Rigidity of contracting map using harmonic map heat flow

Motivated by a question of Tsai-Tsui-Wang, we consider the rigidity of map from manifolds with positive Ricci curvature to manifolds with positive sectional curvature. We show that if the Ricci curvature of the domain dominates that of the target, then distance non-increasing maps must be either Riemannian submersion or isometry. The rigidity result also holds on a wider class of manifolds with positive curvature and weaker contracting property on the map. This is based on a new long-time existence of harmonic map heat flow.