  We study harmonic map heat flow along ancient super Ricci flow, and derive
several Liouville theorems with controlled growth from Perelman's reduced
geometric viewpoint. For non-positively curved target spaces, our growth
condition is sharp. For positively curved target spaces, our Liouville theorem
is new even in the static case (i.e., for harmonic maps); moreover, we point
out that the growth condition can be improved, and almost sharp in the static
case. This fills the gap between the Liouville theorem of Choi and the example
constructed by Schoen-Uhlenbeck.
