Non-comparable invariant metrics and failure of Gromov hyperbolicity on a real-analytic pseudoconvex domain

Abstract

We study the explicit bounded pseudoconvex domain \[ Ω=\{(z_1,z_2,w)\in\mathbb C^3: |w|^2+|z_1z_2(z_1-z_2)|^2+|z_1|^8+|z_2|^8<1\}. \] Its boundary is real analytic and has finite D&#39;Angelo type everywhere. We prove two assertions about the invariant metrics of this domain. First, $(Ω,k_Ω)$ is not Gromov hyperbolic; in fact arbitrarily fat Kobayashi geodesic triangles occur in every neighborhood of the type-$8$ point $(0,0,1)$. Second, the Carathéodory--Reiffen and Kobayashi--Royden metrics are not uniformly comparable. Along an explicit family of tangential directions the Carathéodory metric tends to zero while the Kobayashi metric has a positive limit. For the corresponding distances there are points $x_j,y_j\to(0,0,1)$ such that $c_Ω(x_j,y_j)\to0$ while $k_Ω(x_j,y_j)\to\infty$. This gives, among bounded real-analytic pseudoconvex domains, an affirmative answer to Problem~7.3.2 of the Several Complex Variables Problem List.