Convergence of the weighted Yamabe flow

Abstract

We introduce the weighted Yamabe flow{∂g∂t=(rϕm−Rϕm)g∂ϕ∂t=m2(Rϕm−rϕm) on a smooth metric measure space (Mn,g,e−ϕdvolg,m), where Rϕm denotes the associated weighted scalar curvature, and rϕm 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⩾3.