Integral Current

In mathematics, more particularly in functional analysis, differential topology, and geometric measure theory, a ''k''-current in the sense of Georges de Rham is a functional on the space of compactly supported differential ''k''-forms, on a smooth manifold ''M''. Currents formally behave like Schwartz distributions on a space of differential forms, but in a geometric setting, they can represent integration over a submanifold, generalizing the Dirac delta function, or more generally even directional derivatives of delta functions (multipoles) spread out along subsets of ''M''.

Definition

Let $\Omega_c^m(M)$ denote the space of smooth ''m''-forms with compact support on a smooth manifold $M.$ A current is a linear functional on $\Omega_c^m(M)$ which is continuous in the sense of distributions. Thus a linear functional
$T : \Omega_c^m(M)\to \R$
is an ''m''-dimensional current if it is continuous in the following sense: If a sequence $\omega_k$ of smooth forms, all supported in the same compact set, is such that all derivatives of all their coefficients tend uniformly to 0 when $k$ tends to infinity, then $T(\omega_k)$ tends to 0.

The space $\mathcal D_m(M)$ of ''m''-dimensional currents on $M$ is a real vector space with operations defined by
$(T+S)(\omega) := T(\omega)+S(\omega),\qquad (\lambda T)(\omega) := \lambda T(\omega).$

Much of the theory of distributions carries over to currents with minimal adjustments. For example, one may define the support of a current $T \in \mathcal_m(M)$ as the complement of the biggest open set $U \subset M$ such that
$T(\omega) = 0$ whenever $\omega \in \Omega_c^m(U)$

The linear subspace of $\mathcal D_m(M)$ consisting of currents with support (in the sense above) that is a compact subset of $M$ is denoted $\mathcal E_m(M).$

Homological theory

Integration over a compact rectifiable oriented submanifold ''M'' ( with boundary) of dimension ''m'' defines an ''m''-current, denoted by M:
M(\omega)=\int_M \omega.

If the boundary ∂''M'' of ''M'' is rectifiable, then it too defines a current by integration, and by virtue of Stokes' theorem one has:
\partial M(\omega) = \int_\omega = \int_M d\omega = M(d\omega).

This relates the exterior derivative ''d'' with the boundary operator ∂ on the homology of ''M''.

In view of this formula we can ''define'' a boundary operator on arbitrary currents
$\partial : \mathcal D_ \to \mathcal D_m$
via duality with the exterior derivative by
$(\partial T)(\omega) := T(d\omega)$
for all compactly supported ''m''-forms $\omega.$

Certain subclasses of currents which are closed under $\partial$ can be used instead of all currents to create a homology theory, which can satisfy the Eilenberg–Steenrod axioms in certain cases. A classical example is the subclass of integral currents on Lipschitz neighborhood retracts.

Topology and norms

The space of currents is naturally endowed with the weak-* topology, which will be further simply called ''weak convergence''. A sequence $T_k$ of currents, converges to a current $T$ if
$T_k(\omega) \to T(\omega),\qquad \forall \omega.$

It is possible to define several norms on subspaces of the space of all currents. One such norm is the ''mass norm''. If $\omega$ is an ''m''-form, then define its comass by
$\, \omega\, := \sup\.$

So if $\omega$ is a simple ''m''-form, then its mass norm is the usual L^∞-norm of its coefficient. The mass of a current $T$ is then defined as
$\mathbf M (T) := \sup\.$

The mass of a current represents the ''weighted area'' of the generalized surface. A current such that M(''T'') < ∞ is representable by integration of a regular Borel measure by a version of the Riesz representation theorem. This is the starting point of homological integration.

An intermediate norm is Whitney's ''flat norm'', defined by
$\mathbf F (T) := \inf \.$

Two currents are close in the mass norm if they coincide away from a small part. On the other hand, they are close in the flat norm if they coincide up to a small deformation.

Examples

Recall that
$\Omega_c^0(\R^n)\equiv C^\infty_c(\R^n)$
so that the following defines a 0-current:
$T(f) = f(0).$

In particular every signed regular measure $\mu$ 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 (one of many):
$T(a\,dx\wedge dy + b\,dy\wedge dz + c\,dx\wedge dz) := \int_0^1 \int_0^1 b(x,y,0)\, dx \, dy.$

See also

*Georges de Rham
*Herbert Federer
*Differential geometry
*Varifold

Notes

References

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{{PlanetMath attribution, id=5980, title=Current

Differential topology
Functional analysis
Generalized functions
Generalized manifolds