On complete hypersurfaces with negative Ricci curvature in Euclidean spaces

• Alexandre Paiva Barreto ^[1] ; Francisco Fontenele ^[2]
  1. [1] Universidade Federal de São Carlos

     Universidade Federal de São Carlos

     Brasil

     
  2. [2] Universidade Federal Fluminense

     Universidade Federal Fluminense

     Brasil

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 39, Nº 4, 2023, págs. 1437-1442
• Idioma: inglés
• DOI: 10.4171/RMI/1407
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• Resumen

  • In this paper, we prove that if Mn, n≥3, is a complete Riemannian manifold with negative Ricci curvature and f:Mn→Rn+1 is an isometric immersion such that Rn+1\f(M) is an open set that contains balls of arbitrarily large radius, then infM∣A∣=0, where ∣A∣ is the norm of the second fundamental form of the immersion. In particular, an n-dimensional complete Riemannian manifold with negative Ricci curvature bounded away from zero cannot be properly isometrically immersed in a half-space of Rn+1. This gives a partial answer to a question raised by Reilly and Yau.