Kähler Geometry of Scalar Flat Metrics on Line Bundles Over Polarized Kähler–Einstein Manifolds

In view of a better understanding of the geometry of scalar flat Kähler metrics, this paper studies two families of scalar flat Kähler metrics constructed by Hwang and Singer (Trans Am Math Soc 354(6):2285–2325, 2002) on C^{n+1} and on O(-k). For the metrics in both the families, we prove the existence of an asymptotic expansion for their ϵ-functions and we show that they can be approximated by a sequence of projectively induced Kähler metrics. Further, we show that the metrics on C^{n+1} are not projectively induced, and that the Burns–Simanca metric is characterized among the scalar flat metrics on O(-k) to be the only projectively induced one as well as the only one whose second coefficient in the asymptotic expansion of the ϵ-function vanishes.