In 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 TxM, φ|ξ = λ volξ with λ ≤ 1. Here volξ is the volume form of ξ with respect to g.
Set Gx(φ) = { ξ as above : φ|ξ = volξ }. (In order for the theory to be nontrivial, we need Gx(φ) to be nonempty.) Let G(φ) be the union of Gx(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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de Rham, Georges (1957–1958), On the Area of Complex Manifolds. Notes for the Seminar on Several Complex Variables, Institute for Advanced Study, Princeton, New Jersey.
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Kraines, Vivian Yoh (1965), "Topology of quaternionic manifolds", Bull. Amer. Math. Soc., 71,3, 1: 526–527, doi:10.1090/s0002-9904-1965-11316-7.
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Morgan, Frank (1988), "Area-minimizing surfaces, faces of Grassmannians, and calibrations", Amer. Math. Monthly, 95 (9): 813–822, doi:10.2307/2322896, JSTOR 2322896.
Morgan, Frank (1990), "Calibrations and new singularities in area-minimizing surfaces: a survey In "Variational Methods" (Proc. Conf. Paris, June 1988), (H. Berestycki J.-M. Coron, and I. Ekeland, Eds.)", Prog. Nonlinear Diff. Eqns. Applns, 4: 329–342.
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Wirtinger, W. (1936), "Eine Determinantenidentität und ihre Anwendung auf analytische Gebilde und Hermitesche Massbestimmung", Monatshefte für Mathematik und Physik, 44: 343–365 (§6.5), doi:10.1007/BF01699328.
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