Notes on a slice distance for singular Lp-bundles

Abstract

A slice distance for the class of weak abelian Lp-bundles in 3 dimensions was introduced in a previous article in collaboration with Tristan Rivi\`ere, where it was used to prove the closure of such class of bundles for the weak Lp-convergence. We further investigate this distance here, and we prove more properties of it, for example we show that it is H\"older-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.