Li-Yau gradient bound for collapsing manifolds under integral curvature condition

Let (Mⁿ, g_ij) be a complete Riemannian manifold. For any con-stants p, r > 0, define k(p, r) = (Formula Presented) denotes the negative part of the Ricci curvature tensor. We prove that for any p >ⁿ₂, when k(p, 1) is small enough, a certain Li-Yau type gradient bound holds for the positive solutions of the heat equation on geodesic balls B(O, r) in M with 0 < r ≤ 1. Here the assumption that k(p, 1) is small allows the situation where the manifold is collapsing. Recall that in an earlier paper by Zhang and Zhu, a certain Li-Yau gradient bound was also obtained by the authors, assuming that |Ric⁻ | ∈ L^p (M) and the manifold is noncollapsed. Therefore, to some extent, the results in this paper, as well as the earlier one complete the picture of Li-Yau gradient bounds for the heat equation on manifolds with |Ric⁻ | being L^p integrable, modulo the sharpness of constants.