<p>In this paper, we solve the geodesic equation in the space of Kähler metrics under the setting of asymptotically locally Euclidean (ALE) Kähler manifolds and establish global <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {C}^{1,1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">C</mi> </mrow> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> regularity of the solution. The solution of the geodesic equation is then related to the uniqueness of scalar-flat ALE metrics. To this end, we study the asymptotic behavior of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>-geodesics at spatial infinity. We will prove the convexity of Mabuchi <i>K</i> energy along <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>-geodesics under the assumption that the Ricci curvature of a reference ALE Kähler metric is non-positive. However, by testing the Ricci curvature of ALE Kähler metrics, we find that on the line bundle <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {O}(-k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mo>-</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> over <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {C}\mathbb {P}^{n-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">C</mi> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$n \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$k \ne n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≠</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>, all ALE Kähler metrics cannot have non-positive (or non-negative) Ricci curvature.</p>

Geodesic Equations on asymptotically locally Euclidean Kähler manifolds

期刊论文
发表日期：

2025年10月10日

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Qi Yao

摘要

In this paper, we solve the geodesic equation in the space of Kähler metrics under the setting of asymptotically locally Euclidean (ALE) Kähler manifolds and establish global $\mathcal {C}^{1,1}$ $C^{1, 1}$ regularity of the solution. The solution of the geodesic equation is then related to the uniqueness of scalar-flat ALE metrics. To this end, we study the asymptotic behavior of $\varepsilon $ $ε$ -geodesics at spatial infinity. We will prove the convexity of Mabuchi K energy along $\varepsilon $ $ε$ -geodesics under the assumption that the Ricci curvature of a reference ALE Kähler metric is non-positive. However, by testing the Ricci curvature of ALE Kähler metrics, we find that on the line bundle $\mathcal {O}(-k)$ $O (- k)$ over $\mathbb {C}\mathbb {P}^{n-1}$ $C P^{n - 1}$ with $n \ge 2$ $n \geq 2$ and $k \ne n$ $k \neq n$ , all ALE Kähler metrics cannot have non-positive (or non-negative) Ricci curvature.