On some connections between Kobayashi geometry and pluripotential theory

In this paper, we explore some connections between Kobayashi geometry and the Dirichlet problem for the complex Monge–Ampère equation. Among the results we obtain through these connections are: $(i)$~a theorem on the continuous extension up to $\mathrm{\partial }D$ of a proper holomorphic map $F:D⟶\mathrm{\Omega }$ between domains with ${\dim }_{\mathbb{C}}(D)<{\dim }_{\mathbb{C}}(\mathrm{\Omega })$, and $(ii)$~a result that establishes the existence of bounded domains with ``nice’’ boundary geometry on which Hölder regularity of the solutions to the complex Monge–Ampère equation fails. The first, a result in Kobayashi geometry, relies upon an auxiliary construction that involves solving the complex Monge–Ampère equation with Hölder estimates. The second result relies crucially on a bound for the Kobayashi metric.