I am reading the Geometric Measure Theory book by H. Federer and I have some questions about currents:

  1. Assuming $T \in \mathscr{D}_{m}(U),$ we call $T$ locally normal if and only if $T$ is representable by integration and either $\partial T$ is representable by integration or $m=0 .$ Furthermore call $T$ normal if and only if $T$ is locally normal and $\operatorname{spt}T$ is compact.

Now, I want to construct a locally normal current that is not normal! Does the following current work:

$\mathbf{E}^{n} \llcorner \psi$ corresponding to all weakly differentiable real-valued functions $\psi,$ with $$ \begin{aligned} \left[\partial\left(\mathbf{E}^{n} \llcorner \psi\right)\right](\phi) &=\int\left\langle e_{1} \wedge \cdots \wedge e_{n}, \psi(x) d \phi(x)\right\rangle d\left[\begin{array}{l} n \\ z \end{array}\right) \\ &=(-1)^{n-1} \int \psi(x) \operatorname{div} \xi(x) d \mathfrak{L}^{n} x \end{aligned} $$

  1. $M \subset U$ oriented submanifold, then there is a corresponding $n$-current $[M]$ defined by $$ [M](\omega)=\int_{M}\langle\omega(x), \xi(x)\rangle d H^{n}(x), \omega \in D^{n}(U) $$

Now, I want to calculate $\partial [M]$! Is it $\omega$ on $M$??

  • 1
    $\begingroup$ For the first one, why not just take an $m$-dimensional plane $P \subset U = \mathbf{R}^n$? Pick an orientation for it, then $[P] \in \mathcal{D}_m(U)$ has non-compact support. For the second, if $M$ has a manifold boundary $\partial M$ in $U$, then $\partial [M] = [\partial M]$. (In general the boundary of $[M]$ need not be representable by integration if $M$ were badly behaved.) $\endgroup$ – Leo Moos 2 days ago
  • $\begingroup$ Thank you so much @Leo moos. $\endgroup$ – Zeno cosini 2 days ago

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