Dual flows in hyperbolic space and de Sitter space

TL;DR

This paper studies convex contracting and expanding flows in hyperbolic and de Sitter spaces, revealing their geometric limits and relations via the Gauss map, with detailed convergence behaviors after rescaling.

Contribution

It establishes a connection between contracting and expanding flows through the Gauss map and describes their asymptotic geometric limits in hyperbolic and de Sitter spaces.

Findings

Contracting hypersurfaces shrink to a point in finite time.

Expanding hypersurfaces converge to the maximal slice.

Rescaled hypersurfaces converge to specific geometric shapes exponentially fast.

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 $C^\infty(\mathbb{S}^n)$.