We construct K ̈ahler-Einstein metric with negative curvature near an isolated log canonical singularity by solving Monge-Amp`ere equation with Dirichlet boundary. We continue to consider the geometry of the Kahler-Einstein metric we constructed. In particular, in complex dimension 2, we show that all complete local K ̈ahler-Einstein metrics near isolated singularity are asymptotic the same as the model metric constructed by Kobayashi and Nakamura.

Link:  https://uci.zoom.us/j/91999884028