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Seminar on Geometric Complex Analysis
Seminar information archive ～06/07｜Next seminar｜Future seminars 06/08～
| Date, time & place | Monday 10:30 - 12:00 128Room #128 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | Kengo Hirachi, Shigeharu Takayama |
2015/02/02
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Junjiro Noguchi (University of Tokyo)
Inverse of an Abelian Integral on open Riemann Surfaces and a Proof of Behnke-Stein's Theorem
Junjiro Noguchi (University of Tokyo)
Inverse of an Abelian Integral on open Riemann Surfaces and a Proof of Behnke-Stein's Theorem
[ Abstract ]
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.
