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We now turn to the theory of parabolic Harnack inequalities, which control the variation over space and time of solutions to the scalar heat equation

$u_t = \Delta u$ (1)

which are bounded and non-negative, and (more pertinently to our applications) of the curvature of Ricci flows

$g_t = -2\hbox{Ric}$ (2)

whose Riemann curvature $\hbox{Riem}$ or Ricci curvature $\hbox{Ric}$ is bounded and non-negative. For instance, the classical parabolic Harnack inequality of Moser asserts, among other things, that one has a bound of the form

$u(t_1,x_1) \leq C(t_1,x_1,t_0,x_0,T_-,T_+,M) u(t_0,x_0)$ (3)

whenever $u: [T_-,T_+] \times M \to {\Bbb R}^+$ is a bounded non-negative solution to (1) on a complete static Riemannian manifold M of bounded curvature, $(t_1,x_1), (t_0,x_0) \in [T_-,T_+] \times M$ are spacetime points with $t_1 < t_0$, and $C(t_1,x_1,t_0,x_0,T_-,T_+,M)$ is a constant which is uniformly bounded for fixed $t_1,t_0,T_-,T_+,M$ when $x_1,x_0$ range over a compact set. (The even more classical elliptic Harnack inequality gives (1) in the steady state case, i.e. for bounded non-negative harmonic functions.) In terms of heat kernels, one can view (1) as an assertion that the heat kernel associated to $(t_0,x_0)$ dominates (up to multiplicative constants) the heat kernel at $(t_1,x_1)$.

The classical proofs of the parabolic Harnack inequality do not give particularly sharp bounds on the constant $C(t_1,x_1,t_0,x_0,T_-,T_+,M)$. Such sharp bounds were obtained by Li and Yau, especially in the case of the scalar heat equation (1) in the case of static manifolds of non-negative Ricci curvature, using Bochner-type identities and the scalar maximum principle. In fact, a stronger differential version of (3) was obtained which implied (3) by an integration along spacetime curves (closely analogous to the ${\mathcal L}$-geodesics considered in earlier lectures). These bounds were particularly strong in the case of ancient solutions (in which one can send $T_- \to -\infty$). Subsequently, Hamilton applied his tensor-valued maximum principle together with some remarkably delicate tensor algebra manipulations to obtain Harnack inequalities of Li-Yau type for solutions to the Ricci flow (2) with bounded non-negative Riemannian curvature. In particular, this inequality applies to the $\kappa$-solutions introduced in the previous lecture.

In this current lecture, we shall discuss all of these inequalities (although we will not give the full details for the proof of Hamilton’s Harnack inequality, as the computations are quite involved), and derive several important consequences of that inequality for $\kappa$-solutions. The material here is based on several sources, including Evans’ PDE book, Müller’s book, Morgan-Tian’s book, the paper of Cao-Zhu, and of course the primary source papers mentioned in this article.