Xeometría extrínseca global de superficies

Abstract

[GL] O obxectivo deste traballo é facer un percorrido polos resultados máis importantes da xeometría diferencial dende un punto de vista extrínseco, ata chegar ao célebre Teorema de Alexandrov e ser tamén capaces de responder ao problema isoperimétrico. Comezamos definindo os conceptos básicos de 𝘰𝘷𝘢𝘭𝘰𝘪𝘥𝘦 𝘦 𝘥𝘰𝘮𝘪𝘯𝘪𝘰 𝘪𝘯𝘵𝘦𝘳𝘪𝘰𝘳, que nos permitirán adentrarnos no Teorema de Hadamard-Stoker, o cal nos brinda propiedades das superficies conexas, pechadas en 𝕽³ e con curvatura de Gauss positiva en tódolos puntos. A continuación preséntanse as Fórmulas de Minkowski e, consecuentemente, o Teorema de Hilbert-Liebmann e o de Jellet. Estes resultados, xunto coa Desigualdade de Heintze-Karcher, achégannos ao Teorema de Alexandrov, que nos di que se unha superficie compacta e conexa ten curvatura media constante, entón é unha esfera. Remataremos o noso estudo dando unha demostración do problema isoperimétrico no espazo euclidiano de dimensión tres, cuxa resposta xa era coñecida polos gregos aínda que non souberan xustificalo.

[EN] The aim of this work is to have a glimpse at the most important results of differential geometry from an extrinsic point of view. We will present the famous Alexandrov Theorem and tackle the isoperimetric problem. We start by defining the basic concepts of 𝘰𝘷𝘢𝘭𝘰𝘪𝘥 and 𝘪𝘯𝘯𝘦𝘳 𝘥𝘰𝘮𝘢𝘪𝘯, which will allow us to delve into the Hadamard-Stoker Theorem, that gives us properties of connected and closed surfaces in 𝕽³ with positive Gaussian curvature everywhere. Then we introduce the Minkowski Formulas and, as a consequence, the Hilbert-Liebmann and Jellet Theorems. These results, along with Heintze-Karcher Inequality, bring us closer to the Alexandrov Theorem, which states that a compact and connected surface with constant mean curvature is a sphere. We will finish our study by giving a proof of the isoperimetric problem in the three dimensional Euclidean space, the answer of which was already known to the Greeks even though they did not know how to justify it.