Bounded Plurisubharmonic Exhaustion Functions for Lipschitz Pseudoconvex Domains in $${\mathbb {C}}{\mathbb {P}}^n$$ C P n

Abstract

In this paper, we use Takeuchi’s Theorem to show that for every Lipschitz pseudoconvex domain \(\Omega \) in \({\mathbb {C}}{\mathbb {P}}^n\) there exist a Lipschitz defining function \(\rho \) and an exponent \(0<\eta <1\) such that \(-(-\rho )^\eta \) is strictly plurisubharmonic on \(\Omega \). This generalizes a result of Ohsawa and Sibony for \(C^2\) domains. In contrast to the Ohsawa–Sibony result, we provide a counterexample demonstrating that we may not assume \(\rho =-\delta \), where \(\delta \) is the geodesic distance function for the boundary of \(\Omega \).