Min–Max and Width

How do you find a closed geodesic on a sphere with an arbitrary metric? Minimising length does not work, because a short loop can shrink to a point. George David Birkhoff's answer (1917) was to minimise a maximum. Sweep the sphere out by a family of closed curves, from a point to a point. Some curve in the family is longest. Choose the family to make that longest curve as short as possible. The resulting min–max length, the width, is the length of a closed geodesic. The same idea with 2-spheres sweeping out a 3-manifold produces minimal spheres.

This chapter prepares a step of the proof that the Path does not otherwise cover. The finite extinction theorem says that on a simply connected 3-manifold the Ricci flow with surgery dies out in finite time (12C.2 Finite Extinction). Its proof, by Perelman and independently by Colding and Minicozzi, follows a width under the flow. The width cannot vanish while the manifold survives, and it decreases at a definite rate. So the flow must stop. This is the same logic as the curve shortening flow of 6A.8 Curve Shortening and the First Geometric Flows, where the area enclosed by a curve decreases at the constant rate $2\pi$.

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

• define sweepouts and the width of a sphere, and explain Birkhoff's min–max argument for closed geodesics;
• compute the width of the round sphere and estimate it for an ellipsoid;
• explain why a nontrivial sweepout cannot be pulled tight to a point;
• define sweepouts of a 3-manifold by 2-spheres and the corresponding width, and state the existence of min–max minimal spheres;
• derive the width inequality under the Ricci flow and the extinction time bound that follows from it.

8.1A rubber band on an egg

8.2Birkhoff's min–max

A sweepout of a Riemannian 2-sphere $(S^2, g)$ is a continuous family of closed curves $\gamma_s$, $s \in [0, 1]$, starting and ending at constant curves, that is not homotopic, through such families, to a family of constant curves. Equivalently, the family defines a map $S^2 \to S^2$ of nonzero degree: it covers the sphere (7A.7 Smooth Topology). The standard example is the family of horizontal circles of latitude, from the south pole to the north pole. The width is

the infimum over sweepouts of the length of the longest curve.

Why the width is positive. If every curve of a sweepout were very short, shorter than twice the injectivity radius, each could be contracted to a point along the unique short geodesics to its centre, continuously in $s$. That would homotope the sweepout to constants, contradicting nontriviality (Exercise 8.3). Alternatively, by continuity some curve of the family divides the area of the sphere in half, and an isoperimetric inequality bounds its length from below.

Why the width is attained by a closed geodesic. Take a sweepout whose maximum is close to $W$, and improve it by shortening every curve simultaneously, for instance by the curve shortening flow (6A.8 Curve Shortening and the First Geometric Flows) or by Birkhoff's own procedure of replacing arcs with short geodesic segments. The tightened sweepout's longest curves converge to a curve that cannot be shortened by small deformations, a closed geodesic of length $W$. Lusternik and Schnirelmann (1929) refined the method to find three simple closed geodesics on every Riemannian 2-sphere; Matthew Grayson gave a complete proof with the curve shortening flow in 1989.

On the round unit sphere, $W = 2\pi$, attained by the equator (Exercise 8.2). On an egg-shaped ellipsoid of revolution, the latitude sweepout has maximum the waist, while a sweepout by vertical slices has its maximum at a longer meridian ellipse (Figure 8.1).

8.3Sweeping out a 3-manifold by spheres

In a closed 3-manifold, curves are replaced by 2-spheres. A sweepout is a continuous family of maps $\sigma_s : S^2 \to M$, $s \in [0, 1]$, starting and ending at constant maps; it defines a map $S^3 \to M$, and the sweepout is nontrivial if this map is not null-homotopic, that is, it represents a nonzero element of $\pi_3(M)$. When $M$ is closed and simply connected, $\pi_3(M) \cong H_3(M) \cong \mathbb{Z}$ by the Hurewicz theorem, so nontrivial sweepouts exist (Exercise 8.4). The width is

where Colding and Minicozzi use the energy $\frac12\int|d\sigma_s|^2$ in place of area for technical reasons. As for curves, $W(g) > 0$, and it is attained: there is a min–max minimal sphere, or a finite union of branched minimal (harmonic) spheres, whose total area is $W(g)$. For harmonic maps this follows from the existence theory of Jonathan Sacks and Karen Uhlenbeck (1981); for embedded minimal spheres, from the work of Leon Simon and Francis Smith (1982) and Tobias Colding and Camillo De Lellis (2003), in the tradition of Almgren and Pitts.

8.4The width under the Ricci flow

Let $g(t)$ be a Ricci flow on a closed 3-manifold. For a minimal sphere $\Sigma$, the first variation of area under a change of metric $\partial_tg = -2\operatorname{Ric}$ is

By the traced Gauss equation of 9A.8 Submanifolds and Minimal Surfaces with $H = 0$, $R - \operatorname{Ric}(\nu, \nu) = \frac12(R + R_\Sigma + |A|^2)$, and by Gauss–Bonnet $\int_\Sigma R_\Sigma = 2\int K_\Sigma = 8\pi$ for a sphere. Using the lower bound $R \geq -\frac{3}{2(t + C)}$, which holds for every Ricci flow on a closed 3-manifold (the last exercise of 9B.6 Scalar Curvature and Topology), one finds

in the sense of forward difference quotients (Colding–Minicozzi; Exercise 8.6). The constant term $-4\pi$ comes from the topology of the sphere, through Gauss–Bonnet. Integrating this differential inequality shows that $W$ would become negative in finite time (Exercise 8.7). Since the width of a surviving simply connected component is positive, the flow cannot survive that long: it becomes extinct. Surgery has to be handled too, since the width must not increase across surgeries, and 12C.2 Finite Extinction does this.

8.5History

Birkhoff introduced min–max for closed geodesics in 1917. Lusternik and Schnirelmann's theorem on three closed geodesics dates from 1929, with complete proofs by Ballmann (1978), and by Grayson (1989) via the curve shortening flow. Frederick Almgren (1960s) and Jon Pitts (1981) developed min–max theory for minimal hypersurfaces; Simon and Smith obtained embedded minimal spheres in 3-spheres in 1982, and Sacks and Uhlenbeck harmonic spheres in 1981. Perelman's third preprint (2003) and Colding and Minicozzi's paper (2005) proved finite extinction with these tools.

8.6Exercises

Solution

(a) The area enclosed on one side moves continuously from $0$ to $4\pi$ along the sweepout, so it equals $2\pi$ somewhere. (b) For that curve, $L^2 \geq 2\pi\cdot2\pi$, so $L \geq 2\pi$, and every sweepout has a curve of length at least $2\pi$. The latitude sweepout has maximum length $2\pi$, at the equator.

Solution

Every point of $\gamma_s$ is within $\frac{L(\gamma_s)}{2} < \frac\iota2$ of $\gamma_s(0)$ along the curve, so $\gamma_s$ lies in $B(\gamma_s(0), \frac\iota2)$, where radial geodesics from the centre are unique and depend continuously on the endpoints. Sliding each point to the centre along them is a homotopy, continuous in $s$, to constant curves. A nontrivial sweepout is by definition not homotopic to constants (its degree is nonzero), so this is impossible. Hence every sweepout has a curve of length at least $\iota$, and $W \geq \iota > 0$.

Solution

$H_1 = \pi_1^{ab} = 0$. $M$ is orientable, so $H_2 \cong H^1 = \operatorname{Hom}(H_1, \mathbb{Z}) = 0$, and $H_3 = \mathbb{Z}$. Hurewicz: $\pi_1 = 0$ gives $\pi_2 \cong H_2 = 0$; then $M$ is 2-connected and $\pi_3 \cong H_3 = \mathbb{Z}$, generated by a degree-one map $S^3 \to M$, which is a nontrivial sweepout by spheres (the level spheres of $S^3$).

Solution

The area changes by the integral of the inward normal velocity, $-\oint\kappa\,ds$, and for a simple closed curve the total curvature is $2\pi$ (the turning number is $1$). So $A(t) = A(0) - 2\pi t$, which vanishes at $t = \frac{A(0)}{2\pi}$; the curve must become singular (in fact shrink to a round point) by then. For a minimal sphere, $\int R_\Sigma = 8\pi$ gives the $-4\pi$.

Solution

From the second identity, $\operatorname{Ric}(\nu, \nu) = \frac12(R - R_\Sigma - |A|^2)$, so $R - \operatorname{Ric}(\nu, \nu) = \frac12(R + R_\Sigma + |A|^2)$. Then $\frac{d}{dt}\operatorname{Area} = -\frac12\int R - \frac12\int R_\Sigma - \frac12\int|A|^2 \leq \frac{3}{4(t + C)}\operatorname{Area} - \frac12\cdot8\pi$, using $\int_\Sigma R_\Sigma = 2\int K = 4\pi\chi(S^2) = 8\pi$ and $|A|^2 \geq 0$. Colding and Minicozzi pass from minimal spheres to the width by applying this to the slices of nearly optimal sweepouts.

Solution

(a) $\frac{d}{dt}\big((t + C)^{-3/4}W\big) = (t + C)^{-3/4}\big(W' - \frac{3}{4(t + C)}W\big) \leq -4\pi(t + C)^{-3/4}$. (b) Integrate from $0$ to $t$: $\int_0^t4\pi(s + C)^{-3/4}ds = 16\pi\big((t + C)^{1/4} - C^{1/4}\big)$. (c) The left side is nonnegative while the flow exists, so $(T + C)^{1/4} = C^{1/4} + \frac{C^{-3/4}W(0)}{16\pi}$, that is $T = \Big(C^{1/4} + \frac{W(0)}{16\pi C^{3/4}}\Big)^4 - C$, bounds the extinction time.