How Distances Change

Every argument in Perelman's work compares distances measured at different times. Canonical neighbourhoods are balls of the current metric; the noncollapsing theorem uses cut-off functions of the distance to a point; "bounded curvature at bounded distance" bounds curvature in terms of the distance at the current time; surgery replaces a ball defined by distance. So one needs to know how fast distances can change under the Ricci flow, and in particular how fast they can decrease, which is where positive curvature does its work.

The basic facts are elementary: lengths of fixed curves change at the rate $-\int\operatorname{Ric}(T, T)$, so nonnegative Ricci curvature makes distances shrink and bounded curvature makes all metrics on a time interval uniformly equivalent. The deeper fact, due to Hamilton and sharpened by Perelman, is that distances shrink at a rate controlled only by the Ricci curvature near the two endpoints, however large the curvature is in between. Its proof is a second variation argument along a minimising geodesic, and the same argument bounds the heat operator of the distance function, Perelman's Lemma 8.3. This chapter proves all of it, with the necessary care about distance functions that are not differentiable.

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

• prove that distances are nonincreasing in time under nonnegative Ricci curvature, and that bounded curvature makes metrics and volumes uniformly equivalent;
• work with forward difference quotients, and reduce estimates for $\frac{d}{dt}d_t(x_0, x_1)$ to estimates for the lengths of minimising geodesics;
• prove the two-point estimate $\frac{d}{dt}d_t(x_0, x_1) \geq -2(n - 1)\big(\frac23Kr_0 + r_0^{-1}\big)$ under a Ricci bound near the endpoints only;
• derive the global estimate $\frac{d}{dt}d_t \geq -4(n - 1)\sqrt{2K/3}$ and integrate it;
• prove Perelman's Lemma 8.3(a), $(\partial_t - \Delta)d \geq -(n - 1)\big(\frac23Kr_0 + r_0^{-1}\big)$, in the barrier sense;
• check these on the shrinking sphere and cylinder, and explain how they are used.

6.1Lengths, distances and volumes

For a fixed piecewise smooth curve $\gamma$ and a Ricci flow $g(t)$, 11A.2 How Curvature Evolves showed

with $T$ the unit tangent and $s$ arclength, both for $g(t)$. Two consequences need no further work.

This is used constantly for $\kappa$-solutions, which have nonnegative curvature: going back in time, distances only grow.

Since $|\operatorname{Ric}| \leq (n - 1)|\operatorname{Rm}|$ in a suitable norm, a bound on $|\operatorname{Rm}|$ works as well, with a dimensional factor. This is Morgan and Tian's Lemma 3.15, and it is the reason that, on a parabolic neighbourhood where the curvature is bounded by $r^{-2}$ for a time $r^2$, it does not matter much at which time the balls are measured.

6.2Forward difference quotients

Distance functions are not differentiable in general, in $t$ or in $x$. Following Hamilton, Perelman and Morgan–Tian, time derivatives of such functions are understood through forward difference quotients: for a continuous $f$, "$\frac{df}{dt}(t_0) \geq c$" means $\liminf_{h \downarrow 0}\frac{f(t_0 + h) - f(t_0)}{h} \geq c$, and "$\frac{df}{dt}(t_0) \leq c$" means $\limsup_{h \downarrow 0}\frac{f(t_0 + h) - f(t_0)}{h} \leq c$. Two facts make this notion usable.

• Comparison. If $\frac{df}{dt} \leq \psi(t, f)$ in this sense at every $t \in [a, b)$, with $\psi$ of class $C^1$, and $G' = \psi(t, G)$ with $f(a) \leq G(a)$, then $f \leq G$ on $[a, b]$ (Morgan and Tian's Lemma 2.22; the proof is the usual one for ODE comparison, run with $f - G - \varepsilon e^{Ct}$). Similarly with the inequalities reversed.
• Maxima. The maximum over $x$ of a smooth function $F(x, t)$ on a compact manifold has upper forward derivative at most the maximum of $\partial_tF$ over the points where the maximum is attained (Hamilton's trick, 9B.7 The Heat Equation on a Manifold; Morgan and Tian's Proposition 2.23).

For distances there is a version that reduces everything to the lengths of geodesics.

6.3The two-point estimate

If the Ricci curvature is bounded above by $(n - 1)K$ everywhere, the length of any geodesic of length $d$ decreases at most at rate $(n - 1)Kd$, which is large for long geodesics. The remarkable fact is that minimising geodesics do much better: the second variation inequality, which expresses that a minimising geodesic cannot be shortened, forces the Ricci curvature along its middle to be small on average, so that only the ends contribute.

What happens in the middle of the geodesic has disappeared. If the curvature there is huge and positive, the geodesic is not affected, because a minimising geodesic cannot run through a region of large positive Ricci curvature over a long stretch: the second variation would make it unstable (this is Bonnet–Myers' argument, 9A.7 Jacobi Fields and Curvature versus Topology, localised). If the curvature is negative in the middle, that only makes the length increase. In a region of very high curvature, such as a thin neck, distances across it can still change only at a rate set by the curvature near the endpoints.

So under a Ricci upper bound $K$, distances shrink at a rate of order $\sqrt K$, not $Kd$: they can lose at most about $\sqrt K$ per unit time in absolute terms, whatever their size. On a flow approaching a singularity, with $K(t) \sim \frac{C}{T - t}$, the integral $\int\sqrt{K(t)}\,dt \sim 2\sqrt{C(T - t)}$ converges: distances have limits at the singular time, a fact used to define the limit metric on the region where the curvature stays bounded (12B.9 Surgery).

6.4The heat operator of the distance function

The same computation, with the geodesic ending at a general point $x$ and the second variation bounding the Hessian of the distance, controls the combination $\partial_td - \Delta d$. This is what lets the distance function be used inside maximum principle arguments, through cut-off functions $\varphi(d_t(x_0, \cdot))$.

Two remarks on how the lemma is used.

• The sign. The inequality bounds $\partial_td - \Delta d$ below. For a cut-off function $\eta = \varphi(d)$ with $\varphi$ nondecreasing (equal to $1$ near $x_0$ and growing outwards, as Perelman uses it, or decreasing, as in 11A.5 Shi's Derivative Estimates, with the sign reversed accordingly), $\square\eta = \varphi'\square d - \varphi''|\nabla d|^2$, and a lower bound on $\square d$ is exactly what controls $\square\eta$ in the maximum principle.
• Only local curvature. The hypothesis is a Ricci bound on $B(x_0, r_0)$ only, not near $x$. That is why the lemma can be used in Perelman's pseudolocality theorem and in the noncollapsing argument, where curvature is controlled near one point and nothing is known far away (12A.4 κ-Noncollapsing, 12A.9 Pseudolocality).

6.5Examples

The shrinking sphere. On $S^n$ with $g(t) = (1 - 2(n - 1)t)g_0$, every distance scales: $d_t = \sqrt{1 - 2(n - 1)t}\,d_0$, so $\partial_td = -\frac{(n - 1)d_0}{\sqrt{1 - 2(n - 1)t}} = -(n - 1)Kd$ with $K = \frac{1}{1 - 2(n - 1)t}$ the current curvature, and $\operatorname{Ric} = (n - 1)Kg$. For the antipodal distance $d = \frac{\pi}{\sqrt K}$, $\partial_td = -(n - 1)\pi\sqrt K$, while Corollary 6.5 allows at most $4(n - 1)\sqrt{\frac{2}{3}}\sqrt K \approx 3.27(n - 1)\sqrt K$. The true rate, $\pi(n - 1)\sqrt K \approx 3.14(n - 1)\sqrt K$, is just inside the bound: the estimate is nearly sharp.

The cylinder. On $S^2(\rho(t))\times\mathbb{R}$, two points on the same line $\{\omega\}\times\mathbb{R}$ stay at constant distance: the line is a minimising geodesic along which $\operatorname{Ric}(X, X) = 0$. Two points far apart in $\mathbb{R}$ and on different spheres: the minimising geodesic is nearly parallel to the line, and its length decreases only because of its short sphere component, by an amount bounded independently of the distance between the points, as the theorem predicts.

6.6History

Hamilton proved the two-point estimate in "The formation of singularities in the Ricci flow" (1995, §17), as a tool for comparing distances near singularities. Perelman stated it, with the heat-operator version (a), as Lemma 8.3 of his first preprint (2002), with a four-line proof. Morgan and Tian's Proposition 3.21 and its corollaries write out version (b) in detail; version (a) is proved here in the same way.

6.7Exercises

Solution

$\partial_tdV = -R\,dV$ gives $|\partial_t\log dV| \leq K$ pointwise; integrate over $U$. On $S^2(\rho)\times\Sigma(\sigma)$, with $\Sigma(\sigma)$ a hyperbolic surface of curvature $-\frac{1}{\sigma^2}$, the factors evolve independently, $\rho^2 = \rho_0^2 - 2t$ and $\sigma^2 = \sigma_0^2 + 2t$, and $R = \frac{2}{\rho^2} - \frac{2}{\sigma^2}$. With $\rho_0 = \sigma_0$, $R = 0$ at $t = 0$, yet at that instant lengths in the sphere factor decrease and lengths in the hyperbolic factor increase, at equal and opposite rates; only the volume element is stationary.

Solution

Fix $\varepsilon > 0$, let $f_\varepsilon = f - \varepsilon t$, and $S = \{t \in [a, b] : f_\varepsilon(s) \leq f_\varepsilon(a) \text{ for all } s \in [a, t]\}$, an interval $[a, t^*]$ by continuity. If $t^* < b$, then at $t^*$ the upper forward derivative of $f_\varepsilon$ is at most $-\varepsilon < 0$, so $f_\varepsilon(t^* + h) < f_\varepsilon(t^*) \leq f_\varepsilon(a)$ for small $h > 0$, and $S$ extends beyond $t^*$: a contradiction. So $f_\varepsilon(t) \leq f_\varepsilon(a)$ on $[a, b]$, that is $f(t) \leq f(a) + \varepsilon(t - a)$, for every $\varepsilon$; and the same argument from any starting point gives monotonicity.

Solution

The derivative $\frac23K - r_0^{-2}$ vanishes at $r_0^2 = \frac{3}{2K}$, giving the value $\frac23K\sqrt{\frac{3}{2K}} + \sqrt{\frac{2K}{3}} = 2\sqrt{\frac{2K}{3}}$. As $r_0 \to 0$ the term $r_0^{-1}$, which comes from $|D_uV|^2 = f'^2$ on the short ramps of the test field, blows up: bending the variation field abruptly costs energy. As $r_0 \to \infty$ the curvature term grows, and in any case $r_0 \leq \frac d2$ is needed so that the two ramps do not overlap. The optimal $r_0$ is the curvature scale $K^{-1/2}$, up to a constant.

Solution

On the flat cylinder $\partial_td = 0$, and the universal cover is the plane, where $\Delta|z| = \frac{1}{|z|}$; so away from the cut locus $\Delta d = \frac1d$ and $\partial_td - \Delta d = -\frac1d \geq -\frac{1}{r_0}$ for $d \geq r_0$, which is Lemma 8.3(a) with $K = 0$ and $n - 1 = 1$. On the round cylinder, for $x = (\omega, s)$ with $s$ large and $\omega$ near the point of $x_0$'s sphere, the minimising geodesic is nearly the line, $\operatorname{Ric}(X, X) \approx 0$ along it, so $\partial_td \approx 0$; and $\Delta d \approx 0$ as well, since the level sets of $d$ far away are nearly the cross-sectional spheres, which are totally geodesic. The bound $-(n - 1)(\frac23Kr_0 + r_0^{-1})$ with $K = \frac12$ and $r_0 = \sqrt3$ is about $-2.3$: true but far from sharp here.

Solution

By Proposition 6.1, $d_t$ is nonincreasing and nonnegative, so it has a limit $d_T$, and $d_t \geq d_T$. By Corollary 6.5 with $K(t) = \frac{C}{T - t}$, for $a < b < T$,

$$d_a \leq d_b + 4(n - 1)\int_a^b\sqrt{\frac{2C}{3(T - t)}}\,dt = d_b + 8(n - 1)\sqrt{\frac{2C}{3}}\big(\sqrt{T - a} - \sqrt{T - b}\big).$$

Letting $b \to T$ gives $d_a \leq d_T + c\sqrt{T - a}$. Monotonicity alone gives a limit but says nothing about how fast it is approached: without the upper bound, distances could drop by a definite amount in an arbitrarily short time just before $T$. The upper Ricci bound, through the two-point estimate, limits the total loss after time $t$ to $c\sqrt{T - t}$, uniformly in the points.