## 複素解析幾何セミナー

開催情報 月曜日　10:30～12:00　数理科学研究科棟(駒場) 128号室 平地 健吾, 高山 茂晴, 野村 亮介

### 2015年02月02日(月)

10:30-12:00   数理科学研究科棟(駒場) 126号室

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 $\Omega \Subset X$ 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.