On some connections between Kobayashi geometry and pluripotential theory

Abstract

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 $\partial D$ of a proper holomorphic map $F:D⟶\Omega$ 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 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.

arXiv comment23 pages; restructured Section 1.2; added a new theorem in Section 1.2; to appear in Trans. Amer. Math. Soc. Comments welcome!