On the classification of Ricci solitons and Yamabe solitons

• Ano de defesa: 2023
• Autor(a) principal: Contreras, Jeferson Arley Poveda
• Orientador(a): Leandro Neto, Benedito
• Banca de defesa: Leandro Neto, Benedito, Santos, João Paulo do, Ribeiro Júnior, Ernani de Sousa, Tenenblat, Keti, Barboza, Marcelo Bezerra
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Universidade Federal de Goiás
• Programa de Pós-Graduação: Programa de Pós-graduação em Matemática (IME)
• Departamento: Instituto de Matemática e Estatística - IME (RMG)
• País: Brasil
• Palavras-chave em Português: Ricci soliton; Yamabe soliton; Steady; Shrinking
• Palavras-chave em Inglês: Expanding; Pinching; Soliton de Ricci; Soliton de Yamabe
• Área do conhecimento CNPq: CIENCIAS EXATAS E DA TERRA::MATEMATICA::GEOMETRIA E TOPOLOGIA::TOPOLOGIA ALGEBRICA
• Link de acesso: http://repositorio.bc.ufg.br/tede/handle/tede/12859

In this work, we will study the self-similar solutions of both Ricci flow and Yamabe flow. These solutions are also known as Ricci and Yamabe soliton, respectively. Inspired by the divergence equation used by Robinson in his demonstration of the uniqueness of static black holes and by Brendle’s classification of steady Ricci solitons, we will make some important characterizations of these solitons. We prove that four-dimensional gradient Yamabe solitons must have a Yamabe metric, provided that an asymptotic condition holds. Inspired by the geometry of the cigar soliton, we demonstrate that a gradient steady Ricci soliton is either Ricci flat with a constant potential function or a quotient of the product steady soliton N n−1×R, where N n−1 is Ricci flat, or isometric to the Bryant soliton. In the final Chapter, we prove some rigidity results for shrinking and expanding Ricci solitons.