Abstract
This text reviews certain notions in metric geometry that may have further applications to problems in complex geometry and holomorphic dynamics in several variables. The discussion contains a few new results and formulates a number of questions related to the asymptotic geometry and boundary estimates of bounded complex domains, boundary extensions of biholomorphisms, the dynamics of holomorphic self-maps, Teichmüller theory, and the existence of constant scalar curvature metrics on compact Kähler manifolds.
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