2015/02/05 by Junjirō Noguchi, Junjiro Noguchi, Noguchi, Junjiro
Mathematics · #32E05 #32E40 #32T05 #Abelian group #Complex Variables (math.CV) #Domain (mathematical analysis) #FOS: Mathematics #Geometry #Inverse #Mathematical analysis #Mathematical physics #Mathematics #Meromorphic and Entire Functions #Nonlinear Differential Equations Analysis #Pure mathematics #Riemann hypothesis #Scalar (mathematics) #advanced mathematical theories #math.CV #msc:32E05 #msc:32E40 #msc:32T05
paper · pdf · doi:10.48550/arxiv.1502.01548
20 pages
openalex publication_date 2015/02/05 · arxiv created 2015/04/25 · arxiv updated 2015/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ∈ 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 \Cn. 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 \Cn.