The weighted conformal mean curvature flow

We introduce a Yamabe-type flow$ R^m_{\phi} = 0 \; \; \mbox{ in } M \; \; \mbox{ and }\; \; \left\{ \begin{array}{ll} \frac{\partial g}{\partial t} & = 2(h^m_{\phi}-H^m_{\phi})g \\ \frac{\partial \phi}{\partial t} & = m(H^m_{\phi}-h^m_{\phi}) \end{array} \right. \; \; \mbox{ on } \partial M $on a smooth metric measure space with boundary $ (M, g, e^{-\phi}dV_g, e^{-\phi}dA_g, m) $, where $ R^m_{\phi} $ is the weighted scalar curvature, $ H^m_{\phi} $ is the weighted mean curvature and $ h^m_{\phi} $ is the average of the weighted mean curvature. We prove the long-time existence and convergence of this flow.