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<p>We show that the pluripotential Cauchy-Dirichlet problem for the complex Monge-Ampère flow is solvable for the right-hand side of the form <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2193_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">\(dt \wedge d\mu \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mi>t</mi> <mo>∧</mo> <mi>d</mi> <mi>μ</mi> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2193_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(d\mu \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mi>μ</mi> </mrow> </math></EquationSource> </InlineEquation> is dominated by a Monge-Ampère measure of a bounded plurisubharmonic function. In particular, we remove the strict positivity assumption on <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2193_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(d\mu \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mi>μ</mi> </mrow> </math></EquationSource> </InlineEquation>. We use this result to prove the parabolic version of the bounded subsolution theorem due to Kołodziej in pluripotential theory.</p>

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The Pluripotential Cauchy-Dirichlet Problem for the Complex Monge-Ampère Flow with A General Measure on the Right-Hand Side

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  • 发表日期：

    2025年9月11日

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  • Bowoo Kang

摘要

We show that the pluripotential Cauchy-Dirichlet problem for the complex Monge-Ampère flow is solvable for the right-hand side of the form \(dt \wedge d\mu \) d t ∧ d μ where \(d\mu \) d μ is dominated by a Monge-Ampère measure of a bounded plurisubharmonic function. In particular, we remove the strict positivity assumption on \(d\mu \) d μ . We use this result to prove the parabolic version of the bounded subsolution theorem due to Kołodziej in pluripotential theory.