The isoperimetric problem in the Riemannian manifold admitting a non-trivial conformal vector field

In this article, we will study the isoperimetric problem by introducing a mean curvature type flow in the Riemannian manifold endowed with a non-trivial conformal vector field. This flow preserves the volume of the bounded domain enclosed by a star-shaped hypersurface and decreases the area of hypersurface under certain conditions. We will prove the long time existence and convergence of the flow. As a result, the isoperimetric inequality for such a domain is established. Especially, we solve the isoperimetric problem for the star-shaped hypersurfaces in the Riemannian manifold endowed with a closed, non-trivial conformal vector field, a wide class of warped product spaces studied by Guan, Li and Wang is included.