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Hadamard-type theorems for hypersurfaces in hyperbolic spaces

Gil SolanesTakashi KuroseLuis J. Alías

subject

Pure mathematicsGauss mapMathematics::Dynamical SystemsMathematics::Complex VariablesHyperbolic spaceSecond fundamental formMathematical analysisCauchy–Hadamard theoremGauss–Kronecker curvatureSecond fundamental formHypersurfaceMathematics::Algebraic GeometryComputational Theory and MathematicsBounded functionHadamard theoremTotal curvatureDiffeomorphismGeometry and TopologyMathematics::Differential GeometryAnalysisConvex hypersurfaceMathematics

description

Abstract We prove that a bounded, complete hypersurface in hyperbolic space with normal curvatures greater than −1 is diffeomorphic to a sphere. The completeness condition is relaxed when the normal curvatures are bounded away from −1. The diffeomorphism is constructed via the Gauss map of some parallel hypersurface. We also give bounds for the total curvature of this parallel hypersurface.

10.1016/j.difgeo.2006.02.008http://dx.doi.org/10.1016/j.difgeo.2006.02.008