Abstract
In this article, we prove the following theorem: A complete hypersurface of the hyperbolic space form, which has constant mean curvature and non-negative Ricci curvature Q, has non-negative sectional curvature. Moreover, if it is compact, it is a geodesic distance sphere; if its soul is not reduced to a point, it is a geodesic hypercylinder; if its soul is reduced to a point p, its curvature satisfies ∥▽Q∥<∞, and the geodesic spheres centered at p are convex, then it is a horosphere.
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A part of this work has been done when the second author visited Université Claude Bernard Lyon 1, and was supported by a grant of the People's Republic of China.
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Morvan, JM., Bao-Qiang, W. Hypersurfaces with constant mean curvature in hyperbolic space form. Geom Dedicata 59, 197–222 (1996). https://doi.org/10.1007/BF00155730
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DOI: https://doi.org/10.1007/BF00155730