<p>We extend the classical Plateau–Rayleigh instability criterion in the <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2829_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {E}(\kappa ,\tau )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">E</mi> <mo stretchy="false">(</mo> <mi>κ</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> spaces. We prove the existence of a positive number <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2829_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$L_0>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>L</mi> <mn>0</mn> </msub> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> such that if a truncated circular cylinder of radius <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2829_Article_IEq6.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2829_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {E}(\kappa ,\tau )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">E</mi> <mo stretchy="false">(</mo> <mi>κ</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> has length <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2829_Article_IEq8.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$L>L_0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo>></mo> <msub> <mi>L</mi> <mn>0</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, then it is unstable. This number <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2829_Article_IEq9.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$L_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> depends on <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2829_Article_IEq10.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2829_Article_IEq11.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">$\tau $</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2829_Article_IEq6.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation>. The value <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2829_Article_IEq9.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$L_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> is sharp under axially symmetric variations of the surface. We also extend this result for the partitioning problem in <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2829_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {E}(\kappa ,\tau )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">E</mi> <mo stretchy="false">(</mo> <mi>κ</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

Stability of Cylinders in $\mathbb {E}(\kappa ,\tau )$ Homogeneous Spaces

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2025年3月21日

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Antonio Bueno,
Rafael López

摘要

We extend the classical Plateau–Rayleigh instability criterion in the $\mathbb {E}(\kappa ,\tau )$ $E (κ, τ)$ spaces. We prove the existence of a positive number $L_0>0$ $L_{0} > 0$ such that if a truncated circular cylinder of radius $\rho $ $ρ$ in $\mathbb {E}(\kappa ,\tau )$ $E (κ, τ)$ has length $L>L_0$ $L > L_{0}$ , then it is unstable. This number $L_0$ $L_{0}$ depends on $\kappa $ $κ$ , $\tau $ $τ$ and $\rho $ $ρ$ . The value $L_0$ $L_{0}$ is sharp under axially symmetric variations of the surface. We also extend this result for the partitioning problem in $\mathbb {E}(\kappa ,\tau )$ $E (κ, τ)$ .

Stability of Cylinders in \(\mathbb {E}(\kappa ,\tau )\) Homogeneous Spaces

摘要