Uniformization Theorem of Riemann Surfaces

Ckakraborty, Subhajit (2023) Uniformization Theorem of Riemann Surfaces. Masters thesis, Indian Institute of Science Education and Research Kolkata.

Text (MS dissertation of Subhajit Ckakraborty (18MS063)) Thesis_18MS063.pdf - Submitted Version Restricted to Repository staff only Download (6MB)

Abstract

In this thesis, we will discuss Riemann surfaces that lie at the intersection of many di↵erent fields, such as di↵erential geometry, analysis, algebra, number theory, etc. One of the main reasons for being interested in Riemann surfaces is that the tools used to understand Riemann surfaces motivate much more general theories in di↵erent fields. In this thesis, we will largely focus our attention on those areas that lie at the intersection of analysis and di↵erential geometry, specifically speaking, Hodge decomposition in chapter 4 and the uniformization theorem in chapter 5 which is a kind of classification theorem for Riemann surfaces. For some concrete examples of Riemann surfaces, we would look at algebraic curves in chapter 3.

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