Current 2004-07-02 11:31:50 2004-07-12 10:32:05 Definition mass support % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\R}{\mathbb R} Let $\Lambda_c^m(\R^n)$ denote the space of $C^\infty$ differentiable $m$-forms with compact support in $\R^n$. A continuous linear operator $T\colon \Lambda_c^m(\R^n)\to \R$ is called an $m$-\emph{current}. Let $\mathcal D_m$ denote the space of $m$-currents in $\R^n$. We define a boundary operator $\partial\colon \mathcal D_{m+1}\to \mathcal D_m$ by \[ \partial T(\omega) := T(d\omega). \] We will see that currents represent a generalization of $m$-surfaces. In fact if $M$ is a compact $m$-dimensional oriented manifold with boundary, we can associate to $M$ the current $[[M]]$ defined by \[ [[M]](\omega)=\int_M \omega. \] So the definition of boundary $\partial T$ of a current, is justified by Stokes Theorem: \[ \int_{\partial M} \omega = \int_M d\omega. \] The space $\mathcal D_m$ of $m$-dimensional currents is a real vector space with operations defined by \[ (T+S)(\omega):= T(\omega)+S(\omega),\qquad (\lambda T)(\omega):=\lambda T(\omega). \] The sum of two currents represents the \emph{union} of the surfaces they represents. Multiplication by a scalar represents a change in the \emph{multiplicity} of the surface. In particular multiplication by $-1$ represents the change of orientation of the surface. We define the \emph{support} of a current $T$, denoted by $\mathrm{spt}(T)$, the smallest closed set $C$ such that \[ T(\omega)=0\ \text{whenever $\omega=0$ on $C$}. \] We denote with $\mathcal E_m$ the vector subspace of $\mathcal D_m$ of currents with compact support. \section*{Topology} The space of currents is naturally endowed with the \emph{weak-star} topology, which will be further simply called \emph{weak convergence}. We say that a sequence $T_k$ of currents, weakly converges to a current $T$ if \[ T_k(\omega) \to T(\omega),\qquad \forall \omega. \] A stronger norm on the space of currents is the \emph{mass norm}. First of all we define the mass norm of a $m$-form $\omega$ as \[ ||\omega||:= \sup\{|\langle \omega,\xi\rangle|\colon \text{$\xi$ is a unit, simple, $m$-vector}\}. \] So if $\omega$ is a simple $m$-form, then its mass norm is the usual norm of its coefficient. We hence define the \emph{mass} of a current $T$ as \[ \mathbf M (T) := \sup\{ T(\omega)\colon \sup_x ||\omega(x)||\le 1\}. \] The mass of a currents represents the \emph{area} of the generalized surface. An intermediate norm, is the \emph{flat norm} defined by \[ \mathbf F (T) := \inf \{\mathbf M(A) + \mathbf M(B) \colon T= A + \partial B,\ A\in\mathcal E_m,\ B\in\mathcal E_{m+1}\}. \] Notice that two currents are close in the mass norm if they coincide apart from a small part. On the other hand the are close in the flat norm if they coincide up to a small deformation. \section*{Examples} Recall that $\Lambda_c^0(\R^n)\equiv C^\infty_c(\R^n)$ so that the following defines a $0$-current: \[ T(f) = f(0). \] In particuar every signed measure $\mu$ with finite mass 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: \[ T(a\,dx\wedge dy + b\,dy\wedge dz + c\,dx\wedge dz) = \int_0^1 \int_0^1 b(x,y,0)\, dx \, dy. \]