Some results on calibrated submanifolds in Euclidean space of cohomogeneity one and two
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917270500933632 |
|---|---|
| author | Romshoo, Faisal |
| author_facet | Romshoo, Faisal |
| contents | 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 $\mathbb{T}^2 \subset \mathrm{G}_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_01039 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some results on calibrated submanifolds in Euclidean space of cohomogeneity one and two Romshoo, Faisal Differential Geometry 53C38 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 $\mathbb{T}^2 \subset \mathrm{G}_2$. |
| title | Some results on calibrated submanifolds in Euclidean space of cohomogeneity one and two |
| topic | Differential Geometry 53C38 |
| url | https://arxiv.org/abs/2508.01039 |