Current (mathematics)

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Short description: Distributions on spaces of differential forms

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 [math]\displaystyle{ \Omega_c^m(M) }[/math] denote the space of smooth m-forms with compact support on a smooth manifold [math]\displaystyle{ M. }[/math] A current is a linear functional on [math]\displaystyle{ \Omega_c^m(M) }[/math] which is continuous in the sense of distributions. Thus a linear functional [math]\displaystyle{ T : \Omega_c^m(M)\to \R }[/math] is an m-dimensional current if it is continuous in the following sense: If a sequence [math]\displaystyle{ \omega_k }[/math] of smooth forms, all supported in the same compact set, is such that all derivatives of all their coefficients tend uniformly to 0 when [math]\displaystyle{ k }[/math] tends to infinity, then [math]\displaystyle{ T(\omega_k) }[/math] tends to 0.

The space [math]\displaystyle{ \mathcal D_m(M) }[/math] of m-dimensional currents on [math]\displaystyle{ M }[/math] is a real vector space with operations defined by [math]\displaystyle{ (T+S)(\omega) := T(\omega)+S(\omega),\qquad (\lambda T)(\omega) := \lambda T(\omega). }[/math]

Much of the theory of distributions carries over to currents with minimal adjustments. For example, one may define the support of a current [math]\displaystyle{ T \in \mathcal{D}_m(M) }[/math] as the complement of the biggest open set [math]\displaystyle{ U \subset M }[/math] such that [math]\displaystyle{ T(\omega) = 0 }[/math] whenever [math]\displaystyle{ \omega \in \Omega_c^m(U) }[/math]

The linear subspace of [math]\displaystyle{ \mathcal D_m(M) }[/math] consisting of currents with support (in the sense above) that is a compact subset of [math]\displaystyle{ M }[/math] is denoted [math]\displaystyle{ \mathcal E_m(M). }[/math]

Homological theory

Integration over a compact rectifiable oriented submanifold M (with boundary) of dimension m defines an m-current, denoted by [math]\displaystyle{ M }[/math]: [math]\displaystyle{ M(\omega)=\int_M \omega. }[/math]

If the boundaryM of M is rectifiable, then it too defines a current by integration, and by virtue of Stokes' theorem one has: [math]\displaystyle{ \partial M(\omega) = \int_{\partial M}\omega = \int_M d\omega = M(d\omega). }[/math]

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 [math]\displaystyle{ \partial : \mathcal D_{m+1} \to \mathcal D_m }[/math] via duality with the exterior derivative by [math]\displaystyle{ (\partial T)(\omega) := T(d\omega) }[/math] for all compactly supported m-forms [math]\displaystyle{ \omega. }[/math]

Certain subclasses of currents which are closed under [math]\displaystyle{ \partial }[/math] 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 [math]\displaystyle{ T_k }[/math] of currents, converges to a current [math]\displaystyle{ T }[/math] if [math]\displaystyle{ T_k(\omega) \to T(\omega),\qquad \forall \omega. }[/math]

It is possible to define several norms on subspaces of the space of all currents. One such norm is the mass norm. If [math]\displaystyle{ \omega }[/math] is an m-form, then define its comass by [math]\displaystyle{ \|\omega\| := \sup\{\left|\langle \omega,\xi\rangle\right| : \xi \mbox{ is a unit, simple, }m\mbox{-vector}\}. }[/math]

So if [math]\displaystyle{ \omega }[/math] is a simple m-form, then its mass norm is the usual L-norm of its coefficient. The mass of a current [math]\displaystyle{ T }[/math] is then defined as [math]\displaystyle{ \mathbf M (T) := \sup\{ T(\omega) : \sup_x |\vert\omega(x)|\vert\le 1\}. }[/math]

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 [math]\displaystyle{ \mathbf F (T) := \inf \{\mathbf M(T - \partial A) + \mathbf M(A) : A\in\mathcal E_{m+1}\}. }[/math]

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 [math]\displaystyle{ \Omega_c^0(\R^n)\equiv C^\infty_c(\R^n) }[/math] so that the following defines a 0-current: [math]\displaystyle{ T(f) = f(0). }[/math]

In particular every signed regular measure [math]\displaystyle{ \mu }[/math] is a 0-current: [math]\displaystyle{ T(f) = \int f(x)\, d\mu(x). }[/math]

Let (x, y, z) be the coordinates in [math]\displaystyle{ \R^3. }[/math] Then the following defines a 2-current (one of many): [math]\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. }[/math]

See also

Notes

References