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)\langle\xi(x),\omega(x)\rangle d\mathcal{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 $\mathcal{H}^{m}$ -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)\colon T=A+\partial B,\quad 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 $\mathcal{R}_{m}=\{T\in\mathcal{F}_{m}\colon\mathbf{M}(T)<\infty\}$ 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