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<p>We prove an expansion theorem for scalar-flat asymptotically conical (AC) Kähler metrics. The ADM mass is known to be defined on ALE Riemannian manifolds and can be generalized to AC cases. Consider an AC Kähler manifold asymptotic to a Ricci-flat Kähler metric cone with complex dimension <i>n</i>. Assuming the weak decay conditions required for the mass to be well-defined, each scalar-flat AC Kähler metric can be expanded. The dominant term of the expansion is the standard Kähler metric of the metric cone, and the leading error terms decay as <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(O(r^{2-2n})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>r</mi> <mrow> <mn>2</mn> <mo>-</mo> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with a coefficient only depending on the ADM mass and its dimension. Besides, the mass formula by Hein–LeBrun (Commun Math Phys 347(1):183–221, 2016) can be proved in our setting. As an interesting application, a new version of the positive mass theorem will be discussed in the cases of the resolutions of the Ricci-flat Kähler cones.</p>

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Mass and expansion of asymptotically conical Kähler metrics

  • 期刊论文

  • 发表日期：

    2025年10月13日

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  • Qi Yao

摘要

We prove an expansion theorem for scalar-flat asymptotically conical (AC) Kähler metrics. The ADM mass is known to be defined on ALE Riemannian manifolds and can be generalized to AC cases. Consider an AC Kähler manifold asymptotic to a Ricci-flat Kähler metric cone with complex dimension n. Assuming the weak decay conditions required for the mass to be well-defined, each scalar-flat AC Kähler metric can be expanded. The dominant term of the expansion is the standard Kähler metric of the metric cone, and the leading error terms decay as \(O(r^{2-2n})\) O ( r 2 - 2 n ) with a coefficient only depending on the ADM mass and its dimension. Besides, the mass formula by Hein–LeBrun (Commun Math Phys 347(1):183–221, 2016) can be proved in our setting. As an interesting application, a new version of the positive mass theorem will be discussed in the cases of the resolutions of the Ricci-flat Kähler cones.