Asymptotic behavior of the heat semigroup on certain Riemannian manifolds

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.
Comment: to appear in "From Classical Analysis to Analysis on Fractals - The Robert Strichartz Memorial Volume", Springer