A sharp Li-Yau gradient bound on compact manifolds

Let (ⁿ, g) be a n dimensional, complete ( compact or noncompact) Riemannian manifold whose Ricci curvature is bounded from below by a constant -K ≤ 0. Let u be a positive solution of the heat equation on ⁿ × (0, ∞). The well known Li-Yau gradient bound states that t (|∇ u|²/u² - αₜ u/u) ≤ nα²/2 + t nα²K/2(α-1), ∀ α>1, t>0. The bound with α =1 is sharp if $K=0$. If $-K 1$ but not equal to $1$. An open question ({CLN}, } etc) asks if a sharp bound can be reached. In this short note, we observe that for all complete compact manifolds one can take α=1. Thus a sharp bound, up to computable constants, is found in the compact case. In the noncompact case one can not take α=1 even for the hyperbolic space.