Quasiregular Curves: Removability of Singularities

Abstract We prove a Painlevé 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 injectivity radius bounded from below. As an application, we extend a theorem of Bonk and Heinonen for quasiregular mappings to the setting of quasiregular curves: every non-constant quasiregular $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> -curve from $$\mathbb {R}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> into $$( N, \omega )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where the bounded cohomology class of $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> is in the bounded Künneth ideal, has infinite energy.