Asymptotic behaviour of the Bergman invariant and Kobayashi metric on exponentially flat infinite type domains

We prove the nontangential asymptotic limits of the Bergman canonical invariant, Ricci and Scalar curvatures of the Bergman metric, as well as the Kobayashi–Fuks metric, at exponentially flat infinite type boundary points of smooth bounded pseudoconvex domains in ${\mathbb{C}}^{n+1},n\in \mathbb{N}$. Additionally, we establish the nontangential asymptotic limit of the Kobayashi metric at exponentially flat infinite type boundary points of smooth bounded domains in ${\mathbb{C}}^{n+1},n\in \mathbb{N}$. We first show that these objects satisfy appropriate localizations and then utilize the method of scaling to complete the proofs.