Some results on calibrated submanifolds in Euclidean space of cohomogeneity one and two
Faisal Romshoo
TL;DR
The work develops a symmetry-based framework for constructing calibrated submanifolds in Euclidean spaces by exploiting Lie group actions that preserve calibrations. It reproduces Harvey-Lawson's special Lagrangian results and derives new symmetric instances across SL, associative, coassociative, and Cayley calibrations, including cohomogeneity-one and cohomogeneity-two constructions with maximal tori and Sp(1) actions in $\mathbb{R}^7$ and $\mathbb{R}^8$. A key theme is reducing high-dimensional calibrated conditions to lower-dimensional, tractable equations on a profile $\alpha$ (or complex coordinates $z_i$) under symmetry, yielding rigidity results and explicit families (e.g., $\,\mathbb{T}^2$-invariant coassociatives and associative reductions to SL in $\mathbb{C}^3$). The results illuminate how symmetry constrains calibrated geometry, provide concrete examples, and point to directions for global/global-topological extensions and broader symmetry classes. Overall, the paper advances constructive techniques for symmetric calibrations and deepens connections between different calibrated geometries via symmetry reductions.
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 $\mathbb{T}^2 \subset \mathrm{G}_2$.