Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space

Abstract

We consider the volume preserving flow of smooth, closed and convex hypersurfaces in the hyperbolic space H n + 1 ( n ≥ 2 ) \mathbb {H}^{n+1} (n\geq 2) with the speed given by arbitrary positive power α \alpha of the Gauss curvature. We prove that if the initial hypersurface is convex, then the smooth solution of the flow remains convex and exists for all positive time t ∈ [ 0 , ∞ ) t\in [0,\infty ) . Moreover, we apply a result of Kohlmann which characterises the geodesic ball using the hyperbolic curvature measures and an argument of Alexandrov reflection to prove that the flow converges to a geodesic sphere exponentially in the smooth topology. This can be viewed as the first result for non-local type volume preserving curvature flows for hypersurfaces in the hyperbolic space with only convexity required on the initial data.