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

Abstract: 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.