Current (mathematics)

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 \mathcal 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),\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 T \in \mathcal{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 \mathcal D_m(M) consisting of currents with support (in the sense above) that is a compact subset of M is denoted \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 M: 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: \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 \partial : \mathcal D_{m+1} \to \mathcal 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),\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 \omega is an m-form, then define its comass by \|\omega\| := \sup\{\left|\langle \omega,\xi\rangle\right| : \xi \mbox{ is a unit, simple, }m\mbox{-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 \mathbf M (T) := \sup\{ T(\omega) : \sup_x |\vert\omega(x)|\vert\le 1\}. The mass of a current represents the weighted area of the generalized surface. A current such that M(T) \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 \Omega_c^0(\R^n)\equiv C^\infty_c(\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(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==