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. 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 ${\Omega }_c^m(M)$ 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 $$T:{\Omega }_c^m(M)\to ℝ$$ 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 ${𝒟}_m(M)$ of m-dimensional currents on $M$ is a real vector space with operations defined by $$(T+S)(\omega ):=T(\omega )+S(\omega ),(\lambda T)(\omega ):=\lambda T(\omega ).$$

Much of the theory of distributions carries over to currents with minimal adjustments. For example, one may define the support of a current $T\in {𝒟}_m(M)$ 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 ${𝒟}_m(M)$ consisting of currents with support (in the sense above) that is a compact subset of $M$ is denoted ${ℰ}_m(M).$

Homological theory

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

If the boundary ∂M of M is rectifiable, then it too defines a current by integration, and by virtue of Stokes' theorem one has: $$[[\partial M]](\omega )=\underset{\partial M}{\int }\omega =\underset{M}{\int }d\omega =[[M]](d\omega ).$$

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 $$\partial :{𝒟}_{m+1}\to {𝒟}_m$$ 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 $$T_k(\omega )\to T(\omega ),\mathrm{\forall }\omega .$$

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 $$\Vert \omega \Vert :=\sup \{|⟨\omega ,\xi ⟩|:\xi \text{ is a unit, simple, }m\text{-vector}\}.$$

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 $$𝐌(T):=\sup \{T(\omega ):\underset{x}{\sup }||\omega (x)||\le 1\}.$$

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 $$𝐅(T):=\inf \{𝐌(T-\partial A)+𝐌(A):A\in {ℰ}_{m+1}\}.$$

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 $${\Omega }_c^0({ℝ}^n)\equiv C_c^{\mathrm{\infty }}({ℝ}^n)$$ 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 ${ℝ}^3.$ Then the following defines a 2-current (one of many): $$T(adx\wedge dy+bdy\wedge dz+cdx\wedge dz):=\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}b(x,y,0)dxdy.$$

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.