rectifiable current

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

$$T(\omega )={\int }_S\theta (x)⟨\xi (x),\omega (x)⟩𝑑{\mathscr{H}}^m(x)$$

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 ${ℛ}_m$.

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 ${𝐈}_m$.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

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

The closure of the space ${ℛ}_m$ under the integral flat norm is called the space of integral flat chains and is denoted by ${ℱ}_m$.

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