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Master's Dissertation
Full name
Adam Petzet Rudnik
Knowledge Area
Date of Defense
São Paulo, 2019
Pereira, Antonio Luiz (President)
Barbosa, Ezequiel Rodrigues
Marrocos, Marcus Antonio Mendonça
Title in Portuguese
Análise do fluxo de Ricci em variedades compactas
Keywords in Portuguese
Fluxo de Ricci
Fluxo geométrico
Solitons de Ricci
Variedades compactas
Abstract in Portuguese
Um problema clássico em geometria é o de procurar métricas especiais numa variedade. Dentre os espaços mais simples encontram-se os de curvature (seccional) constante, a saber, os espaços modelo: S N , H N e o R N que são únicos, num certo contexto, a menos de isometrias. Em dimensão 2, nós temos o Teor- ema da Uniformização que afirma que toda superfície fechada admite uma métrica de curvatura constante -1, 0 ou 1, de acordo com seu gênero. Então, uma pergunta natural que surge é se esta conjectura pode ser estendida para dimensões maiores. E aqui o fluxo de Ricci entra em cena para tentar responder a esta pergunta que é conhecida como Conjectura de Geometrização de Thurston. O fluxo de Ricci é um fluxo geométrico, que significa que é definido independente de coordenadas, no qual um começa com uma variedade Riemanniana suave (M, 0 ) e evolui a sua métrica pela equação t = 2Rc(), onde Rc() de- nota o tensor de Ricci da métrica g. O fluxo de Ricci foi introduzido por Hamilton no seu artigo amplamente conhecido de 1982, "Three manifolds with positive Ricci curvature", e pode ser visto como uma equação do calor na métrica Riemanniana.
Title in English
Analysis of the Ricci flow on compact manifolds
Keywords in English
Compact manifolds
Geometric flow
Ricci flow
Ricci solitons
Abstract in English
A classical problem in geometry is to seek special metrics on a manifold. Among the most simple spaces are those of constant (sectional) curvature, namely, the model spaces: S N , H N and R N , which are unique, in a sense, up to isometries. In dimension 2, we have the Uniformization Theorem which states that every closed surface admits a metric of constant curvature -1, 0 or 1, according to its genus. So a natural question that arises is if this conjecture may be extended to higher dimensions. And here the Ricci flow enters the stage to try to answer this question which is known as Thurstons Geometrization Conjecture. The Ricci flow is a geometric flow, meaning that it is defined in a coordinate-independent manner, in which one starts with a differentiable Riemannian manifold (M, 0 ) and evolve its metric by the equation t = 2Rc(), where Rc() denotes the Ricci tensor of the metric g. The Ricci flow was introduced in Hamiltons seminal 1982 paper, "Three manifolds with positive Ricci curvature", and it can be seen as a heat equation for the Riemannian metric on a manifold.
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