Liouville theorems for ancient solutions to the [formula omitted]-harmonic map heat flows

Abstract

We investigate the Liouville properties of ancient solutions to the V-harmonic map heat flows from complete noncompact Riemannian manifolds with nonnegative Bakry–Emery Ricci curvature. When the target is a simply connected complete Riemannian manifold with nonpositive sectional curvature, then a Liouville theorem of ancient solutions holds under certain growth condition near infinity. When the target is a complete Riemannian manifold with sectional curvature bounded above by a positive constant, we show that if the image of the ancient solution u is contained in a regular ball near infinity, then u is a constant.