Liouville theorems for harmonic map heat flow along ancient super Ricci flow via reduced geometry

TL;DR

This paper establishes Liouville theorems for harmonic map heat flow along ancient super Ricci flows using reduced geometry, providing sharp conditions for non-positively curved targets and new results for positively curved targets.

Contribution

It introduces new Liouville theorems for harmonic map heat flow along ancient super Ricci flows, extending static case results and utilizing Perelman's reduced geometry.

Findings

Liouville theorems with controlled growth derived for ancient super Ricci flows

Sharp growth conditions established for non-positively curved target spaces

New Liouville theorem results for positively curved target spaces

Abstract

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.