Some associative submanifolds of the squashed 7-sphere

Abstract

The squashed 7-sphere $S^7$ is a 7-sphere with an Einstein metric given by the canonical variation and its cone ${\mathbb{R}}^8-\{0\}$ has full holonomy $\mathrm{Spin}(7)$. There is a canonical calibrating 4-form $\Phi$ on ${\mathbb{R}}^8-\{0\}$. A minimal 3-submanifold in $S^7$ is called associative if its cone is calibrated by $\Phi$. In this paper, we classify two types of fundamental associative submanifolds in the squashed $S^7$. One is obtained by the intersection with a 4-plane and the other is homogeneous. Then we study their infinitesimal associative deformations and explicitly show that all of them are integrable.

31 pages, v2: minor corrections

Cited by in corpus (5)

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• McLean's second variation formula revisited
• Harmonic $Sp(2)$-invariant $G_2$-structures on the $7$-sphere
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• Associative submanifolds in Joyce's generalised Kummer constructions