A locally constrained mean curvature type flow with free boundary in a hyperbolic ball

In this paper, we study a locally constrained mean curvature flow with free boundary in a hyperbolic ball. Under the flow, the enclosed volume is preserved and the area is decreasing. We prove the long time existence and smooth convergence for such flow under certain star-shaped condition. As an application, we give a flow proof of the isoperimetric problem for the star-shaped free boundary hypersurfaces in a hyperbolic ball.