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<p>Translators in the special linear group <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2376_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">\(\textrm{SL}(2,\mathbb {R})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>SL</mtext> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are surfaces whose mean curvature <i>H</i> and unit normal vector <i>N</i> satisfy <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2376_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="93" /> </InlineMediaObject> <EquationSource Format="TEX">\(H=\langle N,X\rangle \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo>=</mo> <mo stretchy="false">⟨</mo> <mi>N</mi> <mo>,</mo> <mi>X</mi> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation>, where <i>X</i> is a fixed Killing vector field. In this paper we study and classify those translators that are invariant by a one-parameter group of isometries. By the Iwasawa decomposition, there are three types of such groups. The dimension of the Killing vector fields is 4 and an exhaustive discussion is done for each one of the Killing vector fields and each of the invariant surfaces. In some cases, explicit parametrizations of translators are obtained.</p>

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Translators of the Mean Curvature Flow in the Special Linear Group \(\textrm{SL}(2,\mathbb {R})\)

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  • 发表日期：

    2025年2月28日

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  • Rafael López,
  • Marian Ioan Munteanu

摘要

Translators in the special linear group \(\textrm{SL}(2,\mathbb {R})\) SL ( 2 , R ) are surfaces whose mean curvature H and unit normal vector N satisfy \(H=\langle N,X\rangle \) H = ⟨ N , X ⟩ , where X is a fixed Killing vector field. In this paper we study and classify those translators that are invariant by a one-parameter group of isometries. By the Iwasawa decomposition, there are three types of such groups. The dimension of the Killing vector fields is 4 and an exhaustive discussion is done for each one of the Killing vector fields and each of the invariant surfaces. In some cases, explicit parametrizations of translators are obtained.