A constrained mean curvature flow on capillary hypersurface supported on totally geodesic plane

In this paper, we prove a new Minkowski-type formula for capillary hypersurfaces supported on totally geodesic hyperplanes in hyperbolic space. It leads to a volume-preserving flow starting from a star-shaped initial hypersurface. We prove the long-time existence of the flow and its uniform convergence to a [Formula: see text]-totally umbilical cap. Additionally, we establish that a [Formula: see text]-totally umbilical cap is an energy minimizer for a given enclosed volume.