current

Let ${\mathrm{\Lambda }}_c^m({ℝ}^n)$ denote the space of $C^{\mathrm{\infty }}$ differentiable $m$-forms with compact support in ${ℝ}^n$. A continuous linear operator $T:{\mathrm{\Lambda }}_c^m({ℝ}^n)\to ℝ$ is called an $m$-current. Let ${𝒟}_m$ denote the space of $m$-currents in ${ℝ}^n$. We define a boundary operator $\partial :{𝒟}_{m+1}\to {𝒟}_m$ by

$$\partial T(\omega ):=T(d\omega ).$$

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

$$[[M]](\omega )={\int }_M\omega .$$

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

$${\int }_{\partial M}\omega ={\int }_M𝑑\omega .$$

The space ${𝒟}_m$ of $m$-dimensional currents is a real vector space with operations defined by

$$(T+S)(\omega ):=T(\omega )+S(\omega ),(\lambda T)(\omega ):=\lambda T(\omega ).$$

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 support of a current $T$, denoted by $\mathrm{spt}(T)$, the smallest closed set $C$ such that

$$T(\omega )=0\text{whenever }\omega =0\text{ on }C.$$

We denote with ${ℰ}_m$ the vector subspace of ${𝒟}_m$ 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 $T_k$ of currents, weakly converges to a current $T$ if

$$T_k(\omega )\to T(\omega ),\forall \omega .$$

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

$$||\omega ||:=sup\{|⟨\omega ,\xi ⟩|:\xi \text{ is a unit, simple, }m\text{-vector}\}.$$

So if $\omega$ 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

$$𝐌(T):=sup\{T(\omega ):\underset{x}{sup}||\omega (x)||\le 1\}.$$

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

An intermediate norm, is the flat norm defined by

$$𝐅(T):=inf\{𝐌(A)+𝐌(B):T=A+\partial B,A\in {ℰ}_m,B\in {ℰ}_{m+1}\}.$$

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 ${\mathrm{\Lambda }}_c^0({ℝ}^n)\equiv C_c^{\mathrm{\infty }}({ℝ}^n)$ so that the following defines a $0$-current:

$$T(f)=f(0).$$

In particuar every signed measure $\mu$ with finite mass is a $0$-current:

$$T(f)=\int f(x)𝑑\mu (x).$$

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

$$T(adx\wedge dy+bdy\wedge dz+cdx\wedge dz)={\int }_0^1{\int }_0^{1}b(x,y,0)𝑑x𝑑y.$$

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