New Li–Yau–Hamilton Inequalities for the Ricci Flow via the Space-time Approach

i ii Chapter 1 Introduction In [11], Hamilton determined a sharp differential Harnack inequality of Li–Yau type for complete solutions of the Ricci flow with non-negative curvature operator. This Li–Yau–Hamilton inequality (abbreviated as LYH inequality below) is of critical importance to the understanding of singularities of the Ricci flow, as is evident from its numerous applications in [10], [12], [13], and [14]. Moreover, it has been informally claimed by Hamilton that the discovery of a LYH inequality in dimension 3 valid without any hypothesis on curvature is the main unresolved step in his program of approaching Thurston's Geometrization Conjecture by applying the Ricci flow to closed 3-manifolds. See [13] for some of the reasons why such an inequality is believed to hold. (One may also consult the survey paper [2].) Based on unpublished research of Hamil-ton and Hamilton–Yau, the search for such a LYH inequality appears to be an extremely complex and delicate problem. Roughly speaking, their approach is to start with the 3-dimensional LYH inequality for solutions with nonnegative sectional curvature and try to perturb that estimate so that it holds for solutions with arbitrary initial data. Because of an estimate of Hamilton [13] and Ivey [15] which shows that the curvature operator of 3-dimensional solutions tends in a sense to become nonneg-ative, there is hope that such a procedure will work. Some unpublished work of Hamilton and Yau appears close to establishing a general LYH inequality in dimension 3. However, so far no such inequality is known. Due to the perturbational nature of the existing approaches, it is also of interest to understand how general a LYH inequality one can prove under the original hypothesis of nonnegative curvature operator. In this direction, Hamilton and one of the authors [6] obtained a lin-1 2 CHAPTER 1. INTRODUCTION ear trace LYH inequality for a system consisting of a solution of the Lichnerowicz-Laplacian heat equation for symmetric 2-tensors coupled to a solution of the Ricci flow. Since the pair of the Ricci and metric tensors of a solution to the Ricci flow forms such a system, their linear trace inequality generalizes the traced case of Hamilton's tensor (matrix) inequality. In [10] Hamilton had already observed the formal similarity between his proof of the 2-dimensional trace LYH inequality for the Ricci flow and Li and Yau's proof [16] of their Harnack inequality for the heat equation on Riemannian manifolds. In a sense, …