each point as φ = e127 + e347 + e567 + e135 − e146 − e236 − e245, (1)with respect to some local coframe {e1, . . . , e7 } on M
A G2 -structure φ induces a Riemannian metric g φ and a volume form dVg φ on M given by g φ ( X, Y ) dVg φ =
If the 3-form φ is closed and coclosed, the holonomy group of g φ is a subgroup of the exceptional Lie group G2 [2], and the metric g φ is Ricci-flat [3]
Summary
A G2 -structure on a 7-manifold M can be characterized by the existence of a globally defined. = ∆ t φ ( t ), d φ(t) = 0, φ(0) = φ, where φ(t) is a closed G2 form on M, and ∆t = d d∗ + d∗ d is the Hodge Laplacian operator associated with the metric g(t) = g φ(t) induced by the 3-form φ(t) This flow was introduced by Bryant in [27] as a tool to find torsion-free G2 -structures on compact manifolds. Lie groups K and S with a locally conformal calibrated G2 -structure, we show that the solution of the before Laplacian flow is ancient, that is it is defined on a time interval of the form (−∞, T ), where T > 0 is a real number. We prove that the solution φ(t) of the flow on S induces an Einstein metric for all time t where φ(t) is defined
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
