Hamilton's Matrix Harnack Inequality

Ancient Solutions with Non-Negative Curvature

Hamilton's Harnack inequality applies to complete solutions of Ricci flow with non-negative curvature operator. It is most powerful for ancient solutions, i.e., solutions defined on $(-\infty, T)$, where the $R/t$ term vanishes in the limit $t \to -\infty$. Such solutions arise naturally as blow-up limits of finite-time singularities, thanks to Hamilton-Ivey pinching.

The Harnack inequality is a differential inequality that controls how fast the curvature can decrease along the flow, playing the role of a Li-Yau gradient estimate in the Ricci flow setting.

The Harnack Tensor

Define the Harnack tensor by

$$\mathcal{H}_{ij} = \frac{\partial}{\partial t}R_{ij} + \frac{1}{2t}\,R_{ij} + \nabla_i \nabla_j R + R_{ikjl}\, V^k V^l + \nabla_i R_{jk}\, V^k + \nabla_j R_{ik}\, V^k$$

for any vector field $V$. This is a symmetric 2-tensor that combines the time derivative, spatial Hessian, and curvature terms in a way that respects the diffeomorphism invariance of the flow.

Theorem (Hamilton, 1993)

Let $(M^n, g(t))$ be a complete solution of Ricci flow on $(0, T]$ with bounded non-negative curvature operator. Then

$$\boxed{\mathcal{H}_{ij} \geq 0}$$

as a symmetric 2-tensor, for all vector fields $V$ and all $t > 0$.

The Trace Harnack Inequality

Taking the trace of $\mathcal{H}_{ij} \geq 0$ with respect to the metric gives the scalar Harnack inequality:

$$\frac{\partial R}{\partial t} + \frac{R}{t} + 2\langle \nabla R, V \rangle + 2\,\mathrm{Rc}(V, V) \geq 0$$

for all vector fields $V$. Optimising over $V$ by setting the gradient to zero yields $V^i = -g^{ij}\nabla_j R / (2R_{ij})$. In the scalar curvature direction, this simplifies to the optimal inequality:

$$\boxed{\frac{\partial R}{\partial t} + \frac{R}{t} \geq \frac{|\nabla R|^2}{2R}.}$$

For ancient solutions ($t \to -\infty$), the $R/t$ term drops out, and we obtain the clean inequality $\partial_t R \geq |\nabla R|^2/(2R)$, which says the scalar curvature is pointwise non-decreasing in time.

Proof Strategy

The proof introduces the quantity

$$Z = g^{ij}\mathcal{H}_{ij} = \frac{\partial R}{\partial t} + \frac{R}{t} + 2\langle \nabla R, V \rangle + 2\,\mathrm{Rc}(V, V)$$

and computes its evolution under the Ricci flow. The key computation shows

$$\left(\frac{\partial}{\partial t} - \Delta\right) Z = B(V, V) + \frac{2}{t}\, Z$$

where $B$ is a quadratic form satisfying

$$B(V, V) \geq 0 \quad \text{whenever } \mathrm{Rm} \geq 0.$$

The term $(2/t)\,Z$ on the right-hand side has the crucial sign: if $Z$ were to become negative, this term would push it back toward zero. Combined with $B \geq 0$, the scalar maximum principle ensures $Z \geq 0$ for all $t > 0$, provided $Z \geq 0$ at $t = 0$ (which holds in the limit by the initial data assumptions).

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