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(Definition)

Let $\Lambda_c^m(\mathbb{R}^n)$ denote the space of $C^\infty$ differentiable -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 -current. Let $\mathcal D_m$ denote the space of -currents in $\mathbb{R}^n$ . We define a boundary operator $\partial\colon \mathcal D_{m+1}\to \mathcal D_m$ by

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

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

$\displaystyle [[M]](\omega)=\int_M \omega.$

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

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

The space $\mathcal D_m$ of -dimensional currents is a real vector space with operations defined by

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

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

represents the change of orientation of the surface.

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

$\displaystyle T(\omega)=0\$ whenever $\omega=0$ on

$\displaystyle .$

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

of currents, weakly converges to a current

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

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

$\displaystyle \vert\vert\omega\vert\vert:= \sup\{\vert\langle \omega,\xi\rangle\vert\colon$ $\xi$ is a unit, simple,

-vector $\displaystyle \}.$

So if $\omega$ is a simple

-form, then its mass norm is the usual norm of its coefficient. We hence define the mass of a current

$\displaystyle \mathbf M (T) := \sup\{ T(\omega)\colon \sup_x \vert\vert\omega(x)\vert\vert\le 1\}.$

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

An intermediate norm, is the flat norm defined by

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

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:

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

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

$\displaystyle T(f) = \int f(x)\, d\mu(x).$

Let be the coordinates in $\mathbb{R}^3$ . Then the following defines a -current:

$\displaystyle T(a\,dx\wedge dy + b\,dy\wedge dz + c\,dx\wedge dz) = \int_0^1 \int_0^1 b(x,y,0)\, dx \, dy.$

"current" is owned by paolini.

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Also defines:

mass, support

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Cross-references: coordinates, finite, signed measure, deformation, flat, area, coefficient, simple, norm, converges, sequence, weak convergence, topology, vector subspace, closed set, orientation, multiplicity, scalar, multiplication, surfaces, union, sum, operations, vector space, real, Stokes theorem, associate, boundary, oriented manifold, represent, boundary operator, linear operator, continuous, compact, differentiable
There are 80 references to this entry.

This is version 4 of current, born on 2004-07-02, modified 2004-07-12.
Object id is 5980, canonical name is Current.
Accessed 7616 times total.

Classification:

AMS MSC:

58A25 (Global analysis, analysis on manifolds :: General theory of differentiable manifolds :: Currents)

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