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

ξ

∈

Λ

p

T

x

M

{\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 Gx(φ) to be the subset of such planes in the Grassmannian of p-planes in TxM. In cases of interest, Gx(φ) is always nonempty. Let G(φ) be the union of Gx(φ) for all

x

∈

M

{\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, but the subject has a long prehistory.

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

Soon afterwards, the other main examples were introduced.