*Preprint*

**Inserted:** 5 feb 2012

**Last Updated:** 5 feb 2012

**Year:** 2012

**Abstract:**

A slice distance for the class of weak abelian $L^p$-bundles in $3$ dimensions was introduced in previous work with Tristan Rivière, where it was used to prove the closure of such class of bundles for the weak $L^p$-convergence. We further investigate this distance here, and we prove more properties of it, for example we show that it is Hölder-continuous on the slices. Using the same distance, we give here a notion of a boundary trace, giving a suitable setting for minimization problems on weak bundles. We then state some conjectures and some open questions.

**Keywords:**
Slice distance, trace definition, vectorfields with integer fluxes, topological

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