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).
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. 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).
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Acknowledgements
H. F. de Lima is partially supported by CNPq, Brazil, Grant 303977/2015-9.
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de Lima, E.L., de Lima, H.F. A sharp scalar curvature estimate for CMC hypersurfaces satisfying an Okumura type inequality. Ann. Math. Québec 42, 255–265 (2018). https://doi.org/10.1007/s40316-017-0095-9
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DOI: https://doi.org/10.1007/s40316-017-0095-9
Keywords
- Locally symmetric spaces
- Stochastically complete hypersurfaces
- Constant mean curvature hypersurfaces
- Scalar curvature
- Isoparametric hypersurfaces