On the weighted forward reduced Entropy of Ricci flow

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.Comment: 10 page

Nonsingular Ricci flow on a noncompact manifold in dimension three

We consider the Ricci flow $\frac{\partial}{\partial t}g=-2Ric$ on the 3-dimensional complete noncompact manifold $(M,g(0))$ with non-negative curvature operator, i.e., $Rm\geq 0, |Rm(p)|\to 0, ~as ~d(o,p)\to 0.$ We prove that the Ricci flow on such a manifold is nonsingular in any finite time.Comment: 6 page

Yamabe flow and ADM Mass on asymptotically flat manifolds

In this paper, we investigate the behavior of ADM mass and Einstein-Hilbert functional under the Yamabe flow. Through studying the Yamabe flow by weighted spaces, we show that ADM mass and Einstein-Hilbert functional are well-defined and monotone non-increasing under the Yamabe flow on $n$-dimensional, $n\geq 3$, asymptotically flat manifolds. In the case of dimension $n=3$ or 4, we obtain that the ADM mass is invariant under the Yamabe flow and the Yamabe flow is the gradient flow of Einstein-Hilbert functional on asymptotically flat manifold