Liouville theorems for ancient solutions to the V-harmonic map heat flows II

TL;DR

This paper establishes improved Liouville theorems for ancient solutions to V-harmonic map heat flows on certain noncompact manifolds, using refined gradient estimates and growth conditions.

Contribution

It provides new Liouville theorems for ancient solutions to V-harmonic map heat flows with refined gradient estimates and growth conditions, extending previous results.

Findings

Enhanced Liouville theorem for ancient solutions

Liouville theorem for quasi-harmonic maps under growth conditions

Refined gradient estimates improve previous results

Abstract

When the domain is a complete noncompact Riemannian manifold with nonnegative Bakry--Emery Ricci curvature and the target is a complete Riemannian manifold with sectional curvature bounded above by a positive constant, by carrying out refined gradient estimates, we obtain a better Liouville theorem for ancient solutions to the V-harmonic map heat flows. Furthermore, we can also derive a Liouville theorem for quasi-harmonic maps under an exponential growth condition.