Locally constrained flows and geometric inequalities in hyperbolic space and sphere

In this paper, we prove some families of geometric inequalities in hyperbolic space and sphere by some kinds of locally constrained flows. In hyperbolic space, we obtain the geometric inequalities involving two weighted curvature integrals for static convex domains. While in sphere, a new family of ``three terms'' geometric inequalities involving two weighted curvature integrals and one quermassintegral are proved. Unlike hyperbolic spaces, we also obtain an inverse weighted geometric inequality in sphere.