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

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

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

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 over a suitably weighted rectifiable submanifold. 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.

References

de Rham, G. (1973), Variétés Différentiables, Actualites Scientifiques et Industrielles (in French) 1222 (3rd ed.), Paris: Hermann, pp. X+198, Zbl 0284.58001 .
Federer, Herbert (1969), Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften 153, Berlin–Heidelberg–New York: Springer-Verlag, pp. xiv+676, ISBN 978-3-540-60656-7, MR 0257325, Zbl 0176.00801 .
Whitney, H. (1957), Geometric Integration Theory, Princeton Mathematical Series 21, Princeton, NJ and London: Princeton University Press and Oxford University Press, pp. XV+387, MR 0087148, Zbl 0083.28204 .
Lin, Fanghua; Yang, Xiaoping (2003), Geometric Measure Theory: An Introduction, Advanced Mathematics (Beijing/Boston) 1, Beijing/Boston: Science Press/International Press, pp. x+237, ISBN 978-1-57146-125-4, MR 2030862, Zbl 1074.49011