Abstract
We consider a parametrized family of compact G2-calibrated solvmanifolds, and construct associative (so volume-minimizing submanifolds) 3-tori with respect to the closed G2-structure. We also study the Laplacian flow of this closed G2 form on the solvable Lie group underlying to each of these solvmanifolds, and show long time existence of the solution.
Highlights
Open AccessA G2-structure on a seven-dimensional manifold M is defined by a positive3-form φ on M, which induces a Riemannian metric gφ and a volume form dVφ on M such that gφ ( X,Y ) dVφ= ιXφ ∧ ιYφ φ, (1)for any vector fields X, Y on M
If the 3-form φ is covariantly constant with respect to the Levi-Civita connection of the metric gφ or, equivalently, the 3-form φ is closed and coclosed [1], the holonomy group of gφ is a subgroup of the exceptional Lie group G2, and the metric gφ is Ricci-flat
For the metric determined by the invariant closed G2 form φk on M 7 (k ) mentioned before, we show that if H (k ) is the connected solvable Lie group underlying to M 7 (k ), φk induces a solsoliton on H (k )
Summary
3-form φ (the G2 form) on M, which induces a Riemannian metric gφ and a volume form dVφ on M such that gφ ( X ,Y ) dVφ=. The first example of a compact G2-calibrated manifold, which does not admit any torsion-free G2-structure, was obtained in [8]. For the metric determined by the invariant closed G2 form φk on M 7 (k ) mentioned before, we show that if H (k ) is the connected solvable (non-nilpotent) Lie group underlying to M 7 (k ) , φk induces a solsoliton on H (k ) Laplacian operator associated with the metric gφ(t) induced by the 3-form φ (t ) This geometric flow was introduced by Bryant in [16] as a tool to find torsion-free G2-structures on compact manifolds. Of H (k ) .) We show that the Ricci endomorphism Ric ( gk (t )) of the underlying metric gk (t ) of φk (t ) is independent of the time t, and so the solution φk (t ) does not converge to a torsion-free G2-structure as t goes to infinity
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