Dual flows in hyperbolic space and de Sitter space

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 $x_0$ in finite time while the expanding hypersurfaces converge to the maximal slice $\{ \tau =0\}$. After rescaling, by the same scale factor, the resclaed contracting hypersurfaces converge to a unit geodesic sphere, while the rescaled expanding hypersufaces converge to slice $\{ \tau = -1\}$ exponential fast in $C^\infty(\mathbb{S}^n)$.