Boundary behavior of the Kobayashi metric near a point of infinite type

Abstract

Under a potential-theoretical hypothesis named $f$-Property with $f$ satisfying $\displaystyle\int_t^\infty \dfrac{da}{a f(a)}<\infty$, we show that the Kobayashi metric $K(z,X)$ on a weakly pseudoconvex domain $\Om$, satisfies the estimate $K(z,X)\ge Cg(\delta_\Om(x)^{-1})|X|$ for any $X\in T^{1,0}\Om$ where $(g(t))^{-1}$ denotes the above integral and $\delta_\Om(z)$ is the distance from $z$ to $b\Om$.