Rotationally Symmetric Extremal Kähler Metrics on \mathbb Cⁿ and \mathbb C²∖ \0\

2021/05/30 by Selin Taşkent, Taskent, Selin
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2105.14561

openalex publication_date 2021/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study rotationally symmetric extremal Kähler metrics on \mathbb Cⁿ (n≥ 2) and \mathbb C² \backslash\0\. We present a classification of such metrics based on the zeros of the polynomial appearing in Calabi's Extremal Equation. As applications, we prove that there are no U(n) invariant complete extremal Kähler metrics on \mathbb Cⁿ with positive bisectional curvature, and we give a smooth extension lemma for U(n) invariant extremal Kähler metrics on ℂⁿ\backslash\0\. We retrieve known examples of smooth or singular extremal Kähler metrics on Hirzebruch surfaces, bundles over \mathbbCP¹, and weighted complex projective spaces. We also show that certain solutions on \mathbb C²\backslash\0\ correspond to new complete families of constant-scalar-curvature Kähler and strictly extremal Kähler metrics on complex line bundles over \mathbbCP¹ and on \mathbb C²\backslash\0\.

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