Convergence of the Weighted Yamabe Flow

We introduce the weighted Yamabe flow $\frac{\partial g}{\partial t}=(r^m_{\phi}-R^m_{\phi})g$, $\frac{\partial \phi}{\partial t}=\frac{m}{2}(R^m_{\phi}-r^m_{\phi})$ on a smooth metric measure space $(M^n, g, e^{-\phi}{\rm dvol}_g, m)$, where $R^m_{\phi}$ denotes the associated weighted scalar curvature, and $r^m_{\phi}$ denotes the mean value of the weighted scalar curvature. We prove long-time existence and convergence of the weighted Yamabe flow if the dimension $n$ satisfies $n\geqslant 3$.