The weighted Yamabe problem with boundary

Abstract

We introduce a Yamabe-type flow \begin{align*} \left\{ \begin{array}{ll} \frac{\partial g}{\partial t} &=(r^m_ϕ-R^m_ϕ)g \\ \frac{\partial ϕ}{\partial t} &=\frac{m}{2}(R^m_ϕ-r^m_ϕ) \end{array} \right. ~~\mbox{ in }M ~~\mbox{ and }~~ H^m_ϕ=0 ~~\mbox{ on }\partial M \end{align*} on a smooth metric measure space with boundary $(M,g,v^{m}d{V}_g,v^{m}d{A}_g,m)$, where $R_{\varphi }^m$ is the associated weighted scalar curvature, $r_{\varphi }^m$ is the average of the weighted scalar curvature, and $H_{\varphi }^m$ is the weighted mean curvature. We prove the long-time existence and convergence of this flow.