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The Uniformization Theorem

Zozoulenko, Nikita LU (2021) In Bachelor's Theses in Mathematical Sciences MATK11 20211
Mathematics (Faculty of Engineering)
Mathematics (Faculty of Sciences)
Abstract
In this bachelor's thesis we provide a self contained proof of the Uniformization Theorem. It states that a simply connected Riemann surface is conformally equivalent to either the complex plane C, the unit disk D, or the Riemann sphere C^*. The proof given in this thesis is purely analytic and is based on the construction of Green's function and bipolar Green's function of a Riemann surface through the Perron method.
Popular Abstract (Swedish)
En myra som lever på en sfär kan förflytta sig i två dimensioner, framlänges och i sidled. Utan några kunskaper om den globala topologin av myrans värld, så kan den inte urskilja om den lever på en sfär eller i det tvådimensionella talplanet. På ett liknande sätt kan man inom matematiken formalisera en abstrakt topologisk yta som lokalt har samma egenskaper som det komplexa talplanet: En så kallad Riemannyta. Dessa ytor är viktiga objekt inom komplex analys och de används för att studera analytiska funktioner. I detta verk presenteras ett bevis för en känd sats som i stora drag säger att en Riemannyta utan ``hål'' antingen måste vara det komplexa talplanet, en cirkular skiva eller en sfär.
Please use this url to cite or link to this publication:
author
Zozoulenko, Nikita LU
supervisor
organization
course
MATK11 20211
year
type
M2 - Bachelor Degree
subject
keywords
complex analysis, riemann surface, analytic functions, holomorphic functions, differential geometry, harmonic, subharmonic, perron, analysis
publication/series
Bachelor's Theses in Mathematical Sciences
report number
LUNFMA-4110-2021
ISSN
1654-6229
other publication id
2021:K13
language
English
id
9045229
date added to LUP
2022-08-08 17:43:39
date last changed
2022-08-08 17:43:39
@misc{9045229,
  abstract     = {{In this bachelor's thesis we provide a self contained proof of the Uniformization Theorem. It states that a simply connected Riemann surface is conformally equivalent to either the complex plane C, the unit disk D, or the Riemann sphere C^*. The proof given in this thesis is purely analytic and is based on the construction of Green's function and bipolar Green's function of a Riemann surface through the Perron method.}},
  author       = {{Zozoulenko, Nikita}},
  issn         = {{1654-6229}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Bachelor's Theses in Mathematical Sciences}},
  title        = {{The Uniformization Theorem}},
  year         = {{2021}},
}