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# Calibration Form

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

1. is a closed form.

2. The comass of ,

 (1)

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

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

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

 (2)

Since

 (3)

and

 (4)

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

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

 (5)

the Kähler form is

 (6)

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

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. https://mathworld.wolfram.com/CalibrationForm.html