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 \(\mathbb {C}^{n+1}\) and on \({\mathcal {O}}(-k)\) . For the metrics in both the families, we prove the existence of an asymptotic expansion for their \(\epsilon \) -functions and we show that they can be approximated by a sequence of projectively induced Kähler metrics. Further, we show that the metrics on \(\mathbb {C}^{n+1}\) are not projectively induced, and that the Burns–Simanca metric is characterized among the scalar flat metrics on \({\mathcal {O}}(-k)\) to be the only projectively induced one as well as the only one whose second coefficient in the asymptotic expansion of the \(\epsilon \) -function vanishes.
