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Calibrated geometry

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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 ≤ pn) which is a calibration, meaning that:

  • φ is closed: dφ = 0, where d is the exterior derivative
  • for any xM 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

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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

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

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References

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  • 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.