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.
