On an alternate proof of Hamilton’s matrix Harnack inequality for the Ricci flow

In [LY] a differential Harnack inequality was proved for solutions to the heat equation on a Riemannian manifold. Inspired by this result, Hamilton first proved trace and matrix Harnack inequalities for the Ricci flow on compact surfaces [H0] and then vastly generalized his own result to all higher dimensions for complete solutions of the Ricci flow with nonnegative curvature operator [ H2]. Soon afterwards, a matrix Harnack inequality for the Kähler-Ricci flow under the assumption of nonnegative bisectional curvature was proved by HuaiDong Cao [C]. In this paper, following a suggestion of Richard Hamilton, we give an alternate proof of the matrix Harnack inequality for the Ricci flow originally proved by him. In [ H2], Hamilton proved that if (M, g (t)) is a complete solution to the Ricci flow with nonnegative curvature operator, then