current
Let denote the space of differentiable -forms with compact support in . A continuous linear operator is called an -current. Let denote the space of -currents in . We define a boundary operator by
We will see that currents represent a generalization of -surfaces. In fact if is a compact -dimensional oriented manifold with boundary, we can associate to the current defined by
So the definition of boundary of a current, is justified by Stokes Theorem:
The space of -dimensional currents is a real vector space with operations defined by
The sum of two currents represents the union of the surfaces they represents. Multiplication by a scalar represents a change in the multiplicity of the surface. In particular multiplication by represents the change of orientation of the surface.
We define the support of a current , denoted by , the smallest closed set such that
We denote with the vector subspace of of currents with compact support.
Topology
The space of currents is naturally endowed with the weak-star topology, which will be further simply called weak convergence. We say that a sequence of currents, weakly converges to a current if
A stronger norm on the space of currents is the mass norm. First of all we define the mass norm of a -form as
So if is a simple -form, then its mass norm is the usual norm of its coefficient. We hence define the mass of a current as
The mass of a currents represents the area of the generalized surface.
An intermediate norm, is the flat norm defined by
Notice that two currents are close in the mass norm if they coincide apart from a small part. On the other hand the are close in the flat norm if they coincide up to a small deformation.
Examples
Recall that so that the following defines a -current:
In particuar every signed measure with finite mass is a -current:
Let be the coordinates in . Then the following defines a -current:
Title | current |
---|---|
Canonical name | Current |
Date of creation | 2013-03-22 14:27:39 |
Last modified on | 2013-03-22 14:27:39 |
Owner | paolini (1187) |
Last modified by | paolini (1187) |
Numerical id | 7 |
Author | paolini (1187) |
Entry type | Definition |
Classification | msc 58A25 |
Defines | mass |
Defines | support |