Current (mathematics)

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. Formally currents behave like Schwartz distributions on a space of differential forms. 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

$T\colon \Omega_c^m(M)\to \mathbb{R}$

is an m-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

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:

$[[\partial M]](\omega) = \int_{\partial M}\omega = \int_M d\omega = [[M]](d\omega).$

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

$\partial\colon \mathcal D_{m+1} \to \mathcal D_m$

via duality with the exterior derivative by

$(\partial T)(\omega) := T(d\omega)\,$

for all compactly supported (m−1)-forms ω.

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 ω is an m-form, then define its comass by

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

So if ω 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

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

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

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

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.

Examples

Recall that

$\Omega_c^0(\mathbb{R}^n)\equiv C^\infty_c(\mathbb{R}^n)\,$

so that the following defines a 0-current:

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

In 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 ℝ³. 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

References

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This article incorporates material from Current on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.