Some results on calibrated submanifolds in Euclidean space of cohomogeneity one and two

Abstract

We construct calibrated submanifolds in Euclidean space invariant under the action of a Lie group $G$. We first demonstrate the method used in this paper by reproducing the results about special Lagrangians due to Harvey-Lawson. We then show explicitly that an associative submanifold in $\mathbb{R}^7$ invariant under the action of a maximal torus $\mathbb{T}^2 \subset \mathrm{G}_2$ has to be a special Lagrangian submanifold in $\mathbb{C}^3$. Similarly, we also show that a Cayley submanifold in $\mathbb{R}^8$ invariant under the action of a maximal torus $\mathbb{T}^3 \subset \mathrm{Spin}(7)$ has to be a special Lagrangian submanifold in $\mathbb{C}^4$. We construct coassociative submanifolds in $\mathbb{R}^7$ invariant under the action of $\mathrm{Sp}(1)\subset \mathbb{H}$ with a more general ansatz than the one in Harvey-Lawson but we recover exactly the $\mathrm{Sp}(1)$-invariant coassociatives in Harvey-Lawson, giving us a rigidity result. Finally, we construct cohomogeneity two examples of coassociative submanifolds in $\mathbb{R}^7$ which are invariant under the action of a maximal torus $\mathbb{T}^2 \subset \mathrm{G}_2$.