PlanetMath: rectifiable current

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rectifiable current

(Definition)

An -dimensional rectifiable current is a current whose action against a -form $\omega$ can be written as

$\displaystyle T(\omega) = \int_S \theta(x)\langle \xi(x),\omega(x)\rangle d\mathcal H^m(x)$

where

is an

-dimensional bounded rectifiable set, $\xi$ is an orientation of

i.e. $\xi(x)$ is a unit

-vector representing the approximate tangent plane of

for $\mathcal H^m$ -a.e. $x\in S$ and, finally, $\theta(x)$ is an integer valued measurable function defined a.e. on

(called multiplicity). The space of

-dimensional rectifiable currents is denoted by $\mathcal R_m$ .

An -dimensional rectifiable current such that the boundary $\partial T$ is itself an -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

$\displaystyle \mathcal F(T) := \inf \{\mathbf M(A) + \mathbf M(B)\colon T=A+\partial B,\quad A\in \mathcal R_m,\ B\in\mathcal R_{m+1}\}.$

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.

"rectifiable current" is owned by paolini.

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Also defines:

integral current, integral flat norm, integral flat chains

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Cross-references: norm, mass, consequence, closure, compactness, number, real number, multiplication, sum, vector subspaces, rectifiable, point, boundary, multiplicity, measurable function, integer, approximate tangent plane, unit, orientation, rectifiable set, bounded, action, current
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This is version 1 of rectifiable current, born on 2004-07-14.
Object id is 6001, canonical name is RectifiableCurrent.
Accessed 3748 times total.

Classification:

AMS MSC:

58A25 (Global analysis, analysis on manifolds :: General theory of differentiable manifolds :: Currents)

Pending Errata and Addenda

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