# rectifiable current

An $m$-dimensional *rectifiable current ^{}* is a current $T$ whose action against a $m$-form $\omega $ can be written as

$$T(\omega )={\int}_{S}\theta (x)\u27e8\xi (x),\omega (x)\u27e9\mathit{d}{\mathscr{H}}^{m}(x)$$ |

where $S$ is an $m$-dimensional bounded rectifiable set, $\xi $ is an orientation of $S$ i.e. $\xi (x)$ is a unit $m$-vector representing the approximate tangent plane of $S$ at $x$ for ${\mathscr{H}}^{m}$-a.e. $x\in S$ and, finally, $\theta (x)$ is an integer valued measurable function^{} defined a.e. on $S$ (called *multiplicity*).
The space of $m$-dimensional rectifiable currents is denoted by ${\mathcal{R}}_{m}$.

An $m$-dimensional rectifiable current $T$ such that the boundary $\partial T$ is itself an $(m-1)$-dimensional rectifiable current, is called *integral current*. The space of integral currents is denoted by ${\mathbf{I}}_{m}$.
We point out that the word “integral^{}” refers to the fact that the multiplicity $\theta $ is integer valued.

Also notice that rectifiable and integral currents are not vector subspaces of the space of currents. In fact while the sum of two rectifiable currents is again a rectifiable current, the multiplication by a real number gives a rectifiable current only if the number is an integer.

The compactness theorem makes the space of integral currents a good space where geometric problems can be ambiented.

On rectifiable currents one can define an *integral flat norm*

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

The closure of the space ${\mathcal{R}}_{m}$ under the integral flat norm is called the space of *integral flat chains* and is denoted by ${\mathcal{F}}_{m}$.

As a consequence of the closure theorem, one finds that $$ where $\mathbf{M}$ is the mass norm of a current.

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

Canonical name | RectifiableCurrent |

Date of creation | 2013-03-22 14:28:34 |

Last modified on | 2013-03-22 14:28:34 |

Owner | paolini (1187) |

Last modified by | paolini (1187) |

Numerical id | 4 |

Author | paolini (1187) |

Entry type | Definition |

Classification | msc 58A25 |

Defines | integral current |

Defines | integral flat norm |

Defines | integral flat chains |