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