Curves between Lipschitz and $C^1$ and their relation to geometric knot theory

TL;DR

This paper explores the properties of regular curves with derivatives of vanishing mean oscillation, demonstrating smoothing techniques, approximations in energy spaces, and extending key results in geometric knot theory.

Contribution

It introduces methods to smooth such curves, proves approximation results for M"obius energy, and extends existing theorems in geometric knot theory.

Findings

Curves with finite M"obius energy can be approximated by smooth curves with energy convergence.

Extended $ ext{Gamma}$-convergence results for discrete M"obius energies.

Proved conjectures on decomposition of M"obius energy and extended inscribed polygon theorems.

Abstract

In this article we investigate regular curves whose derivatives have vanishing mean oscillations. We show that smoothing these curves using a standard mollifier one gets regular curves again. We apply this result to solve a couple of open problems. We show that curves with finite M\"obius energy can be approximated by smooth curves in the energy space $W^{\frac 32,2}$ such that the energy converges which answers a question of He. Furthermore, we extend the result of Scholtes on the $\Gamma$-convergence of the discrete M\"obius energies towards the M\"obius energy and prove conjectures of Ishizeki and Nagasawa on certain parts of a decomposition of the M\"obius energy. Finally, we extend a theorem of Wu on inscribed polygons to curves with derivatives with vanishing mean oscillation