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

Junjiro Noguchi
2015 arXiv   pre-print
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
more » ... nn 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.
arXiv:1502.01548v2 fatcat:i2c5ngvo2fh4ln6undsmewqcva