Mean curvature type flow in rotationally symmetric spaces

In this paper, the mean curvature type flow of star-shaped closed hypersurfaces in rotationally symmetric spaces is investigated. We prove that the flow exists for all times and converges exponentially to a sphere in the $C^{\infty}$ topology, enclosing the same volume as the initial hypersurfaces. This extends the corresponding result of space forms by Guan and Li (2015) to a large class of rotationally symmetric spaces.