Conformal Completion of $\mathbb{U}(n)$-invariant Ricci-Flat Kãhler Metrics at Infinity

Abstract

For every $n\ge 2$ we give an example of a complete $\mathbb{U}(n)$-invariant cohomogeneity one metric on ${\mathbb{R}}^{2n}$ which is not conformally flat and which carries twistor spinors with zeros. The construction uses a conformal completion at infinity of a $\mathbb{U}(n)$-invariant Ricci-flat Kähler metric on ${\mathbb{R}}^{2n}\backslash \{0\}$ given by Calabi [2] and by Freedman and Gibbons [4]. This extends our results in [6) for $n=2$ to all even dimensions.