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 $\xi \in {\Lambda }^p{T}_{x}M$, 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 bundle over M formed by the union of G_x(φ) for x in M.

The theory and terminology of calibrations was introduced by R. Harvey and B. Lawson in 1982. However, the main examples were introduced much earlier. Edmond Bonan introduced G₂-manifolds and Spin(7)-manifolds in 1966, constructing all the parallel forms and showing that those manifolds were Ricci-flat. Quaternion-Kähler manifolds were studied simultaneously in 1965 by Edmond Bonan and Vivian Yoh Kraines, each of whom 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 φ|_Σ = d vol_Σ. Equivalently, TΣ lies in the bundle 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 $$\underset{\Sigma }{\int }{\mathrm{v}\mathrm{o}\mathrm{l}}_{\Sigma }=\underset{\Sigma }{\int }\phi =\underset{{\Sigma }^{\prime }}{\int }\phi \le \underset{{\Sigma }^{\prime }}{\int }{\mathrm{v}\mathrm{o}\mathrm{l}}_{{\Sigma }^{\prime }},$$ 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.

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.

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