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Abstract

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