Characterization of Lipschitz normally embedded complex curves and Lipschitz trivial values of polynomial mappings

• Ano de defesa: 2023
• Autor(a) principal: Costa, André Luiz Araújo da
• Orientador(a): Grandjean, Vincent Jean-Henri
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Palavras-chave em Português: Geometria Lipschitz; Lipschitz normalmente mergulhado; Valores Lipschitz triviais; Lipschitz geometry; Lipschitz normally embedded; Lipschitz trivial values
• Link de acesso: http://www.repositorio.ufc.br/handle/riufc/70235

We study Lipschitz geometry of fibers of complex polynomial mappings from two points of view: the equivalence of inner and outer metrics of an algebraic curve and the existence of a locally bi-Lipschitz trivial fibre bundle structure over a subset of values of polynomial mappings. We prove that the affi ne part of a connected projective algebraic curve is Lipschitz normally embedded if and only if the following three conditions are satisfi ed: its affi ne part is connected, its affi ne part is locally Lipschitz normally embedded at each of its singular points; and its degree equals to the number of its points at infi nity. Moreover, we show that any Lipschitz trivial value of a real or complex polynomial mapping is a suspension of a regular value of properness of a polynomial mapping in fewer variables. Last, we show that this result cannot extend to rational mappings.