An interesting open problem in complex geometry is to find Kähler metrics with prescribed curvature properties. A typical example of this problem is to find constant scalar curvature metrics on a Kähler manifold, which defines a fourth order elliptic PDE. A natural generalization of this problem is to study metrics with prescribed coefficients of the TYCZ-expansion of the epsilon-function. The talk introduces the epsilon function and its asymptotic expansion for compact and noncompact manifolds, focusing on the geometric properties related to the vanishing of the third coefficient for Kähler-Einstein metrics.