Inverse of Abelian integrals and ramified Riemann domains

Abstract

We deal with the Levi problem (Hartogs’ inverse problem) for ramified Riemann domains by introducing a positive scalar function $$\rho (a, X)$$ for a complex manifold X with a global frame of the holomorphic cotangent bundle by closed Abelian differentials, which is an analogue of Hartogs’ radius. We obtain some geometric conditions in terms of $$\rho (a, X)$$ which imply the validity of the Levi problem for finitely sheeted ramified Riemann domains over $${\mathbf {C}}^n$$ . On the course, we give a new proof of the Behnke–Stein Theorem.