Calibrated geometry

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 in the sense 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 1965 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

$\int_\Sigma \mathrm{vol}_\Sigma = \int_\Sigma \phi = \int_{\Sigma'} \phi \leq \int_{\Sigma'} \mathrm{vol}_{\Sigma'}$

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

• Bonan, Edmond (1965), "Structure presque quaternale sur une variété différentiable", C. R. Acad. Sci. Paris 261: 5445–5448.
• Bonan, Edmond (1966), "Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)", C. R. Acad. Sci. Paris 262: 127–129.
• Berger, M. (1970), "Quelques problemes de geometrie Riemannienne ou Deux variations sur les espaces symetriques compacts de rang un", Enseignement Math. 16: 73–96.
• Brakke, Kenneth A. (1991), "Minimal cones on hypercubes", J. Geom. Anal.: 329–338 (§6.5).
• Brakke, Kenneth A. (1993), Polyhedral minimal cones in R4.
• 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.
• Federer, Herbert (1965), "Some theorems on integral currents", Trans. AMS (Transactions of the American Mathematical Society, Vol. 117) 117: 43–67, doi:10.2307/1994196, JSTOR 1994196.
• Joyce, Dominic D. (2007), Riemannian Holonomy Groups and Calibrated Geometry, Oxford Graduate Texts in Mathematics, Oxford: Oxford University Press, ISBN 978-0-19-921559-1.
• Harvey, F. Reese (1990), Spinors and Calibrations, Academic Press, ISBN 978-0-12-329650-4.
• Kraines, Vivian Yoh (1965), "Topology of quaternionic manifolds", Bull. Amer. Math. Soc, 71,3, 1: 526–527.
• Lawlor, Gary (1998), "Proving area minimization by directed slicing", Indiana U. Math. J. 47: 1547–1592, doi:10.1512/iumj.1998.47.1341.
• Morgan, Frank, Lawlor, Gary (1996), "Curvy slicing proves that triple junctions locally minimize area", J. Diff. Geom. 44: 514–528.
• Morgan, Frank, Lawlor, Gary (1994), "Paired calibrations applied to soap films, immiscible fluids, and surfaces or networks minimizing other norms", Pac. J. Math. 166: 55–83.
• McLean, R. C. (1998), "Deformations of calibrated submanifolds", Communications in Analysis and Geometry 6: 705–747.
• Morgan, Frank (1988), "Area-minimizing surfaces, faces of Grassmannians, and calibrations", Amer. Math. Monthly (The American Mathematical 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.
• Morgan, Frank (2009), Geometric Measure Theory: a Beginner's Guide (4th ed.), London: Academic Press.
• Thi, Dao Trong (1977), "Minimal real currents on compact Riemannian manifolds", Izv. Akad. Nauk. SSSR Ser. Mat 41: 807–820.
• Van, Le Hong (1990), "Relative calibrations and the problem of stability of minimal surfaces", Lecture Notes in Mathematics (New York: Springer-Verlag) 1453: 245–262.
• Wirtinger, W. (1936), "Eine Determinantenidentität und ihre Anwendung auf analytische Gebilde und Hermitesche Massbestimmung", Monatsh. Math. Phys. 44: 343–365 (§6.5), doi:10.1007/BF01699328.