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 $\mathbb{T}^3 \subset \mathrm{Spin}(7)$ 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 $\mathbb{T}^2 \subset \mathrm{G}_2$.

💡 Research Summary

This paper develops a systematic method for constructing calibrated submanifolds in Euclidean spaces that are invariant under the action of a Lie group G. The author first reviews the basic notions of calibrations, special Lagrangian (SL) submanifolds in ℂⁿ, associative and coassociative submanifolds in ℝ⁷, and Cayley submanifolds in ℝ⁸, recalling the classic results of Harvey‑Lawson (HL) and setting up the notation used throughout.

The core technique, described in Section 3, proceeds as follows. One selects a group G that preserves the relevant calibration form. The orbits of G are parametrised by angular variables θ₁,…,θ_k. A “profile” map α(t₁,…,t_l) (a curve for cohomogeneity‑one cases, a surface for cohomogeneity‑two) is chosen, and the full immersed submanifold is defined by
 F(t,θ)=A_θ·α(t),
where A_θ∈G. The tangent vectors V_i=∂F/∂θ_i and V_j=∂F/∂t_j are computed; immersion requires these to be linearly independent, which translates into algebraic conditions on α. Imposing the calibration condition (e.g. ω|_F=0 and Im Ω|_F=0 for SL, φ|_F=0 for associatives, ψ|_F=0 for coassociatives, Φ|_F=0 for Cayleys) yields a system of first‑order ODEs or PDEs for the components of α. Solving these equations produces explicit G‑invariant calibrated submanifolds.

Sections 4 and 5 apply the method to the classic T^{n‑1}‑invariant and SO(n)‑invariant special Lagrangian n‑folds, reproducing the Harvey‑Lawson families. The T^{n‑1} case leads to the familiar constraints
 |α_k|²−|α_n|² = c_k (k=1,…,n‑1)
and a constant real or imaginary part of the product ∏α_k, depending on the parity of n. The SO(n) case yields an analogous condition expressed in terms of the norms of the real and imaginary parts of α.

Section 6 treats associative 3‑folds in ℝ⁷ that are invariant under the maximal torus T²⊂G₂. By writing the profile as a two‑parameter surface and enforcing φ|_F=0, the author shows that the resulting equations are precisely those characterising special Lagrangian 3‑folds in ℂ³. Consequently, any T²‑invariant associative must be a special Lagrangian submanifold of ℂ³ (Theorem 6.1).

Section 7 performs the analogous analysis for Cayley 4‑folds in ℝ⁸ with T³⊂Spin(7) symmetry. The calibration Φ splits as Re Ω + ½ ω² on ℂ⁴, and the T³‑invariance forces the Cayley condition to reduce to the special Lagrangian condition on ℂ⁴. Hence every T³‑invariant Cayley is a special Lagrangian 4‑fold in ℂ⁴ (Theorem 7.1).

Section 8 investigates Sp(1)⊂ℍ‑invariant coassociative 4‑folds. The author adopts a more general Ansatz than in HL, allowing four independent real functions in the profile. After imposing ψ|_F=0, the resulting ODE system collapses to exactly the equations found by Harvey‑Lawson for Sp(1)‑invariant coassociatives. This yields a rigidity theorem: despite the broader Ansatz, no new Sp(1)‑invariant coassociatives exist beyond those already known (Theorem 8.1).

Section 9 presents the paper’s most novel contribution: cohomogeneity‑two coassociative 4‑folds in ℝ⁷ that are invariant under the maximal torus T²⊂G₂. Here the profile α(s,t) depends on two variables, and the calibration conditions ψ|_F=0 and φ|_F=0 give a coupled nonlinear PDE system. The author solves this system explicitly, obtaining families expressed in terms of trigonometric functions and polynomial relations. These examples are new; they are not reducible to the previously known T¹‑invariant families and illustrate that higher cohomogeneity can produce genuinely new calibrated geometries.

The concluding remarks emphasise that symmetry methods not only recover known calibrated families but also provide a powerful framework for discovering new examples, especially in higher cohomogeneity settings. The paper suggests future directions such as exploring non‑compact or non‑abelian symmetry groups, extending the technique to manifolds with non‑flat metrics, and investigating connections with special holonomy and gauge theory.

Overall, the work offers a clear, unified approach to constructing and classifying calibrated submanifolds with Lie‑group symmetries, establishes rigidity results for certain invariant families, and expands the catalogue of explicit examples by introducing cohomogeneity‑two coassociatives.