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<p>We present a renormalization lemma for certain maps defined on the unit disc of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2024_1185_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb {C}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">C</mi> </math></EquationSource> </InlineEquation> and taking values in some metric space. We show that the classical renormalization lemmas of Zalcman and Miniowitz can be deduced from our lemma. We also use it to establish a general normality statement for the Pinchuk’s scaling method in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2024_1185_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb {C}}^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> and, incidentally, reprove the Catlin’s estimates for the Kobayashi metric in finite type domains.</p>

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Zalcman’s renormalization lemma, Pinchuk’s rescaling method, and Catlin’s estimates revisited

  • 期刊论文

  • 发表日期：

    2025年1月8日

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  • François Berteloot

摘要

We present a renormalization lemma for certain maps defined on the unit disc of \({\mathbb {C}}\) C and taking values in some metric space. We show that the classical renormalization lemmas of Zalcman and Miniowitz can be deduced from our lemma. We also use it to establish a general normality statement for the Pinchuk’s scaling method in \({\mathbb {C}}^2\) C 2 and, incidentally, reprove the Catlin’s estimates for the Kobayashi metric in finite type domains.