rectifiable current

An m-dimensional rectifiable currentMathworldPlanetmath is a current T whose action against a m-form ω can be written as


where S is an m-dimensional bounded rectifiable set, ξ is an orientation of S i.e. ξ(x) is a unit m-vector representing the approximate tangent plane of S at x for m-a.e. xS and, finally, θ(x) is an integer valued measurable functionMathworldPlanetmath defined a.e. on S (called multiplicity). The space of m-dimensional rectifiable currents is denoted by m.

An m-dimensional rectifiable current T such that the boundary T is itself an (m-1)-dimensional rectifiable current, is called integral current. The space of integral currents is denoted by 𝐈m. We point out that the word “integralDlmfPlanetmath” refers to the fact that the multiplicity θ 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


The closure of the space m under the integral flat norm is called the space of integral flat chains and is denoted by m.

As a consequence of the closure theorem, one finds that m={Tm:𝐌(T)<} where 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