Sharp differential estimates of Li‐Yau‐Hamilton type for positive (p, p) forms on Kähler manifolds

In this paper we study the heat equation (of Hodge Laplacian) deformation of (p, p)‐forms on a Kähler manifold. After identifying the condition and establishing that the positivity of a (p, p)‐form solution is preserved under such an invariant condition, we prove the sharp differential Harnack (in the sense of Li‐Yau‐Hamilton) estimates for the positive solutions of the Hodge Laplacian heat equation. We also prove a nonlinear version coupled with the Kähler‐Ricci flow and some interpolating matrix differential Harnack‐type estimates for both the Kähler‐Ricci flow and the Ricci flow. © 2011 Wiley Periodicals, Inc.