Given a noncompact disconnected periodic curve <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_9993_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> of infinite length with two components and no self-intersection in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_9993_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb R^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="double-struck">R</mi> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation>, it is proved that there exists a noncompact simply connected periodic minimal surface spanning <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_9993_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>. As an application, it is shown that for any tetrahedron T with dihedral angles <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_9993_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$\le 90^\circ $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>≤</mo> <msup> <mn>90</mn> <mo>∘</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>, there exist four embedded minimal annuli in T, which are perpendicular to <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_9993_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$\partial T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation> along their boundary. It is also proved that every Platonic solid of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_9993_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb R^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="double-struck">R</mi> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> contains a free boundary embedded minimal surface of genus zero.

The periodic Plateau problem and its application

期刊论文
发表日期：

2025年4月13日

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Jaigyoung Choe

摘要

Given a noncompact disconnected periodic curve $\Gamma $ $Γ$ of infinite length with two components and no self-intersection in $\mathbb R^3$ $R^{3}$ , it is proved that there exists a noncompact simply connected periodic minimal surface spanning $\Gamma $ $Γ$ . As an application, it is shown that for any tetrahedron T with dihedral angles $\le 90^\circ $ $\leq 90^{\circ}$ , there exist four embedded minimal annuli in T, which are perpendicular to $\partial T$ $\partial T$ along their boundary. It is also proved that every Platonic solid of $\mathbb R^3$ $R^{3}$ contains a free boundary embedded minimal surface of genus zero.