<p>Fukaya and Oh studied the correspondence between pseudoholomorphic disks in <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1127_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(T^{*}M\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mi>T</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mi>M</mi> </mrow> </math></EquationSource> </InlineEquation> which are bounded by Lagrangian sections <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1127_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(\{L_{i}^{\epsilon }\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msubsup> <mi>L</mi> <mrow> <mi>i</mi> </mrow> <mi>ϵ</mi> </msubsup> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> and gradient trees in <i>M</i> which consist of gradient curves of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1127_Article_IEq7.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">\(\{f_{i}-f_{j}\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>f</mi> <mi>j</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. Here, <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1127_Article_IEq8.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(L_{i}^{\epsilon }\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>L</mi> <mrow> <mi>i</mi> </mrow> <mi>ϵ</mi> </msubsup> </math></EquationSource> </InlineEquation> is defined by <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1127_Article_IEq9.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">\(L_{i}^{\epsilon }=\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mrow> <mi>i</mi> </mrow> <mi>ϵ</mi> </msubsup> <mo>=</mo> </mrow> </math></EquationSource> </InlineEquation>&#xa0;graph<InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1127_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">\((\epsilon df_{i})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mi>d</mi> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. They constructed approximate pseudoholomorphic disks in the case <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1127_Article_IEq11.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">\(\epsilon &gt;0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is sufficiently small. When <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1127_Article_IEq12.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">\(M=\mathbb {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo>=</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> and Lagrangian sections are affine, pseudoholomorphic disks <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1127_Article_IEq13.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(w_{\epsilon }\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>w</mi> <mi>ϵ</mi> </msub> </math></EquationSource> </InlineEquation> can be constructed explicitly. In this paper, we show that pseudoholomorphic disks <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1127_Article_IEq13.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(w_{\epsilon }\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>w</mi> <mi>ϵ</mi> </msub> </math></EquationSource> </InlineEquation> converges to the gradient tree in the limit <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1127_Article_IEq15.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">\(\epsilon \rightarrow +0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> when the number of Lagrangian sections is three and four.</p>

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Explicit correspondences between gradient trees in \(\mathbb {R}\) and holomorphic disks in \(T^{*}\mathbb {R}\)

  • Hidemasa Suzuki

摘要

Fukaya and Oh studied the correspondence between pseudoholomorphic disks in \(T^{*}M\) T M which are bounded by Lagrangian sections \(\{L_{i}^{\epsilon }\}\) { L i ϵ } and gradient trees in M which consist of gradient curves of \(\{f_{i}-f_{j}\}\) { f i - f j } . Here, \(L_{i}^{\epsilon }\) L i ϵ is defined by \(L_{i}^{\epsilon }=\) L i ϵ =  graph \((\epsilon df_{i})\) ( ϵ d f i ) . They constructed approximate pseudoholomorphic disks in the case \(\epsilon >0\) ϵ > 0 is sufficiently small. When \(M=\mathbb {R}\) M = R and Lagrangian sections are affine, pseudoholomorphic disks \(w_{\epsilon }\) w ϵ can be constructed explicitly. In this paper, we show that pseudoholomorphic disks \(w_{\epsilon }\) w ϵ converges to the gradient tree in the limit \(\epsilon \rightarrow +0\) ϵ + 0 when the number of Lagrangian sections is three and four.