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

We introduce a positive scalar function $\rho (a,\mathrm{\Omega })$ for a domain $\mathrm{\Omega }$ of a complex manifold $X$ with a global holomorphic frame of the cotangent bundle by closed Abelian differentials, which heuristically measure the distance from $a\in \mathrm{\Omega }$ to the boundary $\text{del}\mathrm{\Omega }$. We prove an {\em estimate of Cartan–Thullen type with $\rho (a,\mathrm{\Omega })$} for holomorphically convex hulls of compact subsets. In one dimensional case, we apply the obtained estimate of $\rho (a,\mathrm{\Omega })$ to give a new proof of Behnke-Stein’s Theorem for the Steiness of open Riemann surfaces. We then use the same idea to deal with the Levi problem for ramified Riemann domains over ${\text{C}}^n$. We obtain some geometric conditions in terms of $\rho (a,X)$ which imply the validity of the Levi problem for a finitely sheeted Riemann domain over ${\text{C}}^n$.