current

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

 $\int_{\partial M}\omega=\int_{M}d\omega.$

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

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

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

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

We denote with $\mathcal{E}_{m}$ the vector subspace of $\mathcal{D}_{m}$ 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

 $T_{k}(\omega)\to T(\omega),\qquad\forall\omega.$

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

 $||\omega||:=\sup\{|\langle\omega,\xi\rangle|\colon\text{\xi is a unit, % simple, m-vector}\}.$

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

 $\mathbf{M}(T):=\sup\{T(\omega)\colon\sup_{x}||\omega(x)||\leq 1\}.$

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

An intermediate norm, is the flat norm defined by

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

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 $\Lambda_{c}^{0}(\mathbb{R}^{n})\equiv C^{\infty}_{c}(\mathbb{R}^{n})$ 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 $\mathbb{R}^{3}$. Then the following defines a $2$-current:

 $T(a\,dx\wedge dy+b\,dy\wedge dz+c\,dx\wedge dz)=\int_{0}^{1}\int_{0}^{1}b(x,y,% 0)\,dx\,dy.$
Title current Current 2013-03-22 14:27:39 2013-03-22 14:27:39 paolini (1187) paolini (1187) 7 paolini (1187) Definition msc 58A25 mass support