Current (mathematics)

In

multipoles

) spread out along subsets of M.

Definition

Let $\Omega _{c}^{m}(M)$ denote the space of smooth m-

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 \mathbb {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 $T\in {\mathcal {D}}_{m}(M)$ 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

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 _{\partial M}\omega =\int _{M}d\omega =[[M]](d\omega ).

This relates 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}}_{m+1}\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\{\left|\langle \omega ,\xi \rangle \right|:\xi {\mbox{ is a unit, simple, }}m{\mbox{-vector}}\}.

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\{T(\omega ):\sup _{x}|\vert \omega (x)|\vert \leq 1\}.

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\{\mathbf {M} (T-\partial A)+\mathbf {M} (A):A\in {\mathcal {E}}_{m+1}\}.

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.