A Scalar Associated with the Inverse of Some Abelian Integrals and a Ramified Riemann Domain release_akettv65dzce5nymew3eczfvda

by Junjiro Noguchi

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2015  

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 ∈Ω to the boundary Ω. We prove an 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 ^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 ^n.
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Date   2015-02-05
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arXiv  1502.01548v1
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