Curve Shortening and the First Geometric Flows

Before the Ricci flow moves metrics, there are flows that move shapes you can draw. The simplest is curve shortening: a closed curve in the plane moves at each point with velocity equal to its curvature vector, so bends straighten and the curve shrinks. It is a heat equation for the curve, and almost every phenomenon of the Ricci flow can be seen in it first, in a picture. Round curves shrink self-similarly to a point, as round spheres do under Ricci flow. There are solitons that translate without changing shape. There are ancient solutions that exist for all negative time. Arbitrary embedded curves become convex and then round before they disappear, which is the planar version of Hamilton's 1982 theorem. And when the same flow acts on surfaces, as mean curvature flow, a dumbbell develops a thin neck that pinches off: the picture that gave the "neckpinch" its name.

Huisken's monotonicity formula, at the end of the chapter, is the first of the Gaussian-weighted monotone quantities of which Perelman's reduced volume is the Ricci flow version.

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

• define the curvature of a plane curve and the curve shortening flow, and write it as a quasilinear heat equation;
• compute the evolution of length and enclosed area, and deduce a bound on the extinction time;
• verify the shrinking circle, the grim reaper and the paperclip as exact solutions;
• state the theorems of Gage–Hamilton and Grayson, and the avoidance principle;
• describe mean curvature flow, the dumbbell neckpinch, and Huisken's monotonicity formula.

8.1Grain boundaries move by curvature

8.2Curves, curvature and the flow

Let $\gamma : \mathbb{R}/\mathbb{Z} \to \mathbb{R}^2$ be a smooth closed curve with $|\gamma'| > 0$, and $s$ its arc length. The unit tangent is $T = \gamma_s$, and the unit normal $N$ is $T$ rotated by $+90°$. If $\theta$ is the angle of $T$, the curvature is

For a circle of radius $r$ traversed counterclockwise, $\kappa = 1/r$ and $N$ points to the centre. For any closed embedded curve traversed counterclockwise, the tangent turns once, so

(Hopf's Umlaufsatz, the theorem of turning tangents; 8A.9 The Curvature of Surfaces treats it with Gauss–Bonnet). The vector $\kappa N = \gamma_{ss}$ is the curvature vector.

The second form looks exactly like the heat equation, $\gamma_t = \gamma_{ss}$, but it is not linear: arc length $s$ depends on the curve itself. Writing the curve locally as a graph $y = u(x, t)$ and allowing tangential motion, the flow becomes (Exercise 8.5)

a quasilinear parabolic equation, strictly parabolic since $\frac{1}{1 + u_x^2} > 0$. So short-time existence follows from 6A.7 Nonlinear Parabolic Equations, and the comparison principle of 6A.4 Maximum Principles applies. Its geometric form is the avoidance principle: two disjoint closed curves moving by curve shortening stay disjoint, and an embedded curve stays embedded. A curve inside a circle stays inside the shrinking circle, and so must disappear no later than the circle does.

Tangential motion only reparametrises the curve, so the flow is really about the normal velocity, which equals the curvature. This is the first appearance of the theme that becomes DeTurck's trick for the Ricci flow (11A.3 Short-Time Existence and Uniqueness): a geometric flow is determined only up to reparametrisation, and choosing a good parametrisation turns a degenerate equation into a strictly parabolic one.

8.3Length and area

The first formula says that curve shortening decreases length as fast as possible: it is the gradient flow of length (6A.9 Calculus of Variations and Gradient Flows), which is where its name comes from. The second has a startling consequence: the area decreases at exactly $2\pi$ per unit time, for every embedded curve, so a curve enclosing area $A$ cannot exist beyond time $\frac{A}{2\pi}$. Grayson's theorem below says that it exists exactly until then, shrinking to a point at $T = \frac{A}{2\pi}$ (Figure 8.1).

8.4Exact solutions

The shrinking circle. A circle of radius $r(t)$ centred at the origin moves inward at speed $\kappa = 1/r$, so $r' = -1/r$ and

It shrinks homothetically and disappears at $t = r_0^2/2$, exactly the area bound $\pi r_0^2/2\pi$. It is a shrinking soliton: it changes only by scaling. Compare the round sphere under Ricci flow, $g(t) = (1 - 2(n - 1)t)g_0$ (6A.1 What a PDE Is).

The grim reaper. The graph $y = t - \log\cos x$, $|x| < \frac\pi2$, satisfies $u_t = 1 = \frac{u_{xx}}{1 + u_x^2}$, since $u_x = \tan x$ and $u_{xx} = \sec^2x$ (Exercise 8.7). It moves upward at unit speed without changing shape: a translating soliton. It is the curve-shortening analogue of the steady solitons of the Ricci flow, such as the cigar (5A.5 Uniformization and the Two-Dimensional Ricci Flow, 11B.1 Ricci Solitons).

The paperclip. For $t < 0$, the curves

form a solution of curve shortening (Exercise 8.8). For very negative $t$ the curve is long and thin, made of two grim reapers joined end to end; as $t \to 0$ it becomes round and shrinks to the origin. It exists for all negative times, so it is an ancient solution, and it is not self-similar. It is the model of the ancient ovals of the Ricci flow and mean curvature flow, and an example of the kind of solution that blow-up limits produce (6A.7 Nonlinear Parabolic Equations, 11B.2 Ancient Solutions and the Harnack Inequality).

8.5The theorems

Together: every embedded closed curve becomes round before it disappears, with nothing else possible. The proofs combine maximum principles for the curvature, which evolves by $\kappa_t = \kappa_{ss} + \kappa^3$ (Exercise 8.9), monotone quantities, and blow-up analysis. Gage–Hamilton is the precise analogue of Hamilton's 1982 theorem that a three-manifold with positive Ricci curvature flows to a round metric (11A.6 Hamilton’s 1982 Theorem), and the reason is shared: positivity of curvature is preserved, and the flow then makes curvature more uniform. Grayson's theorem has no exact Ricci flow analogue in dimension three: there, as the next section's picture shows, singularities can form before anything becomes round, and that is why surgery is needed.

8.6Mean curvature flow and the neckpinch

The same flow makes sense for surfaces in $\mathbb{R}^3$, and hypersurfaces in $\mathbb{R}^{n+1}$: each point moves with normal velocity equal to the mean curvature $H$, the sum of the principal curvatures (8A.9 The Curvature of Surfaces). This is mean curvature flow, the gradient flow of area, and it is a quasilinear parabolic system for the surface. A round sphere of radius $r$ in $\mathbb{R}^{n+1}$ has $H = n/r$ and shrinks with $r^2 = r_0^2 - 2nt$; a round cylinder $S^{n-1}\times\mathbb{R}$ shrinks with $r^2 = r_0^2 - 2(n - 1)t$, keeping its axis. Gerhard Huisken proved in 1984 that convex surfaces shrink to round points, the analogue of Gage–Hamilton.

But Grayson's theorem fails for surfaces. Matthew Grayson observed in 1989 that a dumbbell, two large spheres joined by a thin neck, does not become convex: the neck has large curvature around it and pinches off first, while the two bulbs are still large (Figure 8.3). Near the pinch the surface looks like a shrinking cylinder. After the pinch, the two pieces become convex and shrink to round points separately.

8.7Huisken's monotonicity formula

How can one analyse the moment of pinching? Gerhard Huisken found in 1990 a quantity that is monotone along every mean curvature flow. Fix a point $x_0$ in space and a time $t_0$, and weight the surface $M_t$ by the backward heat kernel centred there:

for an $n$-dimensional surface. Then

where $\vec H$ is the mean curvature vector and $^\perp$ the normal component. The Gaussian-weighted area never increases, and it is constant exactly when the surface is a self-similarly shrinking solution centred at $(x_0, t_0)$. At a singular point, rescaling and passing to the limit therefore produces a self-shrinker; in the neckpinch it is the round cylinder.

8.8History

Curve shortening was studied by Mullins (1956) as a model of grain growth, and its mathematics by Michael Gage, Richard Hamilton and Matthew Grayson in the 1980s: Gage–Hamilton's convex theorem appeared in 1986 and Grayson's embedded theorem in 1987. Kenneth Brakke introduced mean curvature flow of general surfaces in geometric measure theory in 1978. Gerhard Huisken's convex theorem dates from 1984 and his monotonicity formula from 1990. Grayson's dumbbell example appeared in 1989. Panagiota Daskalopoulos, Richard Hamilton and Nataša Šešum proved in 2010 that the shrinking circle and the paperclip are the only compact convex ancient solutions. Osher and Sethian's level-set method dates from 1988, and the weak level-set theory of mean curvature flow from 1991. Von Neumann's $n - 6$ rule was published in 1952.

8.9Exercises

Solution

$\partial_t|\gamma_p|^2 = 2\gamma_p\cdot\partial_p(VN) = 2\gamma_p\cdot(V_pN + VN_p) = 2V\gamma_p\cdot N_p = 2V|\gamma_p|T\cdot(-\kappa T|\gamma_p|) = -2\kappa V|\gamma_p|^2$, using $\gamma_p\cdot N = 0$. Divide by $2|\gamma_p|$.

Solution

$u_x = \tan x$, $u_{xx} = \sec^2x = 1 + u_x^2$. For translators $u = t + w(x)$: $\frac{w''}{1 + w'^2} = 1$, so $\arctan w' = x + c$, and symmetry gives $c = 0$, $w' = \tan x$, $w = -\log\cos x$ on $(-\frac\pi2, \frac\pi2)$.

Solution

$F_t = e^t\cosh x$, $F_x = e^t\sinh x$, $F_y = \sin y$, $F_{xx} = e^t\cosh x$, $F_{yy} = \cos y$, $F_{xy} = 0$. Left: $e^t\cosh x\,(e^{2t}\sinh^2x + \sin^2y)$. Right: $e^t\cosh x\sin^2y + \cos y\,e^{2t}\sinh^2x$, and on the curve $\cos y = e^t\cosh x$, so the two agree.

Solution

(a) As in Proposition 8.2, with $V = M\gamma\kappa$. (b) The interior angle at each junction is $120°$, so the tangent turns by $180° - 120° = 60° = \frac\pi3$ there. So $\int\kappa\,ds + \frac{n\pi}{3} = 2\pi$, and $\frac{dA}{dt} = -M\gamma(2\pi - \frac{n\pi}{3}) = \frac{\pi}{3}M\gamma(n - 6)$.

Solution

On the circle $|x|^2 = r^2 = 2\tau$ with $\tau = t_0 - t$, so the integrand is constant: $\frac{e^{-1/2}}{\sqrt{4\pi\tau}}\cdot2\pi r = \frac{e^{-1/2}\,2\pi\sqrt{2\tau}}{2\sqrt{\pi\tau}} = \sqrt{2\pi}\,e^{-1/2}$. The curvature vector points inward with length $\frac1r$, so $\vec H = -\frac{x}{r^2} = -\frac{x}{2\tau}$, and $x$ is normal to the circle, so $\vec H + \frac{x^\perp}{2\tau} = 0$.