LAUSR

In this paper, we consider the Yamabe flow on Lie groups with left invariant metrics in particular case on the higher dimensional classical Heisenberg nilpotent Lie groups and the higher dimensional quaternion Lie groups. We construct a explicit solution of the Yamabe flow on each group. Then we investigate the some properties on these Lie groups under the Yamabe flow. At the end of paper, we study the deformation of some feature of compact nilmanifolds $$\Gamma {\setminus } N$$ Γ \ N under the Yamabe flow, where N is a simply connected two-step nilpotent Lie group with a left invariant metric, and $$\Gamma $$ Γ is a discrete cocompact subgroup of N , in particular Heisenberg and quaternion Lie groups.