El problema de Hopf para superficies de Weingarten

Abstract

The aim of this work is to study Hopf’s theorem, a classical result which characterizes constant mean curvature spheres in R 3 , as well as its generalizations. Namely, the extension to special Weingarten surfaces by Hartman and Wintner and the generalization to spheres immersed in the 3-dimensional spaces E 3 (κ, τ ) by Abresch and Rosenberg. We will also discuss the Hopf problem for Weingarten surfaces in E 3 (κ, τ ). In the following chapter, we will review concepts from Riemannian geometry, introduce Weingarten and Riemann surfaces and define the concept of line fields on a manifold. Additionally, we will present the 2-parameter family of E 3 (κ, τ ) spaces. We will prove Hopf’s theorem in the second chapter. In order to do so, we will use two different techniques: on the one hand, we will study the line field of curvatures of a CMC sphere. On the other hand, we will introduce the Hopf differential, a 2-form whose zeros correspond to the set of umbilical points of a surface. The third chapter will deal with the extension of Hopf’s theorem to special Weingarten surfaces by Hartman and Wintner. Additionally, we will show an alternative proof by Shiing Shen Chern. In the fourth chapter we will show the key steps of the extension of Hopf’s theorem to E 3 (κ, τ ) spaces by Abresch and Rosenberg. In addition, we will show that Hopf’s original proof can be applied to space forms M3 (κ). Finally, in the last chapter we will discuss the latest achievements in the Hopf problem for special Weingarten surfaces in E 3 (κ, τ ) spaces, emphasizing a recent result by Jos´e Antonio G´alvez and Pablo Mira.

Quisiera aprovechar este espacio para dar las gracias a dos personas que han sido fundamentales en la realizaci´on de esta memoria. En primer lugar, quer´ıa agradecer a Isabel Fern´andez su implicaci´on, amabilidad y dedicaci´on no solo como tutora en este trabajo, sino tambi´en como directora en los distintos proyectos acad´emicos en los que he tenido la oportunidad de participar en los ´ultimos a˜nos. La considero una referente como investigadora, docente y por supuesto, como persona. De igual manera, quer´ıa agradecer a Pablo Mira, a quien he tenido el placer de conocer recientemente, su compromiso y cercan´ıa, a pesar de las dificultades que supone orientar un Trabajo de Fin de M´aster desde la distancia. Considero que sus conocimientos y sugerencias han sido claves para permitirme obtener una visi´on m´as global de esta l´ınea de investigaci´on. Asimismo, quiero agradecer a ambos su enorme predisposici´on a la hora de orientarme de cara a mi pr´oxima etapa como estudiante predoctoral. Unos a˜nos atr´as, me habr´ıa parecido imposible imaginar que tendr´ıa la suerte de trabajar con personas tan brillantes.