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.