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We introduce a positive scalar function $\rho(a, \Omega)$ for a domain$\Omega$ of a complex manifold $X$ with a global holomorphic frame of thecotangent bundle by closed Abelian differentials, which heuristically measurethe distance from $a \in \Omega$ to the boundary $\del\Omega$. We prove an {\emestimate of Cartan--Thullen type with $\rho(a, \Omega)$} for holomorphicallyconvex hulls of compact subsets. In one dimensional case, we apply the obtainedestimate of $\rho(a, \Omega)$ to give a new proof of Behnke-Stein's Theorem forthe Steiness of open Riemann surfaces. We then use the same idea to deal withthe Levi problem for ramified Riemann domains over $\C^n$. We obtain somegeometric conditions in terms of $\rho(a, X)$ which imply the validity of theLevi problem for a finitely sheeted Riemann domain over $\C^n$.

A Scalar Associated with the Inverse of Some Abelian Integrals and a Ramified Riemann Domain