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

Abstract

We introduce a positive scalar function $ρ(a, Ω)$ for a domain $Ω$ of a complex manifold $X$ with a global holomorphic frame of the cotangent bundle by closed Abelian differentials, which heuristically measure the distance from $a \in Ω$ to the boundary $\delΩ$. We prove an {\em estimate of Cartan--Thullen type with $ρ(a, Ω)$} for holomorphically convex hulls of compact subsets. In one dimensional case, we apply the obtained estimate of $ρ(a, Ω)$ 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 $\C^n$. We obtain some geometric conditions in terms of $ρ(a, X)$ which imply the validity of the Levi problem for a finitely sheeted Riemann domain over $\C^n$.