Estimation of the Carathéodory distance on pseudoconvex domains of finite type whose boundary has Levi form of corank at most one (Q2859328)

For a bounded domain \(D\subset\mathbb C^n\), let \(c_D\), \(b_D\), and \(k_D\) denote, respectively, the Carathéodory distance, the Bergman distance, and the Kobayashi distance on \(D\). In the paper under review the author establishes precise estimates for \(c_D\), \(b_D\), and \(k_D\), where \(D\subset\mathbb C^n\) is a smoothly bounded pseudoconvex domain such that all its boundary points are of regular type and the Levi form of the boundary of \(D\) at each boundary point has at least \(n-2\) positive eigenvalues. He proves that there is a constant \(C>0\) such that \(C\rho\leq d_D\leq C^{-1}\rho\), where \(d\in\{c,b,k\}\) and the function \(\rho\) is given effectively in terms of the defining function of \(D\). The article is a continuation of the author's investigations from [Math. Z. 251, No. 3, 673--703 (2005; Zbl 1081.32007)], where similar estimates in \(\mathbb C^2\) were obtained.