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We give an elementary proof to the asymptotic expansion formula of Rochon-Zhang for the unique complete Kähler-Einstein metric of Cheng-Yau, Kobayashi, Tian-Yau and Bando on quasi-projective manifolds. The main tools are the solution formula for second order ODE's with constant coefficients and spectral theory for Laplacian operator on a closed manifold.

We study the boundary behavior of solutions to the Loewner-Nirenberg problem in domains with conic singularities. To analyze the boundary behavior of solutions with respect to multiple normal directions, we first derive certain eigenvalue growth estimates for singular elliptic operators.