Contact foliations: closed leaves and generalised Weinstein conjectures

• Ano de defesa: 2023
• Autor(a) principal: Barbosa, Douglas Luiz Finamore
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Biblioteca Digitais de Teses e Dissertações da USP
• Palavras-chave em Português: Ações de grupos; Contact dynamics; Dinâmica de contato; Estruturas de contato generalizadas; Folheações; Foliations; Generalised contact structures; Group actions
• Link de acesso: https://www.teses.usp.br/teses/disponiveis/55/55135/tde-30082023-163143/

The Weinstein conjecture, which regards the existence of periodic orbits for Reeb flows, is a classic problem in Contact Geometry. In his doctoral dissertation, Almeida (ALMEIDA, 2018) introduced a novel geometric structure, which generalises contact structures and provides a notion of contact foliation, i.e., higher dimensional analogues for the Reeb flow. In this work, inspired by the classical Weinstein conjecture, we seek to find closed leaves for such contact foliations. By generalising ideas already employed successfully in proving the Weinstein conjecture in the past, we obtain the existence of closed leaves in particular cases when the foliation is either hyperbolic or C1-equicontinuous. This later class encompasses those of quasiconformal, conformal, isometric, and Riemannian contact foliations. Moreover, using techniques from Morse Theory, we were able to relate the closed leaves of a C1-equicontinuous contact foliation to its basic cohomology, obtaining a lower bound for the number of closed orbits, as a function of the foliations codimension.