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

TL;DR

This paper introduces a positive scalar function related to Abelian integrals to measure boundary distance in complex manifolds, providing new estimates and applications to the Levi problem and Steinness of Riemann surfaces.

Contribution

It defines a new scalar function $ ho(a, \Omega)$ associated with Abelian integrals and applies it to estimate holomorphic convexity and solve the Levi problem for ramified Riemann domains.

Findings

Established a Cartan--Thullen type estimate using $ ho(a, \Omega)$

Provided a new proof of Behnke-Stein's theorem on Stein Riemann surfaces

Derived geometric conditions for the Levi problem in ramified Riemann domains

Abstract

We introduce a positive scalar function $\rho(a, \Omega)$ for a domain $\Omega$ of a complex manifold $X$ with a global holomorphic frame of the cotangent bundle by closed Abelian differentials, which heuristically measure the distance from $a \in \Omega$ to the boundary $\del\Omega$. We prove an {\em estimate of Cartan--Thullen type with $\rho(a, \Omega)$} for holomorphically convex hulls of compact subsets. In one dimensional case, we apply the obtained estimate of $\rho(a, \Omega)$ 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 $\C^n$. 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 $\C^n$.