current


Let Λcm(n) denote the space of C differentiableMathworldPlanetmathPlanetmath m-forms with compact support in n. A continuous linear operator T:Λcm(n) is called an m-current. Let 𝒟m denote the space of m-currents in n. We define a boundary operatorMathworldPlanetmath :𝒟m+1𝒟m by

T(ω):=T(dω).

We will see that currents represent a generalizationPlanetmathPlanetmath of m-surfaces. In fact if M is a compactPlanetmathPlanetmath m-dimensional oriented manifold with boundary, we can associate to M the current [[M]] defined by

[[M]](ω)=Mω.

So the definition of boundary T of a current, is justified by Stokes Theorem:

Mω=M𝑑ω.

The space 𝒟m of m-dimensional currents is a real vector space with operationsMathworldPlanetmath defined by

(T+S)(ω):=T(ω)+S(ω),(λT)(ω):=λT(ω).

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 -1 represents the change of orientation of the surface.

We define the supportMathworldPlanetmath of a current T, denoted by spt(T), the smallest closed setPlanetmathPlanetmath C such that

T(ω)=0whenever ω=0 on C.

We denote with m the vector subspace of 𝒟m of currents with compact support.

Topology

The space of currents is naturally endowed with the weak-star topologyMathworldPlanetmath, which will be further simply called weak convergence. We say that a sequencePlanetmathPlanetmath Tk of currents, weakly convergesPlanetmathPlanetmath to a current T if

Tk(ω)T(ω),ω.

A stronger norm on the space of currents is the mass norm. First of all we define the mass norm of a m-form ω as

||ω||:=sup{|ω,ξ|:ξ is a unit, simple, m-vector}.

So if ω is a simple m-form, then its mass norm is the usual norm of its coefficient. We hence define the mass of a current T as

𝐌(T):=sup{T(ω):supx||ω(x)||1}.

The mass of a currents represents the area of the generalized surface.

An intermediate norm, is the flat norm defined by

𝐅(T):=inf{𝐌(A)+𝐌(B):T=A+B,Am,Bm+1}.

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 Λc0(n)Cc(n) so that the following defines a 0-current:

T(f)=f(0).

In particuar every signed measure μ with finite mass is a 0-current:

T(f)=f(x)𝑑μ(x).

Let (x,y,z) be the coordinates in 3. Then the following defines a 2-current:

T(adxdy+bdydz+cdxdz)=0101b(x,y,0)𝑑x𝑑y.
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