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

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

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

${\displaystyle \partial \colon {\mathcal {D}}_{m+1}\to {\mathcal {D}}_{m}}$

via duality with the exterior derivative by

${\displaystyle (\partial T)(\omega ):=T(d\omega )\,}$

for all compactly supported m-forms ω.

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 T_k of currents, converges to a current 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 ω is an m-form, then define its comass by

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

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

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

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

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

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

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

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 ℝ³. 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, G. (1973), Variétés Différentiables, Actualites Scientifiques et Industrielles (in French), 1222 (3rd ed.), Paris: Hermann, pp. X+198, Zbl 0284.58001.
• Federer, Herbert (1969), Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, 153, Berlin–Heidelberg–New York: Springer-Verlag, pp. xiv+676, ISBN 978-3-540-60656-7, MR 0257325, Zbl 0176.00801.
• Whitney, H. (1957), Geometric Integration Theory, Princeton Mathematical Series, 21, Princeton, NJ and London: Princeton University Press and Oxford University Press, pp. XV+387, MR 0087148, Zbl 0083.28204.
• 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, MR 2030862, Zbl 1074.49011

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