# current

Let ${\mathrm{\Lambda}}_{c}^{m}({\mathbb{R}}^{n})$ denote the space of ${C}^{\mathrm{\infty}}$ differentiable^{} $m$-forms with compact support in ${\mathbb{R}}^{n}$. A continuous linear operator $T:{\mathrm{\Lambda}}_{c}^{m}({\mathbb{R}}^{n})\to \mathbb{R}$ is called an $m$-*current*. Let ${\mathcal{D}}_{m}$ denote the space of $m$-currents in ${\mathbb{R}}^{n}$.
We define a boundary operator^{} $\partial :{\mathcal{D}}_{m+1}\to {\mathcal{D}}_{m}$ 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:

$${\int}_{\partial M}\omega ={\int}_{M}\mathit{d}\omega .$$ |

The space ${\mathcal{D}}_{m}$ of $m$-dimensional currents is a real vector space with operations^{} defined by

$$(T+S)(\omega ):=T(\omega )+S(\omega ),(\lambda T)(\omega ):=\lambda T(\omega ).$$ |

The sum of two currents represents the *union* of the surfaces they represents. Multiplication by a scalar represents a change in the *multiplicity* of the surface. In particular multiplication by $-1$ represents the change of orientation of the surface.

We define the *support ^{}* of a current $T$, denoted by $\mathrm{spt}(T)$, the smallest closed set

^{}$C$ such that

$$T(\omega )=0\text{whenever}\omega =0\text{on}C.$$ |

We denote with ${\mathcal{E}}_{m}$ the vector subspace of ${\mathcal{D}}_{m}$ of currents with compact support.

## Topology

The space of currents is naturally endowed with the *weak-star* topology^{}, which will be further simply called *weak convergence*. We say that a sequence^{} ${T}_{k}$ of currents, weakly converges^{} to a current $T$ if

$${T}_{k}(\omega )\to T(\omega ),\forall \omega .$$ |

A stronger norm on the space of currents is the *mass norm*. First of all we define the mass norm of a $m$-form $\omega $ as

$$||\omega ||:=sup\{|\u27e8\omega ,\xi \u27e9|:\xi \text{is a unit, simple,}m\text{-vector}\}.$$ |

So if $\omega $ is a simple $m$-form, then its mass norm is the usual norm of its coefficient. We hence define the *mass* of a current $T$ as

$$\mathbf{M}(T):=sup\{T(\omega ):\underset{x}{sup}||\omega (x)||\le 1\}.$$ |

The mass of a currents represents the *area* of the generalized surface.

An intermediate norm, is the *flat norm* defined by

$$\mathbf{F}(T):=inf\{\mathbf{M}(A)+\mathbf{M}(B):T=A+\partial B,A\in {\mathcal{E}}_{m},B\in {\mathcal{E}}_{m+1}\}.$$ |

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.

## Examples

Recall that ${\mathrm{\Lambda}}_{c}^{0}({\mathbb{R}}^{n})\equiv {C}_{c}^{\mathrm{\infty}}({\mathbb{R}}^{n})$ 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:

$$T(f)=\int f(x)\mathit{d}\mu (x).$$ |

Let $(x,y,z)$ be the coordinates in ${\mathbb{R}}^{3}$. Then the following defines a $2$-current:

$$T(adx\wedge dy+bdy\wedge dz+cdx\wedge dz)={\int}_{0}^{1}{\int}_{0}^{1}b(x,y,0)\mathit{d}x\mathit{d}y.$$ |

Title | current |
---|---|

Canonical name | Current |

Date of creation | 2013-03-22 14:27:39 |

Last modified on | 2013-03-22 14:27:39 |

Owner | paolini (1187) |

Last modified by | paolini (1187) |

Numerical id | 7 |

Author | paolini (1187) |

Entry type | Definition |

Classification | msc 58A25 |

Defines | mass |

Defines | support |