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Liouville theorems for the ancient solution of heat flows

Liouville theorems for the ancient solution of heat flows

Let $M$ be a complete Riemannian manifold with Ricci curvature bounded from below: $Ric(M)\ge -\kappa$. Let $N$ be a simply connected complete Riemannian manifold with nonpositive sectional curvature. Using a gradient estimate, we prove Liouville’s theorem for the ancient solution of heat flows.