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 θ-totally umbilical cap. Additionally, we establish that a θ-totally umbilical cap is an energy minimizer for a given enclosed volume.
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Chai et al. (2024) studied this question.
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