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 in Harvey-Lawson (Harvey, R. and Blaine Lawson, H.: Acta Math. 148, 47–157 1982). We then show explicitly that an associative submanifold in \(\mathbb {R}^7\) invariant under the action of a maximal torus \(\mathbb {T}^2 \subset \textrm{G}_2\) 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 \(\mathbb {T}^3 \subset \text {Spin}(7)\) has to be a special Lagrangian submanifold in \(\mathbb {C}^4\). We construct coassociative submanifolds in \(\mathbb {R}^7\) invariant under the action of \(\textrm{Sp}(1)\subset \mathbb {H}\) with a more general ansatz than the one in (Harvey, R. and Blaine Lawson, H.: Acta Math. 148, 47–157 1982) but we recover exactly the \(\textrm{Sp}(1)\)-invariant coassociatives in (Harvey, R. and Blaine Lawson, H.: Acta Math. 148, 47–157 1982), 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 \(\mathbb {T}^2 \subset \textrm{G}_2\).