Ancient Solutions and the Harnack Inequality

Singularity models live forever in the past. When you blow up a flow near a singularity, rescaling by a large factor $Q$, the time interval before the singularity is stretched by $Q$ as well. In the limit it becomes infinitely long: the model is an ancient solution, defined for all $t \in (-\infty, T)$. Ancient solutions are rigid, because they have had infinite time to smooth out, and Hamilton's main tool for exploiting this is his Harnack inequality for the Ricci flow (1993). It is a differential inequality for the curvature, the analogue of the Li–Yau inequality for positive solutions of the heat equation, and it lets one compare curvature at different points and times. For ancient solutions with nonnegative curvature operator it gives $\partial_tR \geq 0$: curvature can only increase in time at each point.

By the end of this chapter you will be able to:

• define ancient, eternal and immortal solutions, and explain why blow-up limits are ancient;
• state the trace form of Hamilton's Harnack inequality and describe the matrix form;
• deduce $\partial_tR \geq 0$ for ancient solutions with nonnegative curvature operator;
• integrate the inequality along a space-time path to compare curvatures;
• state Hamilton's rigidity theorem for eternal solutions, and check it on the cigar.

2.1The ancestor

2.2Ancient solutions

A Ricci flow defined for $t \in (-\infty, T)$ is ancient; one defined for all $t \in \mathbb{R}$ is eternal; one defined on $(0, \infty)$, or $[0, \infty)$, is immortal. Examples:

• shrinking round spheres and cylinders, and all shrinking solitons, are ancient;
• steady solitons, such as the cigar and the Bryant soliton (11B.1 Ricci Solitons), are eternal;
• the King–Rosenau sausage on $S^2$ (11A.7 Ricci Flow on Surfaces) is ancient but not a soliton;
• hyperbolic metrics and expanding solitons are immortal.

Why blow-up limits are ancient. If $g(t)$ exists on $[0, T)$ and $Q_k \to \infty$, the rescaled flows $g_k(s) = Q_kg(t_k + \frac{s}{Q_k})$ exist for $s \in [-Q_kt_k, Q_k(T - t_k))$. If $t_k$ stays away from $0$, the left end $-Q_kt_k \to -\infty$, and any limit flow (11B.3 Compactness of Ricci Flows) is defined for all negative times (Exercise 2.5).

2.3Hamilton's Harnack inequality

This is the trace of the matrix Harnack inequality: a certain quadratic form, built from $\operatorname{Ric}$, its first and second derivatives, the curvature tensor and $\frac{1}{2t}\operatorname{Ric}$, is nonnegative on all pairs (a vector $W$ and a 2-form $U$). Hamilton proved it with the maximum principle for systems (11A.4 Maximum Principles under Ricci Flow), applied to the evolution equation of this quadratic form. The proof is a long computation, and the guide does not reproduce it.

Equality. For an expanding gradient soliton with nonnegative curvature operator, after shifting time so that the soliton emerges at $t = 0$, the trace inequality holds with equality for the right choice of $V$, the gradient of the soliton potential up to sign and normalisation. Solitons are the equality cases of Harnack inequalities, as the Gaussian is for Li–Yau.

Ancient solutions. If $g(t)$ is ancient, apply the theorem on $(t_0, T)$, so that $\frac1t$ becomes $\frac{1}{t - t_0}$, and let $t_0 \to -\infty$. With $V = 0$:

2.4Comparing curvature across space-time

Integrating the trace inequality along a path, with $V$ chosen as half the velocity of the path, gives (Exercise 2.7):

Curvature cannot be large at one place and time and then disappear immediately nearby: wherever it was large, it stays comparably large a little later, at nearby points. This is one of the tools for showing that blow-up limits have bounded curvature on their whole past, and in Perelman's work it is replaced by the more powerful reduced distance (12A.5 Reduced Distance and Reduced Volume), whose definition is modelled on the path integral in this estimate.

2.5Eternal solutions and rigidity

The idea: at the space-time maximum, the Harnack quantity attains its minimum value $0$, and a strong maximum principle for the matrix Harnack form makes it vanish identically; the vanishing is exactly the soliton equation. The cigar is the example: in coordinates where it moves by diffeomorphisms, it is $g(t) = \frac{dx^2 + dy^2}{e^{4t} + x^2 + y^2}$, with $R = \frac{4e^{4t}}{e^{4t} + x^2 + y^2}$. At each fixed point $R$ increases in time, and $R(0, t) = 4$ is the maximum over all of space-time (Figure 2.2, Exercise 2.8). Hamilton used the theorem in his attack on the cigar problem: a Type II blow-up limit in dimension three with these properties would split off a cigar factor (11B.4 Singularities).

2.6History

Peter Li and Shing-Tung Yau proved their inequality for the heat equation in 1986. Hamilton proved the Harnack inequality for the Ricci flow on surfaces in 1988, in general dimension in 1993, and the rigidity of eternal solutions the same year. Bennett Chow and Sun-Chin Chu (1995) interpreted the Harnack quantity geometrically, as the curvature of a connection on space-time. Ancient solutions were studied systematically after Perelman, who made them central.

2.7Exercises

Solution

$t_k + \frac{s}{Q_k} \in [0, T)$ iff $s \in [-Q_kt_k, Q_k(T - t_k))$; parabolic rescaling preserves the equation. Since $t_k \to T > 0$, $Q_kt_k \to \infty$. The limit is eternal if also $Q_k(T - t_k) \to \infty$, which happens for Type II singularities with suitable point picking (11B.4 Singularities); for Type I one has $Q_k(T - t_k)$ bounded and the limit is ancient only.

Solution

On $(t_0, T)$, the theorem gives $\partial_tR + \frac{R}{t - t_0} \geq 0$. As $t_0 \to -\infty$, $\frac{R}{t - t_0} \to 0$ (since $R$ is bounded at each time). For the sphere, $R = \frac{n}{2(T - t)}$ and $\partial_tR = \frac{n}{2(T - t)^2} > 0$.

Solution

(a) Chain rule. (b) The inequality with $V = \frac12\gamma'$: $\partial_tR \geq -\frac Rt - \langle\nabla R, \gamma'\rangle - \frac12\operatorname{Ric}(\gamma', \gamma')$, so $\partial_tR + \langle\nabla R, \gamma'\rangle \geq -\frac Rt - \frac12R|\gamma'|^2$. Divide by $R$. (c) $L(t_2) - L(t_1) \geq -\log\frac{t_2}{t_1} - \frac12\int_{t_1}^{t_2}|\gamma'|^2_{g(t)}dt$. Since $\partial_tg = -2\operatorname{Ric} \leq 0$, $|\gamma'|_{g(t)} \leq |\gamma'|_{g(t_1)} = \frac{d_{t_1}(x_1, x_2)}{t_2 - t_1}$, so the integral is at most $\frac{d_{t_1}^2}{t_2 - t_1}$. Exponentiate.

Solution

With $x = \sqrt a\,y$, $\frac{|dx|^2}{a + |x|^2} = \frac{|dy|^2}{1 + |y|^2}$, the cigar, with $R = \frac{4}{1 + |y|^2} = \frac{4a}{a + r^2}$. With $a = e^{4t}$: $\partial_tg = -\frac{4e^{4t}}{(e^{4t} + r^2)^2}|dx|^2$ and $-2\operatorname{Ric} = -Rg = -\frac{4e^{4t}}{e^{4t} + r^2}\cdot\frac{|dx|^2}{e^{4t} + r^2}$: equal. $\partial_tR = \frac{16e^{4t}r^2}{(e^{4t} + r^2)^2} \geq 0$, and $R \leq 4$ with equality at $r = 0$.

Solution

$\nabla R = 0$, $\partial_tR = \frac{1}{(T - t)^2}$, and $\operatorname{Ric}(V, V) \geq 0$. So the left side is $\frac{1}{(T - t)^2} + \frac{1}{(T - t)(t - t_0)} + 2\operatorname{Ric}(V, V) > 0$. It is smallest for $V$ along the $\mathbb{R}$ factor (where $\operatorname{Ric}(V, V) = 0$) and $t_0 \to -\infty$, giving $\frac{1}{(T - t)^2} > 0$: strict. The cylinder is a shrinking soliton, and the trace inequality is an equality only for expanding ones.