Current (mathematics)

Let $\backslash Omega\_c^m(M)$ denote the space of smooth m-forms with compact support on a smooth manifold $M.$ A current is a linear functional on $\backslash Omega\_c^m(M)$ which is continuous in the sense of distributions. Thus a linear functional

$$T\; :\; \backslash Omega\_c^m(M)\backslash to\; \backslash R$$

is an m-dimensional current if it is continuous in the following sense: If a sequence $\backslash omega\_k$ 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 $T(\backslash omega\_k)$ tends to 0.

The space $\backslash mathcal\; D\_m(M)$ of m-dimensional currents on $M$ is a real vector space with operations defined by

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

Much of the theory of distributions carries over to currents with minimal adjustments. For example, one may define the support of a current $T\; \backslash in\; \backslash mathcal\{D\}\_m(M)$ as the complement of the biggest open set $U\; \backslash subset\; M$ such that

$$T(\backslash omega)\; =\; 0$$ whenever $\backslash omega\; \backslash in\; \backslash Omega\_c^m(U)$

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

Integration over a compact rectifiable oriented submanifold M (with boundary) of dimension m defines an m-current, denoted by $M$:

$$M(\backslash omega)=\backslash int\_M\; \backslash omega.$$

If the boundary ∂M of M is rectifiable, then it too defines a current by integration, and by virtue of Stokes' theorem one has:

$$\backslash partial\; M(\backslash omega)\; =\; \backslash int\_\{\backslash partial\; M\}\backslash omega\; =\; \backslash int\_M\; d\backslash omega\; =M(d\backslash omega).$$

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

$$\backslash partial\; :\; \backslash mathcal\; D\_\{m+1\}\; \backslash to\; \backslash mathcal\; D\_m$$

via duality with the exterior derivative by

$$(\backslash partial\; T)(\backslash omega)\; :=\; T(d\backslash omega)$$

for all compactly supported m-forms $\backslash omega.$

Certain subclasses of currents which are closed under $\backslash partial$ 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.

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

$$T\_k(\backslash omega)\; \backslash to\; T(\backslash omega),\backslash qquad\; \backslash forall\; \backslash omega.$$

It is possible to define several norms on subspaces of the space of all currents. One such norm is the mass norm. If $\backslash omega$ is an m-form, then define its comass by

$$\backslash |\backslash omega\backslash |\; :=\; \backslash sup\backslash \{\backslash left|\backslash langle\; \backslash omega,\backslash xi\backslash rangle\backslash right|\; :\; \backslash xi\; \backslash mbox\{\; is\; a\; unit,\; simple,\; \}m\backslash mbox\{-vector\}\backslash \}.$$

So if $\backslash omega$ 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

$$\backslash mathbf\; M\; (T)\; :=\; \backslash sup\backslash \{\; T(\backslash omega)\; :\; \backslash sup\_x\; |\backslash vert\backslash omega(x)|\backslash vert\backslash le\; 1\backslash \}.$$

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

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

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.

Recall that

$$\backslash Omega\_c^0(\backslash R^n)\backslash equiv\; C^\backslash infty\_c(\backslash R^n)$$

so that the following defines a 0-current:

$$T(f)\; =\; f(0).$$

In particular every signed regular measure $\backslash mu$ is a 0-current:

$$T(f)\; =\; \backslash int\; f(x)\backslash ,\; d\backslash mu(x).$$

Let (x, y, z) be the coordinates in $\backslash R^3.$ Then the following defines a 2-current (one of many):

$$T(a\backslash ,dx\backslash wedge\; dy\; +\; b\backslash ,dy\backslash wedge\; dz\; +\; c\backslash ,dx\backslash wedge\; dz)\; :=\; \backslash int\_0^1\; \backslash int\_0^1\; b(x,y,0)\backslash ,\; dx\; \backslash ,\; dy.$$

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

{{reflist|20em}}

• {{cite book|mr=0760450|author-link1=Georges de Rham|last1=de Rham|first1=Georges|title=Differentiable manifolds. Forms, currents, harmonic forms|translator-first1=F. R.|translator-last1=Smith|others=With an introduction by S. S. Chern.|edition=Translation of 1955 French original|series=Grundlehren der mathematischen Wissenschaften|volume=266|publisher=Springer-Verlag|location=Berlin|year=1984|isbn=3-540-13463-8|zbl=0534.58003|doi=10.1007/978-3-642-61752-2}}
• {{cite book| last = Federer| first = Herbert|author-link1=Herbert Federer| title = Geometric measure theory| place= Berlin–Heidelberg–New York| publisher = Springer-Verlag| series = Die Grundlehren der mathematischen Wissenschaften| volume = 153| year = 1969| isbn = 978-3-540-60656-7| mr=0257325| zbl= 0176.00801 | doi=10.1007/978-3-642-62010-2}}
• {{cite book|last1=Griffiths|first1=Phillip|last2=Harris|first2=Joseph|title=Principles of algebraic geometry|series=Pure and Applied Mathematics|publisher=John Wiley & Sons|location=New York|year=1978|isbn=0-471-32792-1|mr=0507725|author-link1=Phillip Griffiths|author-link2=Joe Harris (mathematician)|zbl=0408.14001|doi=10.1002/9781118032527}}
• {{Cite book

| last = Simon|author-link1=Leon Simon

| first = Leon

| title = Lectures on geometric measure theory

| location = Canberra

| publisher = Centre for Mathematical Analysis at Australian National University

| series = Proceedings of the Centre for Mathematical Analysis

| volume = 3

| year = 1983

| url = https://projecteuclid.org/proceedings/proceedings-of-the-centre-for-mathematics-and-its-applications/Lectures-on-Geometric-Measure-Theory/toc/pcma/1416406261

| isbn = 0-86784-429-9

| mr = 0756417

| zbl = 0546.49019

}}

• {{cite book

|last=Whitney

|first=Hassler

|author-link=Hassler Whitney

|doi=10.1515/9781400877577

|title=Geometric integration theory

|isbn = 9780691652900

|series=Princeton Mathematical Series

|volume=21

|publisher=Princeton University Press and Oxford University Press

|place=Princeton, NJ and London

|year=1957

|mr=0087148

|zbl=0083.28204

}}.

• {{Citation

| last1=Lin

| first1=Fanghua

| last2=Yang

| first2=Xiaoping

| title=Geometric Measure Theory: An Introduction

| series=Advanced Mathematics (Beijing/Boston)

| volume=1

| place=Beijing/Boston

| pages=x+237

| publisher=Science Press/International Press

| isbn=978-1-57146-125-4

| year=2003

| mr=2030862

| zbl=1074.49011

}}

{{PlanetMath attribution|id=5980|title=Current}}

Category:Differential topology

Category:Functional analysis

Category:Generalized functions

Category:Generalized manifolds

Category:Schwartz distributions