  We show that, on a complete, connected and non-compact Riemannian manifold of
non-negative Ricci curvature, the solution to the heat equation with $L^{1}$
initial data behaves asymptotically as the mass times the heat kernel. In
contrast to the previously known results in negatively curved contexts, the
radiality assumption on the initial data is not required. Similar long-time
convergence results remain valid on more general manifolds satisfying the
Li-Yau two-sided estimate of the heat kernel. Moreover, we provide a
counterexample such that this asymptotic phenomenon fails in sup norm on
manifolds with two Euclidean ends.
