In this paper, we first introduce the weighted forward reduced volume of Ricci flow. The weighted forward reduced volume, which related to expanders of Ricci flow, is well-defined on noncompact manifolds and monotone non-increasing under Ricci flow. Moreover, we show that, just the same as the Perelman's reduced volume, the weighted reduced volume entropy has the value $(4\pi)^{ \frac{ n}{ 2}}$ if and only if the Ricci flow is the trivial flow on flat Euclidean space.
Categories: math.DG