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

TL;DR

This paper constructs and classifies calibrated submanifolds in Euclidean space with symmetry under Lie group actions, revealing their geometric structures and relationships to special Lagrangian submanifolds.

Contribution

It introduces methods for constructing invariant calibrated submanifolds and demonstrates their properties, including new examples and rigidity results for cohomogeneity one and two cases.

Findings

Associative submanifolds invariant under maximal tori are special Lagrangians.

Constructed coassociative submanifolds with symmetry groups, recovering known examples.

Provided new cohomogeneity two coassociative submanifold examples.

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 $\mathbb{T}^2 \subset \mathrm{G}_2$.