<p>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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2146_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation>-curve from <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2146_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> into <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2146_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$( N, \omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>,</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where the bounded cohomology class of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2146_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation> is in the bounded Künneth ideal, has infinite energy.</p>

Quasiregular Curves: Removability of Singularities

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2025年8月18日

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Toni Ikonen

摘要

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 $ $ω$ -curve from $\mathbb {R}^n$ $R^{n}$ into $( N, \omega )$ $(N, ω)$ , where the bounded cohomology class of $\omega $ $ω$ is in the bounded Künneth ideal, has infinite energy.