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

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

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

The linear subspace of ${\displaystyle {\mathcal{D}}_m(M)}$ consisting of currents with support (in the sense above) that is a compact subset of ${\displaystyle M}$ is denoted ${\displaystyle {\mathcal{E}}_m(M).}$

Homological theory

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

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

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 ${\displaystyle \partial :{\mathcal{D}}_{m+1}\to {\mathcal{D}}_m}$ via duality with the exterior derivative by ${\displaystyle (\partial T)(\omega ):=T(d\omega )}$ for all compactly supported m-forms ${\displaystyle \omega .}$

Certain subclasses of currents which are closed under ${\displaystyle \partial }$ 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 ${\displaystyle T_k}$ of currents, converges to a current ${\displaystyle T}$ if ${\displaystyle T_k(\omega )\to T(\omega ),\mathrm{\forall }\omega .}$

It is possible to define several norms on subspaces of the space of all currents. One such norm is the mass norm. If ${\displaystyle \omega }$ is an m-form, then define its comass by ${\displaystyle \Vert \omega \Vert :=sup\{|⟨\omega ,\xi ⟩|:\xi \text{ is a unit, simple, }m\text{-vector}\}.}$

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

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

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 ${\displaystyle {\mathrm{\Omega }}_c^0({\text{R}}^n)\equiv C_c^{\mathrm{\infty }}({\text{R}}^n)}$ so that the following defines a 0-current: ${\displaystyle T(f)=f(0).}$

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

Let (x, y, z) be the coordinates in ${\displaystyle {\text{R}}^3.}$ Then the following defines a 2-current (one of many): ${\displaystyle T(adx\wedge dy+bdy\wedge dz+cdx\wedge dz):={\int }_0^1{\int }_0^{1}b(x,y,0)dxdy.}$

See also

• Georges de Rham
• Herbert Federer
• Differential geometry
• Varifold

Notes

References

• de Rham, Georges (1984). Differentiable manifolds. Forms, currents, harmonic forms. Grundlehren der mathematischen Wissenschaften. 266. With an introduction by S. S. Chern. (Translation of 1955 French original ed.). Berlin: Springer-Verlag. doi:10.1007/978-3-642-61752-2. ISBN 3-540-13463-8. 
• Federer, Herbert (1969). Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften. 153. Berlin–Heidelberg–New York: Springer-Verlag. doi:10.1007/978-3-642-62010-2. ISBN 978-3-540-60656-7. 
• Griffiths, Phillip; Harris, Joseph (1978). Principles of algebraic geometry. Pure and Applied Mathematics. New York: John Wiley & Sons. doi:10.1002/9781118032527. ISBN 0-471-32792-1. 
• Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis. 3. Canberra: Centre for Mathematical Analysis at Australian National University. 1983. ISBN 0-86784-429-9. https://projecteuclid.org/proceedings/proceedings-of-the-centre-for-mathematics-and-its-applications/Lectures-on-Geometric-Measure-Theory/toc/pcma/1416406261. 
• Whitney, Hassler (1957). Geometric integration theory. Princeton Mathematical Series. 21. Princeton, NJ and London: Princeton University Press and Oxford University Press. doi:10.1515/9781400877577. ISBN 9780691652900. .
• 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 

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