Strong Localization of the Kobayashi-Eisenman Volume Element and Its Boundary Asymptotics

We establish a quantitative version of strong localization of the Kobayashi-Eisenman volume element and the quotient invariant near plurisubharmonic peak points of domains in ${\mathbb{C}}^n$. As an application of this strong localization result, we derive the non-tangential asymptotic limit of the Kobayashi-Eisenman volume element at exponentially flat infinite type boundary points of domains in ${\mathbb{C}}^{n+1}$.