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

TL;DR

This paper demonstrates the existence of Lipschitz plurisubharmonic exhaustion functions for Lipschitz pseudoconvex domains in complex projective space, extending previous results to less regular domains using Takeuchi's Theorem.

Contribution

It generalizes the existence of plurisubharmonic exhaustion functions from smooth to Lipschitz pseudoconvex domains in $ ext{CP}^n$, and provides a counterexample regarding boundary defining functions.

Findings

Existence of Lipschitz defining functions with plurisubharmonic powers

Extension of Ohsawa and Sibony's results to Lipschitz domains

Counterexample showing limitations of boundary distance functions

Abstract

In this paper, we use Takeuchi's Theorem to show that for every Lipschitz pseudoconvex domain $\Omega$ in $\mathbb{CP}^n$ there exists 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 $\Omega$.