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, that is, dφ = 0, where d is the exterior derivative.
• φ has operator norm at most 1. That is, for any x ∈ M and any p-vector ${\displaystyle \xi \in \Lambda ^{p}T_{x}M}$, we have φ(ξ) ≤ vol(ξ), with volume defined with respect to the Riemannian metric g.

A main reason for defining a calibration is that it creates a distinguished set of "directions" (i.e. p-planes) in which φ is actually equal to the volume form, that is, the inequality above is an equality. For x in M, set G_x(φ) to be the subset of such planes in the Grassmannian of p-planes in T_xM. In cases of interest, G_x(φ) is always nonempty. Let G(φ) be the union of G_x(φ) for all ${\displaystyle x\in M}$, viewed as a subspace of the bundle of p-planes in TM.

History

Harvey and Lawson introduced the term calibration and developed the theory in 1982,^[1] but the subject has a long prehistory.^[2]

The first motivating example, that of Kähler manifolds, is due implicitly to Wirtinger in 1936^[3] and explicitly to de Rham in 1957.^[4] In 1965, Federer used this to construct the first examples of singular minimal submanifolds.^[5]

Soon afterwards, the other main examples were introduced. Edmond Bonan studied G₂-manifolds and Spin(7)-manifolds in 1966,^[6] constructing all the parallel forms and showing that such manifolds must be Ricci-flat, although examples of either would not be constructed for another 20 years until the work of Robert Bryant. Quaternion-Kähler manifolds were studied simultaneously in 1965 by Edmond Bonan^[7] and Vivian Yoh Kraines,^[8] each of whom constructed the parallel 4-form. Finally, in 1970, Berger gave the general argument that calibrated submanifolds are minimal and applied it to these cases.^[9]

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 φ|_Σ = d vol_Σ. Equivalently, TΣ lies in G(φ).

A famous one-line argument shows that calibrated closed submanifolds minimize volume within their homology class. Indeed, suppose that Σ is calibrated, and Σ ′ is a 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 has operator norm 1.

The same argument shows that even a noncompact calibrated submanifold is a minimal submanifold in the variational sense, and therefore has zero mean curvature.

In particular, affine complex algebraic varieties are locally area-minimizing. Federer used this to give some of the first examples of singular minimal submanifolds, such as the algebraic curve ${\displaystyle \{w^{2}=z^{3}\}\subset \mathbb {C} ^{2}}$.^[5][2]

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 parallel 3-form and its 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

1. Harvey, Reese; Lawson, H. Blaine (1982). "Calibrated geometries". Acta Mathematica. 148 (0): 47–157. doi:10.1007/BF02392726.
2. Morgan, Frank (2009), Geometric Measure Theory: a Beginner's Guide (4th ed.), London: Academic Press, pp. 74–75, ISBN 978-0-12-374444-9
3. 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.
4. 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.
5. Federer, Herbert (1965), "Some theorems on integral currents", Transactions of the American Mathematical Society, 117: 43–67, doi:10.2307/1994196, JSTOR 1994196.
6. Bonan, Edmond (1966). "Sur les variétés riemanniennes à groupe d'holonomie G₂ ou Spin(7)". C. R. Acad. Sci. Paris. 262: 127–129.
7. Bonan, Edmond (1965). "Structure presque quaternale sur une variété différentiable". C. R. Acad. Sci. Paris. 260: 5445–5448.
8. 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.
9. Berger, Marcel (1970), "Quelques problèmes de géométrie riemannienne ou Deux variations sur les espaces symétriques compacts de rang un", L'Enseignement mathématique, 16: 73–96, §6.

• Bonan, Edmond (1982), "Sur l'algèbre extérieure d'une variété presque hermitienne quaternionique", C. R. Acad. Sci. Paris, 295: 115–118.
• 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.
• 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.
• 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.
• 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.
• 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.

Further reading

• 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.