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Conformal gradient vector fields on Riemannian manifolds with boundary

Volume 159 / 2020

Israel Evangelista, Emanuel Viana Colloquium Mathematicum 159 (2020), 231-241 MSC: Primary 53C20, 53A30. DOI: 10.4064/cm7638-12-2018 Published online: 22 November 2019


Let $(M^n,g)$ be an $n$-dimensional compact connected Riemannian manifold with smooth boundary. We show that the presence of a nontrivial conformal gradient vector field on $M$, with an appropriate control on the Ricci curvature, causes $M$ to be isometric to a hemisphere of $\mathbb {S}^{n}$. We also prove that if an Einstein manifold with boundary admits a nonzero conformal gradient vector field, then its scalar curvature is positive and it is isometric to a hemisphere of $\mathbb {S}^{n}$. Furthermore, we prove that if $ M $ admits a nontrivial conformal vector field and has constant scalar curvature, then the scalar curvature is positive. Finally, a suitable control on the energy of a conformal vector field implies that $M$ is isometric to a hemisphere $\mathbb {S}^n_+$.


  • Israel EvangelistaUFPI, Curso de Matemática
    Universidade Federal do Piauí
    Campus Ministro Reis Velloso
    Parnaíba, PI, Brazil
  • Emanuel VianaInstituto Federal de Educação
    Ciência e Tecnologia do Ceará (IFCE)
    Campus Caucaia
    Caucaia, CE, Brazil
    UFC, Departamento de Matemática
    Universidade Federal do Ceará
    Campus do Pici
    Fortaleza, CE, Brazil

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