THE WEIGHTED CONFORMAL MEAN CURVATURE FLOW

We introduce a Yamabe-type flow ( partial derivative g R m partial derivative t =2( h m phi - H phi m ) g phi = 0 in M and partial derivative phi partial derivative t = m ( H phi m - h m phi ) on partial derivative 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 phi m 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.