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.

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 1936³ 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 Bonan⁷ 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

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 ${\displaystyle \{w^{2}=z^{3}\}\subset \mathbb {C} ^{2}}$.⁵²

• 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.