Curve Shortening and Minimal Discs

The last step of Perelman's proof of the Poincaré conjecture is a computation that fits in a few lines: the area of a minimal disc spanning a loop decreases at least at the rate $2\pi$, when the metric moves by the Ricci flow and the loop moves by the curve-shortening flow. Everything else in the third preprint, and most of Morgan and Tian's chapter 18, is there to make that computation legitimate for whole families of loops, some of them badly behaved. This chapter prepares the computation and its ingredients: the curve-shortening flow in the plane and in a Riemannian manifold with a changing metric, the evolution of length and curvature, the trick of Altschuler and Grayson that keeps families of curves under control, minimal discs and their area, and finally Perelman's estimate and the extinction it forces.

It is the companion of 10A.11 Min–Max and Width. There the min–max objects were 2-spheres sweeping out the manifold, as in Colding and Minicozzi's argument; here they are loops and the discs they bound, as in Perelman's own. The two arguments share the essential mechanism, Gauss–Bonnet supplying a fixed negative rate, and differ in where the technical difficulties lie.

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

• derive the curve-shortening flow as the gradient flow of length, and compute the evolution of length, enclosed area and curvature in the plane;
• write the flow in a Riemannian manifold whose metric moves by the Ricci flow, and derive the evolution of the length element;
• explain why families of curves are a problem, and how Altschuler and Grayson's ramps in $M\times S^1$ solve it;
• define the least area $A(c, g)$ of a disc spanning a loop and Perelman's min–max invariant for families of loops;
• prove Perelman's estimate $\frac{dA}{dt} \leq -2\pi - \frac12R_{\min}A$ for a smooth minimal disc, from the first variation of area with moving boundary, the traced Gauss equation and Gauss–Bonnet;
• integrate the estimate with the lower bound on scalar curvature to obtain finite extinction.

12.1Curves that shrink by their curvature

Let $c : S^1\times[0, T) \to \mathbb{R}^2$ be a family of immersed closed curves, $c_t = c(\cdot, t)$, with arclength $s$, unit tangent $S = \partial_sc$ and curvature vector $H = \partial_sS = \partial_s^2c$, of length $k = |H|$. The curve-shortening flow is

Each point moves towards its local centre of curvature at a speed equal to the curvature; a straight segment does not move, a tight bend moves fast. The flow was introduced by W. W. Mullins in 1956 as a model of the motion of grain boundaries in metals, and studied mathematically from the 1980s; 6A.10 Curve Shortening and the First Geometric Flows met it as the first geometric heat equation.

It is the gradient flow of length. For a variation $c_\epsilon$ of a closed curve with variation field $V = \partial_\epsilon c$, the first variation of length is

integrating by parts around the closed curve. So among velocities $V$ of fixed $L^2$ norm, $V = H$ decreases length fastest, and along the flow

The circle. A circle of radius $r$ has $k = \frac1r$ and $H$ pointing to the centre, so it stays a circle with $r' = -\frac1r$: $r(t)^2 = r_0^2 - 2t$, shrinking to a point at $t = \frac{r_0^2}{2}$. The same law $(\text{size})^2 = \text{const} - 2t$ governs the $S^2$ factor of the shrinking neck (11A.1 The Equation and Its First Solutions): both are self-similar shrinkers.

Enclosed area. For an embedded closed curve bounding a region of area $\mathcal A$, oriented counterclockwise, the inward normal is $\nu$ and $H = k_{\mathrm s}\nu$ with signed curvature $k_{\mathrm s}$. The area changes by the outward normal velocity integrated along the curve:

by the theorem of turning tangents (8A.9 The Curvature of Surfaces): the total signed curvature of a simple closed curve is $2\pi$. Every embedded curve loses area at exactly the rate $2\pi$, whatever its shape, so it must disappear by time $\frac{\mathcal A(0)}{2\pi}$. This is the planar ancestor of Perelman's estimate, and the constant $2\pi$ is the same Gauss–Bonnet constant.

What happens before it disappears is the content of two famous theorems (statements). Michael Gage and Richard Hamilton (1986) proved that a convex curve stays convex and shrinks to a point, becoming round in shape; Matthew Grayson (1987) proved that an embedded curve becomes convex before it becomes singular. So embedded curves in the plane shrink to round points. Immersed curves with self-crossings can instead develop singularities in which a small loop collapses, a cusp-like pinch.

Curvature. In the plane the curvature evolves by

a heat equation with a cubic reaction term (Exercise 12.5): diffusion along the curve, and growth where the curve is already bent. The reaction is what drives the curvature to infinity at a shrinking point, exactly as the reaction $R^2$ drives scalar curvature to infinity on a shrinking surface (11A.12 Ricci Flow on Surfaces).

12.2The flow in a Riemannian manifold, with a moving metric

In Perelman's argument the ambient manifold is $(M, g(t))$ with $g(t)$ a Ricci flow, and the curves move by the curve-shortening flow of the current metric. Let $c : S^1\times[t_0, t_1] \to M$, write $X = \partial_xc$ for the velocity in the fixed parameter $x \in S^1$, $S = \frac{X}{|X|}$ the unit tangent, and

the curvature vector of $c_t$ in $(M, g(t))$, with $k = |H|$. The flow is $\partial_tc = H$. Then:

So length decreases by curvature, as in the plane, and in addition the Ricci flow shrinks it where the Ricci curvature is positive (11A.2 How Curvature Evolves). Perelman records two further consequences in his §2.1 (statements, with the source of each term).

• Curvature. $\partial_tk \leq \partial_s^2k + k^3 + C(k + 1)$, where $C$ bounds the curvature of $g(t)$ and its derivative: the planar equation plus error terms from the curvature of the ambient space and from the motion of the metric.
• Total curvature. $\Theta = \int k\,ds$ satisfies $\frac{d\Theta}{dt} \leq \int C(k + 1)\,ds$, and with $\frac{dL}{dt} \leq \int(C - k^2)\,ds$ this shows that length and total curvature grow at most exponentially in $t$; in a flat ambient space both would be nonincreasing.

Short-time existence holds for smoothly immersed initial curves, and the solution exists as long as the curvature stays bounded (Perelman cites Gage and Hamilton for this). If the curvature does not concentrate at some time, in the sense that each arc of length $r$ has total curvature at most a small $\varepsilon$, then the curvature and its derivatives are controlled for a time of order $\varepsilon r^2$ afterwards (Perelman's §2.2, adapting Ecker and Huisken; statement). That is the local regularity the arguments use.

12.3Families of curves, and ramps

Perelman's invariant is defined for families of loops, a 2-sphere's worth of them (10A.4 Homotopy Groups, Degree and the Hurewicz Theorem), and the flow must be applied to the whole family at once, continuously in the family parameter. Two difficulties appear.

• A loop in the family need not be immersed. It can traverse an arc and come back along it, with zero velocity at the turning point. There is no curvature vector there, and no evident way to define the flow so that it depends continuously on the curve.
• Even immersed curves can become singular, and different curves in a family can become singular at different times.

The solution, due to Steven Altschuler and Matthew Grayson (1992), is to add a dimension. Replace $M$ by $M\times S^1_\lambda$, the product with a circle of small length $\lambda$, with the product metric $g(t) + d\theta^2$ (still a Ricci flow, since the circle is flat), and replace each loop $c$ by a loop $\hat c$ in the product that winds once around the circle factor and projects to $c$: $\hat c(x) = (c(x), \lambda x)$ for $x \in [0, 1)$. Let $U$ be the unit vector field along the circle factor.

A ramp is automatically an embedded, immersed curve: it is a graph over the circle factor. And the lifted curve $\hat c$ above is a ramp even when $c$ is not immersed, because its velocity always has the component $\lambda$ along $U$.

Combining this with the curvature inequality, Altschuler and Grayson estimate the ratio $\frac ku$ by the maximum principle and conclude that the curvature of a ramp stays bounded: a ramp has no singularities, and the flow exists on the whole time interval, depending continuously on the initial curve. As $\lambda \to 0$, the projections of the ramps to $M$ approximate a "regularised" flow of the original family. Perelman's §3 and Morgan and Tian's chapter 18, §4 to §7, carry this out: the hard work is to show that in the limit the estimates for minimal discs survive, which needs control of minimal annuli between nearby ramps (their area grows at most exponentially, by a computation like the one below, Perelman's §2.4), and a lower bound on the lengths of curves bounding small annuli (§2.5).

12.4Discs of least area

Let $c$ be a contractible loop in $(M, g)$. The Plateau problem asks for a disc of least area with boundary $c$. Jesse Douglas and Tibor Radó solved it in $\mathbb{R}^3$ (1930–31), and Charles Morrey (1948) in Riemannian manifolds with bounded geometry: for a rectifiable Jordan curve $c$ there is a map of the disc, conformal and harmonic, of least area among discs spanning $c$, and it is a minimal surface, smooth (even analytic) in the interior, immersed except at isolated branch points (statement; Robert Osserman and Robert Gulliver proved the absence of interior branch points for area minimisers in 3-manifolds). Following Perelman, define

the least area of a disc spanning $c$. For a family $\Gamma$ of contractible loops let $A(\Gamma, g) = \sup_{c \in \Gamma}A(c, g)$, and for a nontrivial homotopy class $\alpha$ of such families let

This is a min–max quantity of the same type as the width of 10A.11 Min–Max and Width. The relevant homotopy classes are elements of $\pi_2(\Lambda M, M)$, where $\Lambda M$ is the space of contractible loops and $M \subset \Lambda M$ the constant loops; a class is nontrivial if its families cannot be deformed into families of constant loops. Perelman notes that such a class exists whenever $M$ is not aspherical, by a classical result of Jean-Pierre Serre on the homotopy of loop spaces; for $M$ with $\pi_2(M) = 0$ and $\pi_3(M) \neq 0$ it is the class of 10A.4 Homotopy Groups, Degree and the Hurewicz Theorem, $\pi_2(\Lambda M) \cong \pi_3(M)$. Nontriviality makes $A(\alpha, g)$ positive: if every loop of a family bounded a disc of tiny area, the family could be contracted through those discs.

12.5Perelman's estimate

The same inequality then holds for $A(\alpha, g(t))$, the min–max over families (Perelman's Lemma 1.2), provided the computation can be made for the loops near the maximum of a nearly optimal family. That is where the difficulties lie: the loops near the maximum need not be embedded, the minimal discs may have branch points (which only help: they contribute with a favourable sign in Gauss–Bonnet), and the flow of the family must be defined continuously. The ramps of the previous section handle the flow, and the estimates of Perelman's §3, expanded in Morgan and Tian's §18.5–18.7, handle the rest.

12.6Extinction

The scalar curvature satisfies $\partial_tR = \Delta R + 2|\operatorname{Ric}|^2 \geq \Delta R + \frac23R^2$ in dimension three, so by the maximum principle $R_{\min}(t) \geq -\frac{3}{2(t + C)}$ for a constant $C$ depending on the initial metric (11A.7 Maximum Principles under Ricci Flow for the maximum principle; the computation is the last exercise of 9B.6 Scalar Curvature and Topology). So $-\frac12R_{\min} \leq \frac{3}{4(t + C)}$ and, writing $A(t) = A(\alpha, g(t))$,

Perelman integrates this in one line. Since $\frac34 \leq 1$, $\frac{dA}{dt} \leq -2\pi + \frac{A}{t + C}$, and then $\hat A = \frac{A}{t + C}$ satisfies

The right side is not integrable at infinity, while $\hat A \geq 0$; so the flow cannot exist for all time with $\alpha$ nontrivial. With the exact exponent one gets an explicit bound: $\phi(t) = (t + C)^{-3/4}$ gives $\frac{d}{dt}(\phi A) \leq -2\pi\phi$, and $\phi A$ would become negative once $8\pi\big((t + C)^{1/4} - C^{1/4}\big) > C^{-3/4}A(0)$.

Through surgery. At a surgery time, Perelman's construction gives maps from the pre-surgery manifold to the post-surgery one that are $(1 + \xi)$-Lipschitz for any prescribed $\xi > 0$ (for irreducible $M$ the surgeries are topologically trivial: one component is diffeomorphic to the old manifold and the others are spheres). Such maps do not increase areas of discs by more than the factor $(1 + \xi)^2$, so $A$ does not jump up, and the differential inequality persists across surgeries. In the general case (prime decomposition without aspherical factors) only finitely many surgeries are topologically nontrivial, by Kneser's finiteness theorem; after them every component satisfies the hypotheses, and the argument applies to each (Perelman's §1.5). The conclusion is Perelman's Theorem 1.1: for a closed orientable 3-manifold whose prime decomposition has no aspherical factors, the Ricci flow with surgery becomes extinct in finite time. For a simply connected $M$ that, with 10A.6 Necks, Caps and Their Topology, is the Poincaré conjecture.

12.7History

W. W. Mullins introduced the curve-shortening flow in 1956, for grain boundaries. Gage and Hamilton proved that convex plane curves shrink to round points (1986), and Grayson that embedded ones become convex (1987); Grayson also studied the flow on surfaces (1989). Altschuler and Grayson introduced the ramp regularisation in "Shortening space curves and flow through singularities" (1992). Douglas and Radó solved the Plateau problem in 1930–31, Morrey in Riemannian manifolds in 1948. Perelman's third preprint, with the disc argument, appeared in July 2003; Morgan and Tian wrote it out in full in 2007, adding the arguments for two points that Perelman had stated briefly.

12.8Exercises

Solution

With $ds = |X|\,dx$, Proposition 12.1 with $\operatorname{Ric} = 0$ gives $\partial_t|X| = -k^2|X|$. Then $\partial_t\partial_s = \partial_t(|X|^{-1}\partial_x) = k^2|X|^{-1}\partial_x + |X|^{-1}\partial_x\partial_t = k^2\partial_s + \partial_s\partial_t$. For a convex curve write $S = (\cos\theta, \sin\theta)$ with $\partial_s\theta = k$ and $H = kN$, $N = (-\sin\theta, \cos\theta)$. Then $\partial_tS = \partial_t\partial_sc = \partial_s\partial_tc + k^2S = \partial_s(kN) + k^2S = (\partial_sk)N - k^2S + k^2S = (\partial_sk)N$, so $\partial_t\theta = \partial_sk$. Finally $\partial_tk = \partial_t\partial_s\theta = \partial_s\partial_t\theta + k^2\partial_s\theta = \partial_s^2k + k^3$.

Solution

The region moves inward at normal speed $k_{\mathrm s}$ (signed curvature with respect to the inward normal for a counterclockwise curve), so $\mathcal A' = -\oint k_{\mathrm s}\,ds = -2\pi$ by the theorem of turning tangents. A curve of area $10$ survives until $t = \frac{10}{2\pi} \approx 1.59$ (it stays embedded and becomes convex by Grayson's theorem, then shrinks to a point exactly then). For a figure-eight the total signed curvature is $0$, not $2\pi$ (the two lobes turn in opposite directions), the "enclosed area" with signs is the difference of the lobes, and the curve is not embedded: the turning-tangents theorem does not apply. In fact a lobe can collapse first, producing a singularity before the whole curve disappears.

Solution

$g(t) = (1 - 4t)g_0$ has the same geodesics as $g_0$ (Christoffel symbols are unchanged by constant scaling), so $c$ is a closed geodesic for all $t$: $H = 0$ and the curve stays put. Its length is $2\pi\sqrt{1 - 4t}$, with derivative $-\frac{4\pi}{\sqrt{1 - 4t}}$. On the other hand $\operatorname{Ric}(g(t)) = 2g_0 = \frac{2}{1 - 4t}g(t)$, so $\operatorname{Ric}(S, S) = \frac{2}{1 - 4t}$, and $-\int\operatorname{Ric}(S, S)\,ds = -\frac{2}{1 - 4t}\cdot2\pi\sqrt{1 - 4t} = -\frac{4\pi}{\sqrt{1 - 4t}}$. They agree, with $k = 0$.

Solution

As in the proof, $\operatorname{tr}_D\operatorname{Ric} = 2K_{12} + K_{1\nu} + K_{2\nu} = \frac R2 + K_{12}$, and $K_D = K_{12} + \det\operatorname{II}$, so $\operatorname{tr}_D\operatorname{Ric} = \frac R2 + K_D - \det\operatorname{II}$. For a minimal surface $\det\operatorname{II} = -\lambda^2 = -\frac12|\operatorname{II}|^2 \leq 0$. Comparing: $\operatorname{tr}_D\operatorname{Ric} = R - \operatorname{Ric}(\nu, \nu)$, and the traced Gauss equation gives $\operatorname{Ric}(\nu, \nu) = \frac12(R - R_D - |\operatorname{II}|^2)$ with $R_D = 2K_D$, so $R - \operatorname{Ric}(\nu, \nu) = \frac12R + K_D + \frac12|\operatorname{II}|^2 = \frac12R + K_D - \det\operatorname{II}$, the same identity.

Solution

$\hat c' = (0, \lambda)$, so $S = U$ and $u = 1 > 0$: a ramp. It is a straight line in the flat torus, hence a geodesic, with $H = 0$, so it does not move. For the second loop, $\hat c'(x) = (c'(x), \lambda)$ with $c'(x) = \pm1$ except at $x = \frac12$, where $c$ turns back; but the second component $\lambda$ never vanishes, so $\hat c$ is immersed (indeed a ramp, $u = \frac{\lambda}{\sqrt{1 + \lambda^2}} > 0$ away from the corner, and it can be smoothed there). The circle direction gives every lifted curve a nonzero velocity, which is the whole point of the trick.

Solution

With $\phi = (t + 1)^{-3/4}$, $(\phi A)' \leq -2\pi(t + 1)^{-3/4}$, so $\phi(T)A(T) \leq A_0 - 8\pi\big((T + 1)^{1/4} - 1\big)$. This is negative once $(T + 1)^{1/4} > 1 + \frac{A_0}{8\pi}$, so $T(A_0) = \big(1 + \frac{A_0}{8\pi}\big)^4 - 1$ works. For $A_0 = 4\pi$: $T = (1.5)^4 - 1 = 4.0625$. The round sphere becomes extinct much sooner, at $\frac14$: the estimate uses only the universal bound on $R_{\min}$, while on the sphere $R = \frac{6}{1 - 4t}$ is large and positive, and for the hemisphere of area $2\pi(1 - 4t)$ the term $-\frac12R_{\min}A = -6\pi$, together with $-2\pi$, gives exactly its rate of decrease, $8\pi$. The universal estimate is crude but needs nothing about the metric.