A gap theorem for constant scalar curvature hypersurfaces

• Lima, Eudes L. de ^[1] ; Lima, Henrique F. de ^[1]
  1. [1] Universidade Federal de Campina Grande

     Universidade Federal de Campina Grande

     Brasil

     
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 73, Fasc. 1, 2022, págs. 1-11
• Idioma: inglés
• DOI: 10.1007/s13348-020-00304-3
• Texto completo no disponible (Saber más ...)
• Resumen

  • We obtain a sharp estimate to the norm of the traceless second fundamental form of complete hypersurfaces with constant scalar curvature immersed into a locally symmetric Riemannian manifold obeying standard curvature constraints (which includes, in particular, the Riemannian space forms with constant sectional curvature). When the equality holds, we prove that these hypersurfaces must be isoparametric with two distinct principal curvatures. Our approach involves a suitable Okumura type inequality which was introduced by Meléndez (Bull Braz Math Soc 45:385–404, 2014) , corresponding to a weaker hypothesis when compared with to the assumption that these hypersurfaces have a priori at most two distinct principal curvatures.