Quasiregular curves: Removability of singularities

TL;DR

This paper extends classical removability results for quasiregular mappings to quasiregular curves, proving a Painlevé theorem, establishing inequalities, and analyzing singularity removability in Euclidean and Riemannian settings.

Contribution

It introduces a Painlevé theorem for quasiregular curves, extending removability results and inequalities from quasiregular mappings to curves with applications in Riemannian geometry.

Findings

Proved a Painlevé theorem for bounded quasiregular curves.

Extended a fundamental volume form inequality to calibrations.

Established a sharp removability theorem for curves into manifolds with curvature bounds.

Abstract

We prove a Painlev\'e theorem for bounded quasiregular curves in Euclidean spaces extending removability results for quasiregular mappings due to Iwaniec and Martin. The theorem is proved by extending a fundamental inequality for volume forms to calibrations and proving a Caccioppoli inequality for quasiregular curves. We also establish a qualitatively sharp removability theorem for quasiregular curves whose target is a Riemannian manifold with sectional curvature bounded from above and an injectivity radius lower bound. As an application, we extend a theorem of Bonk and Heinonen for quasiregular mappings to the setting of quasiregular curves: every non-constant quasiregular $\omega$-curve from $\mathbb{R}^n$ into $( N, \omega )$, where the bounded cohomology class of $\omega$ is in the bounded K\"unneth ideal, has infinite energy.