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Abstract

We consider contracting flows in (n+1)-dimensional hyperbolic space and expanding flows in (n+1)-dimensional de Sitter space. When the flow hypersurfaces are strictly convex we relate the contracting hypersurfaces and the expanding hypersurfaces by the Gauß map. The contracting hypersurfaces shrink to a point in finite time while the expanding hypersurfaces converge to the maximal slice {\tau = 0}. After rescaling, by the same scale factor, the rescaled contracting hypersurfaces converge to a unit geodesic sphere, while the rescaled expanding hypersurfaces converge to slice {\tau = −1} exponentially fast.