On some connections between Kobayashi geometry and pluripotential theory

2025/05/22 by Bharali, Gautam, Masanta, Rumpa
#32F45 #32H35 #32U05 (Primary) 32T40 (Secondary) #Analysis of PDEs (math.AP) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.16949

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 ∂D of a proper holomorphic map F: D\longrightarrow Ω between domains with dim_ℂ(D) < dim_ℂ(Ω), 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.

Citations

• Characterizing domains by the limit set of their automorphism group
• Hölder regularity for solutions to complex Monge-Ampère equations
• On the Third Gap for Proper Holomorphic Maps between Balls

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