<p>Self-shrinkers, originating from the singularities of the mean curvature flows, satisfy a non-linear second order elliptic equation involving the mean curvature. The purpose of this article is to study the solution of the non-linear second order elliptic equation of self-shrinkers. Though developing a concise method for solving the corresponding complicated system of equation concerning self-shrinkers, we are able to show that any linear Weingarten self-shrinkers in <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1714_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {R}}^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> is congruent to a plane <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1714_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb R^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="double-struck">R</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>, a cylinder <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1714_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$S^1(1)\times {\mathbb {R}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>S</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>, the round sphere <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1714_Article_IEq7.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$S^2(\sqrt{2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>S</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msqrt> <mn>2</mn> </msqrt> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> or a generalized cylinder <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1714_Article_IEq8.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$\Gamma \times {\mathbb {R}} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Γ</mi> <mo>×</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1714_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> is a generalized Abresch–Langer curve.</p>

Linear Weingarten self-shrinkers in ${\mathbb {R}}^3$

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发表日期：

2025年3月13日

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Dan Yang,
Yu Fu

摘要

Self-shrinkers, originating from the singularities of the mean curvature flows, satisfy a non-linear second order elliptic equation involving the mean curvature. The purpose of this article is to study the solution of the non-linear second order elliptic equation of self-shrinkers. Though developing a concise method for solving the corresponding complicated system of equation concerning self-shrinkers, we are able to show that any linear Weingarten self-shrinkers in ${\mathbb {R}}^3$ $R^{3}$ is congruent to a plane $\mathbb R^2$ $R^{2}$ , a cylinder $S^1(1)\times {\mathbb {R}}$ $S^{1} (1) \times R$ , the round sphere $S^2(\sqrt{2})$ $S^{2} (\sqrt{2})$ or a generalized cylinder $\Gamma \times {\mathbb {R}} $ $Γ \times R$ , where $\Gamma $ $Γ$ is a generalized Abresch–Langer curve.

Linear Weingarten self-shrinkers in \({\mathbb {R}}^3\)

摘要