Calibrated geometry

The mathematical field of differential geometry, a calibrated manifold is a Riemannian manifold (M,g) of dimension n equipped with a differential p-form φ (for some 0 ≤ p ≤ n) which is a calibration, meaning that:

• φ is closed: dφ = 0, where d is the exterior derivative
• for any x ∈ M and any oriented p-dimensional subspace ξ of T_xM, φ|_ξ = λ vol_ξ with λ ≤ 1. Here vol_ξ is the volume form of ξ with respect to g.

Set G_x(φ) = { ξ as above : φ|_ξ = vol_ξ }. (In order for the theory to be nontrivial, we need G_x(φ) to be nonempty.) Let G(φ) be the union of G_x(φ) for x in M.

The theory of calibrations is due to R. Harvey and B. Lawson and others. Much earlier (in 1966) Edmond Bonan introduced G₂-manifold and Spin(7)-manifold, constructed all the parallel forms and showed that those manifolds were Ricci-flat. Quaternion-Kähler manifold were simultaneously studied in 1967 by Edmond Bonan and Vivian Yoh Kraines and they constructed the parallel 4-form.

Calibrated submanifolds

A p-dimensional submanifold Σ of M is said to be a calibrated submanifold with respect to φ (or simply φ-calibrated) if TΣ lies in G(φ).

A famous one line argument shows that calibrated p-submanifolds minimize volume within their homology class. Indeed, suppose that Σ is calibrated, and Σ ′ is a p submanifold in the same homology class. Then

${\displaystyle \int _{\Sigma }\mathrm {vol} _{\Sigma }=\int _{\Sigma }\varphi =\int _{\Sigma '}\varphi \leq \int _{\Sigma '}\mathrm {vol} _{\Sigma '}}$

where the first equality holds because Σ is calibrated, the second equality is Stokes' theorem (as φ is closed), and the inequality holds because φ is a calibration.

Examples

• On a Kähler manifold, suitably normalized powers of the Kähler form are calibrations, and the calibrated submanifolds are the complex submanifolds. This follows from the Wirtinger inequality.
• On a Calabi–Yau manifold, the real part of a holomorphic volume form (suitably normalized) is a calibration, and the calibrated submanifolds are special Lagrangian submanifolds.
• On a G₂-manifold, both the 3-form and the Hodge dual 4-form define calibrations. The corresponding calibrated submanifolds are called associative and coassociative submanifolds.
• On a Spin(7)-manifold, the defining 4-form, known as the Cayley form, is a calibration. The corresponding calibrated submanifolds are called Cayley submanifolds.

References

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