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Rigidity of volume-minimising hypersurfaces in Riemannian 5-manifolds

Published online by Cambridge University Press:  23 May 2018

ABRAÃO MENDES
ABRAÃO MENDES*
Affiliation:
Federal University of Alagoas, Institute of Mathematics, Lourival Melo Mota Avenue, 57072-970, Tabuleiro do Martins, Maceió, AL, Brazil. e-mail: abraaomr@gmail.com
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Abstract

In this paper we generalise the main result of [4] for manifolds that are not necessarily Einstein. In fact, we obtain an upper bound for the volume of a locally volume-minimising closed hypersurface Σ of a Riemannian 5-manifold M with scalar curvature bounded from below by a positive constant in terms of the total traceless Ricci curvature of Σ. Furthermore, if Σ saturates the respective upper bound and M has nonnegative Ricci curvature, then Σ is isometric to 𝕊4 up to scaling and M splits in a neighbourhood of Σ. Also, we obtain a rigidity result for the Riemannian cover of M when Σ minimises the volume in its homotopy class and saturates the upper bound.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 167 , Issue 2 , September 2019 , pp. 345 - 353
Copyright
Copyright © Cambridge Philosophical Society 2018 

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