• Source: 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, 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-manifolds and Spin(7)-manifolds, constructed all the parallel forms and showed that those manifolds were Ricci-flat. Quaternion-Kähler manifolds 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





∫

Σ




v
o
l


Σ


=

∫

Σ


φ
=

∫


Σ
′



φ
≤

∫


Σ
′





v
o
l



Σ
′





{\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


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.
Bonan, Edmond (1982), "Sur l'algèbre extérieure d'une variété presque hermitienne quaternionique", C. R. Acad. Sci. Paris, 295: 115–118.
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., 1 (4): 329–338 (§6.5), doi:10.1007/BF02921309, S2CID 119606624.
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", Transactions of the American Mathematical Society, 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 (3): 526–527, doi:10.1090/s0002-9904-1965-11316-7.
Lawlor, Gary (1998), "Proving area minimization by directed slicing", Indiana Univ. Math. J., 47 (4): 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{{citation}}: CS1 maint: multiple names: authors list (link).
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, doi:10.2140/pjm.1994.166.55{{citation}}: CS1 maint: multiple names: authors list (link).
McLean, R. C. (1998), "Deformations of calibrated submanifolds", Communications in Analysis and Geometry, 6 (4): 705–747, doi:10.4310/CAG.1998.v6.n4.a4.
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.
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 (4): 807–820, Bibcode:1977IzMat..11..807C, doi:10.1070/IM1977v011n04ABEH001746.
Van, Le Hong (1990), "Relative calibrations and the problem of stability of minimal surfaces", Global analysis—studies and applications, IV, Lecture Notes in Mathematics, vol. 1453, New York: Springer-Verlag, pp. 245–262.
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, S2CID 121050865.

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