El primer valor propio del operador de estabilidad para superficies compactas con curvatura media constante

Abstract

ABSTRACT - Objectives of the thesis The main aim of the thesis consists of obtaining upper bounds for the first eigenvalue of the Jacobi operator for compact surfaces of constant mean curvature into different ambient spaces: homogeneous spaces, Riemannian Killing submersions and warped products. Moreover, we also intend to deep into the geometry of the aforementioned ambient spaces, and we derive some results related to the stability of these surfaces. - Research methodology The methodology followed in preparing this thesis is the standard one in any research work in mathematics. A bibliographic review of the field of the thesis. Once we have known the literature, study and comprehension of the different published papers which determine the base of such a topic. The study, at the same time, of the useful tools and techniques in order to understand different works. Research stays with the end of discussing with experts of this subject, attendance to specific courses and conferences. Study of the problem by using the studied material, reflecting progresses in research papers to publish in international journals. Active participation in the geometry seminar which carries out in the research group in which I am included. In this seminar, we expose our results and problems. Daily discussions of the problems and progresses with the advisors of the thesis. - Results or conclusions of the thesis For the previously stated purpose, the first remarkable progress is that we find two general upper bounds for compact, two-sided surfaces of constant mean curvature in arbitrary 3-dimensional Riemannian manifolds. So, later on, we particularize such bounds to the case in which the ambient manifold is a homogeneous space, a Killing submersion or a warped product. Specifically, for the homogeneous 3-manifolds with isometry group of dimension 6 or 4, we give upper bounds for the first stability eigenvalue, but also we achieve to characterize some surfaces by means of this eigenvalue. For instance, we get a characterization of Hopf tori into certain Berger spheres. After studying homogeneous spaces with isometry group of dimension 4, it is natural to think of extending the obtained results to the case of Riemannian Killing submersions. These ambients are rather unknown, and for this reason, on the one hand we focus on the study of their geometry, giving some general formulae, and on the other hand we devote to our problem, that is, we find out estimates for the aforementioned eigenvalue. As remarkable results, it is important to emphasize that by using our bounds we are able to characterize both horizontal surfaces and Hopf tori. As a consequence, some restrictions over the mean curvature of compact, two-sided surfaces of constant mean curvature are derived when we assume that they are stable. To finish, we think about the same problem in the case in which the ambient is a warped product. In these spaces, we are going to work with a very natural convergence condition. Moreover, we impose certain conditions over the warping function such as being a solution of the Jacobi equation or a concave function. For these cases, we get complete characterizations of stable surfaces. Consequently, we believe that this thesis contributes to the study of constant mean curvature surfaces in different 3-dimensional manifolds, being this topic between the differential geometry and the geometric analysis. In fact, our contribution has been published in four research papers in international journals and a chapter of a book as it is shown in the references ([AMO], [MO1]-[MO4]) of the presented thesis.