A New Characterization of \(L^2\)-Domains of Holomorphy with Null Thin Complements via \(L^2\)-Optimal Conditions

In this paper, we show that the \(L^2\) $L^2$ -optimal condition implies the \(L^2\) $L^2$ -divisibility of \(L^2\) $L^2$ -integrable holomorphic functions. As an application, we offer a new characterization of bounded \(L^2\) $L^2$ -domains of holomorphy with null thin complements using the \(L^2\) $L^2$ -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 Kähler domains, remains Stein. By the way, we construct an \(L^2\) $L^2$ -optimal domain that does not admit any complete Kähler metric.