The weighted geometric inequalities for static convex domains in static rotationally symmetric spaces

We consider a locally constrained curvature flow in a static rotationally symmetric space ${\mathbf{N}}^{n+1}$, which was firstly introduced by Hu and Li in the hyperbolic space. We prove that if the initial hypersurface is graphical, then the smooth solution of the flow remains to be graphical, exists for all positive time $t\in [0,\mathrm{\infty })$ and converges to a slice of ${\mathbf{N}}^{n+1}$ exponentially in the smooth topology. Moreover, we prove that the flow preserves static convexity if the initial hypersurface is close to a slice of ${\mathbf{N}}^{n+1}$ in the $C^1$ sense. As applications, we prove a family of weighted geometric inequalities for static convex domains which is close to a slice of ${\mathbf{N}}^{n+1}$ in the $C^1$ sense.