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 (see The History of Calibrations).

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

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

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

History of Calibrations (Geometric Measure Theory, Chapter 6, Section 6.5, Frank Morgan)

The original example of complex analytic varieties was implicit in Wilhelm Wirtinger (1936), explicit for complex analytic submanifolds in Georges de Rham [1] (1957), and applied to singular complex varieties in the context of rectifiable currents by Herbert Federer [3, §4] (1965). Marcel Berger [2, §6, last paragraph] (1970) was the first to extract the underlying principle and apply it to other examples such as quaternionic varieties, followed by Dao (1977). The term calibration was coined in the landmark paper of Harvey and Lawson, which discovered rich new calibrated geometries of “special Lagrangian,” “associative,” and “Cayley” varieties.

The method has grown in power and applications. Surveys appear in Morgan [1, 2]. Mackenzie and Lawlor use calibrations in the proof (Nance; Lawlor [1]) of the angle conjecture on when a pair of m-planes in Rn is area minimizing. The “vanishing calibrations” of Lawlor [3] actually provide sufficient differential-geometric conditions for area minimization, a classification of all area-minimizing cones over products of m spheres, examples of nonorientable area-minimizing cones, and singularities stable under perturbations. The “paired calibrations” of Lawlor and Morgan [2] and of Brakke [1, 2] and the covering space calibrations of Brakke [3] prove new examples of soap films in R3, in R4, and above. Other developments include Murdoch’s “twisted calibrations” of nonorientable surfaces, Le’s “relative calibrations” of stable surfaces, and Pontryagin calibrations on Grassmannians (Gluck, Mackenzie, and Morgan).

Lawlor [2] has developed a related theory for proving minimization by slicing. Lawlor and Morgan [1] show, for example, that three minimal surfaces meeting at 120 degrees minimize area locally.

References

• 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 .
• Brakke, Kenneth A. (1995), "Soap films and covering spaces", J. Geom. Anal.: 445–514 .
• de Rham, Georges (1957–1958), On the Area of Complex Manifolds. Notes for the Seminar on Several Complex Variables, Institute for Advanced Study, Princeton, NJ .
• 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 978-0199215591 .
• Harvey, F. Reese (1990), Spinors and Calibrations, Academic Press, ISBN 978-0123296504 .
• Harvey, F. Reese; Lawson, H. Blaine (1982), "Calibrated geometries", Acta Mathematica 148: 47–157, doi:10.1007/BF02392726 .
• Lawlor, Gary (1998), "Proving area minimization by directed slicing", Indiana U. Math. J. 47: 1547–1592 .
• Morgan, Frank, Lawlor, Gary (1996 pages = 514–528), "Curvy slicing proves that triple junctions locally minimize area", J. Diff. Geom. 44 .
• Morgan, Frank, Lawlor, Gary (1994 pages = 55–83), "Paired calibrations applied to soap films, immiscible fluids, and surfaces or networks minimizing other norms", Pac. J Math. 166 .
• 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, Vol. 95, No. 9) 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. Academic Press, London .
• 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, Springer-Verlag, New York 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 .