García Vázquez, Juan Carlos2024-12-232024-12-232024-07-10Carrasco Yllanes, V. (2024). Superficies de Riemann. (Trabajo Fin de Grado Inédito). Universidad de Sevilla, Sevilla.https://hdl.handle.net/11441/166117These notes are an introduction to the classic complex analysis object known as Riemann surfaces. The ultimate goal of this dissertation is to state and prove the Riemann Mapping Theorem. This result will allow us to classify simply connected Riemann Surfaces, asserting that every such surface is equivalent to one and only one of the following surfaces: the unit disc, the Riemann sphere or the complex plane. With this aim in mind, first we will define Riemann Surfaces and give some examples. Then we will study differentiable functions over these surfaces as well as some of their properties. We will stop to better understand analytic continutation and we will learn about germs and covering surfaces, giving mayor results about these subjets such as the Monodromy Theorem. Following that, we will dive into the world of subarmonics functions, familiarizing ourselves with Perron families, the Dirichlet problem in Riemann surfaces and Green’s functions. This last part will end with a classification of Riemann surfaces, paving the way for the main result of this work. Then we will finally be able to establish The Riemann Mapping Theorem and give its proof, concluding this dissertation.application/pdf98 p.spaAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/bachelorThesisinfo:eu-repo/semantics/openAccess