K$\ddot{\operatorname{a}}$hlerity of invariant metrics on pseudoconvex domain of dimension two

For a two dimensional bounded pseudoconvex domain of finite type, we prove uniformization theorems via K$\ddot{\operatorname{a}}$hler-Kobayashi metric or K$\ddot{\operatorname{a}}$hler-Carath$\acute{\operatorname{e}}$odory metric with quasi-finite geometry of order three. In particular, a pseudoconvex Reinhardt domain of finite type is the unit ball if and only if the Bergman metric is a scalar multiple of the Kobayashi metric or Carath$\acute{\operatorname{e}}$odory metric. Moreover, we establish a rigidity theorem concerning holomorphic sectional curvature of Bergman metric and Lu constant.