Geometric inequalities and their stabilities for modified quermassintegrals in hyperbolic space

In this paper, we first consider the curve case of Hu-Li-Wei's flow for shifted principal curvatures of h-convex hypersurfaces in $\mathbb{H}^{n+1}$ proposed in [10]. We prove that if the initial closed curve is smooth and strictly h-convex, then the solution exists for all time and preserves strict h-convexity along the flow. Moreover, the evolving curve converges smoothly and exponentially to a geodesic circle centered at the origin. The key ingredient in our proof is the Heintze-Karcher type inequality for h-convex curves proved recently in [14]. As an application, we then provide a new proof of geometric inequalities involving weighted curvature integrals and modified quermassintegrals for h-convex curves in $\mathbb{H}^2$. We finally discuss the stability of these inequalities as well as Alexandrov-Fenchel type inequalities for modified quermassintegrals for strictly h-convex domains in $\mathbb{H}^{n+1}$.