An 
-dimensional varifold on an open
subset 
of Euclidean space is a Radon
measure on pairs 
, where 
and 
is an unoriented 
-dimensional linear subspace.
Its weight measure on 
is obtained by forgetting the plane
coordinate. Unlike an oriented current, a varifold does not encode a choice of orientation
or a boundary operator.
An integral varifold is represented by a countably 
-rectifiable set 
and a positive integer
multiplicity 
, locally integrable
with respect to Hausdorff measure. For a compactly supported continuous test function 
, it acts by
 |
where 
is the approximate tangent
plane at almost every point.
A smooth embedded submanifold with constant positive integer multiplicity gives an example.
Stationary integral varifolds are
the critical objects for the first variation of volume.
