A new characterization of $L^2$-domains of holomorphy with null thin complements via $L^2$-optimal conditions

Abstract

In this paper, we show that the $L^2$-optimal condition implies the $L^2$-divisibility of $L^2$-integrable holomorphic functions. As an application, we offer a new characterization of bounded $L^2$-domains of holomorphy with null thin complements using the $L^2$-optimal condition, which appears to be advantageous in addressing a problem proposed by Deng-Ning-Wang. Through this characterization, we show that a domain in a Stein manifold with a null thin complement, admitting an exhaustion of complete K\"ahler domains, remains Stein. By the way, we construct an $L^2$-optimal domain that does not admit any complete K\"ahler metric.