Harnack quadratics for the Ricci flow via the space - time approach

In [H4], Hamilton determined a sharp tensor Harnack inequality for complete solutions of the Ricci flow with non-negative curvature operator. This Harnack inequality is of critical importance to the understanding of singularities of the Ricci flow, as is evident from its numerous applications in [H3], [H5], [H6], and [H7]. Moreover, according to Hamilton, the discovery of a Harnack 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 [H6] for some of the reasons why such an inequality is believed to hold. (One may also consult the survey paper [CaC].) Based on unpublished research of Hamilton and Hamilton–Yau, the search for such a Harnack appears to be an extremely complex and delicate problem. Roughly speaking, their approach is to start with the 3-dimensional Harnack 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 [H6] and Ivey [I] which shows that the curvature operator of 3-dimensional solutions tends in a sense to become nonnegative, there is hope that such a procedure will work. It appears that the results of Hamilton and Yau thus far just miss being strong enough to prove a general Harnack inequality in dimension 3. Due to the perturbational nature of the existing approaches, it is also of interest to understand how general a Harnack inequality one can prove under the original hypothesis of nonnegative curvature operator. In this direction, Hamilton and one of the authors [CH] obtained what they call a linear trace Harnack inequality for a system consisting of a solution of the LichnerowiczLaplacian 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) Harnack inequality. In [H3] Hamilton had