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<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10209_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">\((X,\omega )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a compact Kähler manifold of dimension <i>n</i> and fix an integer <i>m</i> such that <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10209_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">\(1\le m\le n\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>m</mi> <mo>≤</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>. In this note we give a characterization of finite energy range of <i>m</i>-Hessian operator, which generalizes Darvas-DiNezza-Lu’s characterization theorem on Monge-Ampère operator and also gives a different proof of their theorem. This can be viewed as a solution to a generalized problem of Guedj-Zeriahi on the range of Hessian operator.</p>

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On the Finite Energy Range of m-Hessian Operator

  • 期刊论文

  • 发表日期：

    2025年3月12日

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  • Genglong Lin

摘要

Let \((X,\omega )\) ( X , ω ) be a compact Kähler manifold of dimension n and fix an integer m such that \(1\le m\le n\) 1 ≤ m ≤ n . In this note we give a characterization of finite energy range of m-Hessian operator, which generalizes Darvas-DiNezza-Lu’s characterization theorem on Monge-Ampère operator and also gives a different proof of their theorem. This can be viewed as a solution to a generalized problem of Guedj-Zeriahi on the range of Hessian operator.