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

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 ${\mathbb{T}}^3\subset \mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(7)$ has to be a special Lagrangian submanifold in ${\mathbb{C}}^4$. We construct coassociative submanifolds in ${\mathbb{R}}^7$ invariant under the action of $\mathrm{S}\mathrm{p}(1)\subset \mathbb{H}$ with a more general ansatz than the one in Harvey-Lawson but we recover exactly the $\mathrm{S}\mathrm{p}(1)$-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 ${\mathbb{T}}^2\subset {\mathrm{G}}_2$.