Laplacian Solitons and Symmetry in G_2-geometry

Details Laplacian Solitons and Symmetry in G_2-geometry Laplacian Solitons and Symmetry in G_2-geometry

2012-01-12

arxiv.org/abs/1201.2627v1

Mathematics

Differential Geometry

Scientific paper

In this paper, it is shown that (with no additional assumptions) on a compact 7-dimensional manifold which admits a $G_2$-structure soliton solutions to the Laplacian flow of R. Bryant can only be shrinking or steady. We also show that the space of symmetries (vector fields that annihilate via the Lie derivative) of a torsion-free $G_2$-structure on a compact 7-manifold is canonically isomorphic to $H^1(M,\mathbb{R})$. Some comparisons with Ricci solitons are also discussed, along with some future directions of exploration.

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