Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains
Let \(\Omega\) be a bounded strictly pseudoconvex domain in \({\mathbb C}^n, n\geq 2\), with \(C^2\)-smooth bondary. For any \(x\in\Omega\) choose a point \(\pi(x)\in\partial\Omega\) such that \(|x-\pi(x)|=\text{dist}(x,\partial\Omega)\) and, for \(x,y\in\Omega\), set \[ g(x,y)=2\log \left[{d_H(\pi(x),\pi(y))+ \max(\text{dist}(x,\partial\Omega)^{1/2}, \text{dist}(y,\partial\Omega)^{1/2})\over \sqrt{\text{dist}(x,\partial\Omega)^{1/2}\text{dist}(y,\partial\Omega)^{1/2}}}\right] \] where \(d_H\) is the Carnot-Carathéodory metric on \(\partial\Omega\). The authors prove that there exists a constant \(C\geq 0\) such that the Kobayashi distance \(d_K\) on \(\Omega\) satisfies the global estimates \[ g(x,y)-C\leq d_K(x,y)\leq g(x,y)+ C \tag \(*\) \] for \(x,y\in\Omega\). In fact this is a consequence of a more general result which asserts that if a Finsler metric \(F\) on \(\Omega\) satisfies certain local estimates near \(\partial\Omega\), the associated distance function \(d_F\) satisfies the global estimates \((*)\). Applying the estimates \((*)\), the authors prove that the metric space \((\Omega,d_K)\) is hyperbolic in the sense of M. Gromov and the boundary \(\partial_{G}\Omega\) of \((\Omega,d_K)\) as a Gromov hyperbolic space can be identified with the Euclidean boundary \(\partial\Omega\); moreover the Carnot-Carathéodory metric \(d_H\) on \(\partial\Omega\) lies in the canonical class of snowflake equivalent metrics on \(\partial_{G}\Omega\). This allows one to use general facts of the theory of Gromov hyperbolic space for studying strictly pseudoconvex domains. In particular, one can obtain in this way the well-known result that any proper holomorphic map between bounded strictly pseudoconvex domains extends continuously to the closure of the domains; actually, as the authors observe, the extension is Lipschitz with respect to the Carnot-Carathéodory metric on the boundaries.
scientific article; zbMATH DE number 1542503