Rotationally Symmetric Extremal Kähler Metrics on ${\mathbb{C}}^n$ and ${\mathbb{C}}^2\setminus \{0\}$

Abstract

In this paper, we study rotationally symmetric extremal Kähler metrics on ${\mathbb{C}}^n$ ($n\ge 2$) and ${\mathbb{C}}^2\backslash \{0\}$. 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 ${\mathbb{C}}^n$ with positive bisectional curvature, and we give a smooth extension lemma for $U(n)$ invariant extremal Kähler metrics on ${\mathbb{C}}^n\backslash \{0\}$. We retrieve known examples of smooth or singular extremal Kähler metrics on Hirzebruch surfaces, bundles over ${\mathbb{CP}}^1$, and weighted complex projective spaces. We also show that certain solutions on ${\mathbb{C}}^2\backslash \{0\}$ correspond to new complete families of constant-scalar-curvature Kähler and strictly extremal Kähler metrics on complex line bundles over ${\mathbb{CP}}^1$ and on ${\mathbb{C}}^2\backslash \{0\}$.

References in corpus (1)

• Existence of Gradient Kahler-Ricci Solitons