Calibration Form

A calibration form on a Riemannian manifold M is a differential p-form phi such that

1. phi is a closed form.

2. The comass of phi,

 sup_(v in  ^ ^pTM, |v|=1)|phi(v)|

defined as the largest value of phi on a p vector of p-volume one, equals 1.

A p-dimensional submanifold is calibrated when phi restricts to give the volume form.

It is not hard to see that a calibrated submanifold N minimizes its volume among objects in its homology class. By Stokes' theorem, if N^' represents the same homology class, then






it follows that the volume of N is less than or equal to the volume of N^'.

A simple example is dx on the plane, for which the lines y=c are calibrated submanifolds. In fact, in this example, the calibrated submanifolds give a foliation. On a Kähler manifold, the Kähler form omega is a calibration form, which is indecomposable. For example, on


the Kähler form is

 dx_1 ^ dy_1+dx_2 ^ dy_2.

On a Kähler manifold, the calibrated submanifolds are precisely the complex submanifolds. Consequently, the complex submanifolds are locally volume minimizing.

See also

Kähler Form, Kähler Manifold, Volume Form

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Calibration Form." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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