New weighted Alexandrov-Fenchel type inequalities for hypersurfaces in hyperbolic space

In this paper, we obtain two monotonic quantities under the locally constrained inverse curvature flow. Using the monotonicity, a family of new weighted geometric inequalities for closed static-convex hypersurface in hyperbolic space $H^{n+1}$ is obtained, which implies that, among all closed static-convex hypersurfaces with fixed weighted k-th ($k⩾2$) mean curvature integral, the geodesic ball reaches the maximum value of l-th ($l⩽k$) quermassintegral. We also consider the case of $k=1$ and obtain a geometric inequality which is an improved version of Wei and Zhou's result in Wei and Zhou (2023) [24].