<p>Let <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1962_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega \subset {\mathbb {C}}^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> be a smooth domain. We establish conditions under which a weakly conformal, branched <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1962_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>-free boundary Hamiltonian stationary Lagrangian immersion <i>u</i> of a disc in <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1962_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {C}}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> is a <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1962_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>-free boundary minimal immersion. We deduce that if <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1962_Article_IEq7.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$u$</EquationSource> <EquationSource Format="MATHML"><math> <mi>u</mi> </math></EquationSource> </InlineEquation> is a weakly conformal, branched <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1962_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$B_1(0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>-free boundary Hamiltonian stationary Lagrangian immersion of a disc with Legendrian boundary, then <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1962_Article_IEq9.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$u(D^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <msup> <mi>D</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a Lagrangian equatorial plane disc. Furthermore, we present examples of <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1962_Article_IEq10.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>-free boundary Hamiltonian stationary discs, demonstrating the optimality of our assumptions.</p>

Free Boundary Hamiltonian Stationary Lagrangian Discs in ${\mathbb {C}}^2$

期刊论文
出版模式：

开放获取
发表日期：

2025年4月4日

查看全文

Filippo Gaia

摘要

Let $\Omega \subset {\mathbb {C}}^2$ $Ω \subset C^{2}$ be a smooth domain. We establish conditions under which a weakly conformal, branched $\Omega $ $Ω$ -free boundary Hamiltonian stationary Lagrangian immersion u of a disc in ${\mathbb {C}}^2$ $C^{2}$ is a $\Omega $ $Ω$ -free boundary minimal immersion. We deduce that if $u$ $u$ is a weakly conformal, branched $B_1(0)$ $B_{1} (0)$ -free boundary Hamiltonian stationary Lagrangian immersion of a disc with Legendrian boundary, then $u(D^2)$ $u (D^{2})$ is a Lagrangian equatorial plane disc. Furthermore, we present examples of $\Omega $ $Ω$ -free boundary Hamiltonian stationary discs, demonstrating the optimality of our assumptions.

Free Boundary Hamiltonian Stationary Lagrangian Discs in \({\mathbb {C}}^2\)

摘要