Notes on a slice distance for singular Lp-bundles

TL;DR

This paper explores a slice distance for weak abelian Lp-bundles in 3D, establishing its properties, including Hölder continuity, and introduces a boundary trace concept to facilitate minimization problems.

Contribution

It extends previous work by analyzing the slice distance in detail, proving new properties, and proposing a boundary trace framework for weak bundles.

Findings

The slice distance is Hölder-continuous on slices.

A new boundary trace concept is introduced for minimization problems.

Open conjectures and questions are presented for future research.

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.