Liouville theorems for ancient caloric functions via optimal growth conditions

We prove two Liouville theorems for ancient nonnegative solutions of the heat equation on a complete noncompact Riemannian manifold with Ricci curvature bounded from below by −K, K ≥ 0. If, at any fixed time, such a solution grows subexponentially in space, then it is either constant (when K = 0) or stationary (if K > 0). We also show the optimality of this growth condition through examples.