<p>In this paper, we obtain a rigidity result of 2-dimensional complete lagrangian self-shrinkers with constant squared norm <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1904_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">$|\vec {H}|^{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <mover accent="true"> <mi>H</mi> <mo stretchy="false">→</mo> </mover> <msup> <mrow> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> of the mean curvature vector in the Euclidean space <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1904_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}^{4}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>4</mn> </msup> </math></EquationSource> </InlineEquation>. The same idea is also used to give a similar result of Lagrangian <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1904_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\xi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ξ</mi> </math></EquationSource> </InlineEquation>-submanifolds in <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1904_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}^{4}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>4</mn> </msup> </math></EquationSource> </InlineEquation>.</p>

The Rigidity Theorem for Complete Lagrangian Self-Shrinkers

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发表日期：

2025年1月21日

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Zhi Li,
Ruixin Wang,
Guoxin Wei

摘要

In this paper, we obtain a rigidity result of 2-dimensional complete lagrangian self-shrinkers with constant squared norm $|\vec {H}|^{2}$ $| \vec{H} |^{2}$ of the mean curvature vector in the Euclidean space $\mathbb {R}^{4}$ $R^{4}$ . The same idea is also used to give a similar result of Lagrangian $\xi $ $ξ$ -submanifolds in $\mathbb {R}^{4}$ $R^{4}$ .