Current (mathematics)

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 $\scriptstyle\Omega_c^m(\mathbb{R}^n)$ denote the space of smooth m-forms with compact support on $\mathbb{R}^n$. A current is a linear functional on $\scriptstyle\Omega_c^m(\mathbb{R}^n)$ which is continuous in the sense of distributions. Thus a linear functional

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

is an m-current if it is continuous in the following sense: If a sequence ωn of smooth forms, all supported in the same compact set, is such that all derivatives of all their coefficients tend uniformly to 0 when n tends to infinity, then Tn) tends to 0.

The space $\scriptstyle\mathcal D_m$ of m-dimensional currents on ℝn is a real vector space with operations defined by

$(T+S)(\omega):= T(\omega)+S(\omega),\qquad (\lambda T)(\omega):=\lambda T(\omega).$

Multiplication by a constant scalar represents a change in the multiplicity of the surface[clarification needed]. In particular multiplication by −1 represents the change of orientation of the surface.

Much of the theory of distributions carries over to currents with minimal adjustments. For example, one may define the support of a current T as the complement of the biggest open set U such that T(ω) = 0 whenever the support of ω lies entirely in U.

The linear subspace of $\scriptstyle\mathcal D_m$ consisting of currents with compact support is denoted $\scriptstyle\mathcal E_m$. It can be naturally identified with the dual space to the space of all smooth m-forms on ℝn.

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 boundaryM of M is rectifiable, then it too defines a current by integration, and one has Stokes' theorem:

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

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

More generally, a boundary operator can be defined on arbitrary currents

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

by dualizing the exterior derivative:

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

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 Tk 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 μ 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

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