A sharp scalar curvature estimate for CMC hypersurfaces satisfying an Okumura type inequality
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A sharp scalar curvature estimate for CMC hypersurfaces satisfying an Okumura type inequality Eudes Leite de Lima1 · Henrique Fernandes de Lima2
Received: 8 September 2017 / Accepted: 23 October 2017 © Fondation Carl-Herz and Springer International Publishing AG 2017
Abstract We obtain a sharp estimate to the scalar curvature of stochastically complete hypersurfaces immersed with constant mean curvature in a locally symmetric Riemannian space obeying standard curvature constraints (which includes, in particular, a Riemannian space with constant sectional curvature). For this, we suppose that these hypersurfaces satisfy a suitable Okumura-type inequality recently introduced by Meléndez (Bull Braz Math Soc 45:385–404, 2014), which is a weaker hypothesis than to assume that they have two distinct principal curvatures. Our approach is based on the equivalence between stochastic completeness and the validity of the weak version of the Omori–Yau’s generalized maximum principle, which was established by Pigola et al. (Proc Am Math Soc 131:1283–1288, 2002; Mem Am Math Soc 174:822, 2005). Keywords Locally symmetric spaces · Stochastically complete hypersurfaces · Constant mean curvature hypersurfaces · Scalar curvature · Isoparametric hypersurfaces Résumé Nous obtenons une estimation optimale de la courbure scalaire des hypersurfaces stochastiquement complètes immergées avec courbure moyenne constante dans un espace Riemannien localement symétrique, obéissant aux contraintes de courbure standard (qui comprend, en particulier, un espace Riemannien avec courbure sectionnelle constante). Pour cela, nous supposons que ces hypersurfaces satisfont une inégalité appropriée de type Okumura récemment introduite par Meléndez (Bull Braz Math Soc 45:385–404, 2014), ce qui est une hypothèse plus faible que de supposer qu’elles ont deux courbures principales distinctes.
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Henrique Fernandes de Lima [email protected] Eudes Leite de Lima [email protected]
1
Unidade Acadêmica de Ciências Exatas e da Natureza, Universidade Federal de Campina Grande, Cajazeiras, Paraíba 58900-000, Brazil
2
Departamento de Matemática, Universidade Federal de Campina Grande, Campina Grande 58429-970, Paraíba, Brazil
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E. L. de Lima, H. F. de Lima
Notre approche est basée sur l’équivalence entre la complétude stochastique et la validité de la version faible du principe maximal généralisé de Omori–Yau, qui a été établi par Pigola et al. (Proc Am Math Soc 131:1283–1288, 2002; Mem Am Math Soc 174:822, 2005). Mathematics Subject Classification Primary 53C42; Secondary 53C40
1 Introduction The problem of characterizing hypersurfaces immersed with constant mean curvature in a n+1 Riemannian space form M c of constant sectional curvature c constitutes an important and fruitful topic in the theory of isometric immersions, which has being widely approached by many authors. For instance, Alencar and do Carmo [1] studied compact hypersurfaces immersed with constant mean curvature in the Euclidean sphere. Specifically, they introduced a tensor , t
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