THE WEIGHTED YAMABE FLOW WITH BOUNDARY

We introduce a Yamabe-type flow � partial differential g partial differential t = (rm & phi; - Rm & phi; )g partial differential & phi; partial differential t = m & phi; = 0 on partial differential M 2 (Rm & phi; - rm & phi; ) in M and Hmon a smooth metric measure space with boundary (M, g, vmdVg, vmdAg, m), where Rm & phi; is the associated weighted scalar curvature, rm & phi; is the average of the weighted scalar curvature, and H & phi;m is the weighted mean curvature. We prove the long-time existence and convergence of this flow.