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| 開催情報 | 月曜日 10:30～12:00 数理科学研究科棟(駒場) 128号室 |
|---|---|
| 担当者 | 平地 健吾, 高山 茂晴 |
2015年02月02日(月)
10:30-12:00 数理科学研究科棟(駒場) 126号室
野口潤次郎 氏 (東京大学)
Inverse of an Abelian Integral on open Riemann Surfaces and a Proof of Behnke-Stein's Theorem
野口潤次郎 氏 (東京大学)
Inverse of an Abelian Integral on open Riemann Surfaces and a Proof of Behnke-Stein's Theorem
[ 講演概要 ]
Let X be an open Riemann surface and let Ω⋐ be a relatively compact domain of X. We firstly introduce a scalar function \rho(a, \Omega)>0 for a \in \Omega by means of an Abelian integral, which is a sort of convergence radius of the inverse of the Abelian integral, and heuristically measures the distance from a to the boundary \partial \Omega. We prove a theorem of Cartan-Thullen type with \rho(a, \Omega) for a holomorphically convex hull \hat{K}_\Omega of a compact subset K \Subset \Omega; in particular, -\log \rho(a, \Omega) is a continuous subharmonic function in \Omega. Secondly, we give another proof of Behnke-Stein's Theorem (the Steiness of X), one of the most basic facts in the theory of Riemann surfaces, by making use of the obtained theorem of Cartan--Thullen type with \rho(a, \Omega), and Oka's Jôku-Ikô together with Grauert's Finiteness Theorem which is now a rather easy consequence of Oka-Cartan's Fundamental Theorem, particularly in one dimensional case.
Let X be an open Riemann surface and let Ω⋐ be a relatively compact domain of X. We firstly introduce a scalar function \rho(a, \Omega)>0 for a \in \Omega by means of an Abelian integral, which is a sort of convergence radius of the inverse of the Abelian integral, and heuristically measures the distance from a to the boundary \partial \Omega. We prove a theorem of Cartan-Thullen type with \rho(a, \Omega) for a holomorphically convex hull \hat{K}_\Omega of a compact subset K \Subset \Omega; in particular, -\log \rho(a, \Omega) is a continuous subharmonic function in \Omega. Secondly, we give another proof of Behnke-Stein's Theorem (the Steiness of X), one of the most basic facts in the theory of Riemann surfaces, by making use of the obtained theorem of Cartan--Thullen type with \rho(a, \Omega), and Oka's Jôku-Ikô together with Grauert's Finiteness Theorem which is now a rather easy consequence of Oka-Cartan's Fundamental Theorem, particularly in one dimensional case.
