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

$${\displaystyle T:\Omega _{c}^{m}(M)\to \mathbb {R} }$$

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

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

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

$${\displaystyle T(\omega )=0}$$

whenever ${\displaystyle \omega \in \Omega _{c}^{m}(U)}$

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

$${\displaystyle T_{k}(\omega )\to T(\omega ),\qquad \forall \omega .}$$

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

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

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

$${\displaystyle \mathbf {M} (T):=\sup\{T(\omega ):\sup _{x}|\vert \omega (x)|\vert \leq 1\}.}$$

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

$${\displaystyle \Omega _{c}^{0}(\mathbb {R} ^{n})\equiv C_{c}^{\infty }(\mathbb {R} ^{n})}$$

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 ${\displaystyle \mathbb {R} ^{3}.}$ Then the following defines a 2-current (one of many):

$${\displaystyle T(a\,dx\wedge dy+b\,dy\wedge dz+c\,dx\wedge dz):=\int _{0}^{1}\int _{0}^{1}b(x,y,0)\,dx\,dy.}$$

See also

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

References

• de Rham, Georges (1984). Differentiable manifolds. Forms, currents, harmonic forms. Grundlehren der mathematischen Wissenschaften. Vol. 266. Translated by Smith, F. R. 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. MR   0760450. Zbl   0534.58003.
• Federer, Herbert (1969). Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften. Vol. 153. Berlin–Heidelberg–New York: Springer-Verlag. doi:10.1007/978-3-642-62010-2. ISBN   978-3-540-60656-7. MR   0257325. Zbl   0176.00801.
• 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. MR   0507725. Zbl   0408.14001.
• Simon, Leon (1983). Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis. Vol. 3. Canberra: Centre for Mathematical Analysis at Australian National University. ISBN   0-86784-429-9. MR   0756417. Zbl   0546.49019.
• Whitney, Hassler (1957). Geometric integration theory. Princeton Mathematical Series. Vol. 21. Princeton, NJ and London: Princeton University Press and Oxford University Press. doi:10.1515/9781400877577. ISBN   9780691652900. MR   0087148. Zbl   0083.28204..
• Lin, Fanghua; Yang, Xiaoping (2003), Geometric Measure Theory: An Introduction, Advanced Mathematics (Beijing/Boston), vol. 1, Beijing/Boston: Science Press/International Press, pp. x+237, ISBN   978-1-57146-125-4, MR   2030862, Zbl   1074.49011

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