The epsilon function and the coefficients of its asymptotic expansion for constant scalar curvature Kaehler metrics on noncompact manifolds

An important open problem in Kaehler geometry consists in characterizing projectively induced metrics in view of the properties of their curvatures. In the first part of this thesis we compute the third coefficient arising from the TYCZ-expansion of the epsilon function associated to a Kaehler-Einstein metric and discuss the consequences of its vanishing. In the second part of the thesis, in view of a better understanding of the geometry of scalar flat Kaehler metrics, we study two families of scalar flat Kaehler metrics constructed by A. D. Hwang and M. A. Singer 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 epsilon functions and we show that they can be approximated by a sequence of projectively induced Kaehler 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 TYCZ-expansion of the epsilon function vanishes.