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 byStokes 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 froma 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:

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