Orientador: Caio Jose Colletti Negreiros / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-10T10:04:33Z (GMT). No. of bitstreams: 1 Grama_LinoAndersondaSilva_M.pdf: 862309 bytes, checksum: ac6a88c1ff96ef74d7a840ce591336f5 (MD5) Previous issue date: 2008 / Resumo: O principal objetivo desta dissertação 'e apresentar a construção e a classificação das aplicações harmônicas de S2 em U(n), baseado nas idéias de K.Uhlenbeck. Apresentamos um exemplo de aplicação harmônica em U(4) e provamos que tal exemplo 'e, de fato, uma aplicação harmônica não-holomorfa na variedade de Grassman G2(C4), de 2-planos em C4.Demonstramos o teorema de Valli sobre o espectro da energia e, por fim, parametrizamos o conjunto Harm(S2, U(n)), de todas aplicações harmônicas de S2 em U(n), fornecendo uma classifica¸c¿ao para tais aplicações, seguindo o trabalho de J.C.Wood / Abstract: This dissertation is concerned with the construction and classification of harmonic maps from S2 on U(n), according to K. Uhlenbeck. We construct an example of harmonic map on U(4) and prove that this example is, in fact, a non-holomorphic harmonic map in the Grassmann manifold G2(C4) of 2-plans on C4. We also prove the theorem of Valli on the spectrum of energy and, finally, describe the arametrization of the space Harm(S2, U(n)), of all harmonics maps from S2 in U(n), provide the classification for such maps, following the work of J.C.Wood / Mestrado / Mestre em Matemática

The aim of this dissertation is to give explicit descriptions of the set of proper holomorphic mappings between two complex manifolds with reasonable restrictions on the domain and target spaces. Without any restrictions, this problem is intractable even when posed for do-mains in . We give partial results for special classes of manifolds. We study, broadly, two types of structure results: Descriptive. The first result of this thesis is a structure theorem for finite proper holomorphic mappings between products of connected, hyperbolic open subsets of compact Riemann surfaces. A special case of our result follows from the techniques used in a classical result due to Remmert and Stein, adapted to the above setting. However, the presence of factors that have no boundary or boundaries that consist of a discrete set of points necessitates the use of techniques that are quite divergent from those used by Remmert and Stein. We make use of a finiteness theorem of Imayoshi to deal with these factors. Rigidity. A famous theorem of H. Alexander proves the non-existence of non-injective proper holomorphic self-maps of the unit ball in . ,n >1. Several extensions of this result for various classes of domains have been established since the appearance of Alexander’s result, and it is conjectured that the result is true for all bounded domains in . , n > 1, whose boundary is C2-smooth. This conjecture is still very far from being settled. Our first rigidity result establishes the non-existence of non-injective proper holomorphic self-maps of bounded, balanced pseudo convex domains of finite type (in the sense of D’Angelo) in ,n >1. This generalizes a result in 2, by Coupet, Pan and Sukhov, to higher dimensions. As in Coupet–Pan–Sukhov, the aforementioned domains need not have real-analytic boundaries. However, in higher dimensions, several aspects of their argument do not work. Instead, we exploit the circular symmetry and a recent result in complex dynamics by Opshtein. Our next rigidity result is for bounded symmetric domains. We prove that a proper holomorphic map between two non-planar bounded symmetric domains of the same dimension, one of them being irreducible, is a biholomorphism. Our methods allow us to give a single, all-encompassing argument that unifies the various special cases in which this result is known. Furthermore, our proof of this result does not rely on the fine structure (in the sense of Wolf et al.) of bounded symmetric domains. Thus, we are able to apply our techniques to more general classes of domains. We illustrate this by proving a rigidity result for certain convex balanced domains whose automorphism groups are assumed to only be non-compact. For bounded symmetric domains, our key tool is that of Jordan triple systems, which is used to describe the boundary geometry.