Loops, Disks and Curve Shrinking

10A.12 Curve Shortening and Minimal Discs introduced the curve-shortening flow and minimal disks, and 12C.3 Finite Extinction explained the idea of Perelman's extinction argument: once $\pi_2 = 0$ (12C.2 Two-Spheres Disappear), a nontrivial element of $\pi_3$ is a $2$-sphere's worth of contractible loops; each loop bounds a minimal disk; let the loops move by the curve-shortening flow while the metric moves by the Ricci flow; then the area of the minimal disk decreases at rate at least $2\pi$, by the same Gauss–Bonnet computation as for spheres, but now with a boundary term that the curve-shortening flow cancels exactly. The difficulty, which Perelman explains in one paragraph and resolves in two pages, is that the curve-shortening flow of a family of loops is not well behaved: loops need not be embedded or even immersed, and the flow of an immersed loop can develop singularities. Perelman's remedy, from Altschuler and Grayson, is to add a dimension: replace the loop $c$ in $M$ by a "ramp" $c_\lambda$ in $M\times S^1_\lambda$ that winds once around the circle. Ramps are embedded, stay ramps, and their curve-shortening flow exists for all time. Project back to $M$, and let $\lambda \to 0$.

Morgan and Tian devote fifty pages to making this rigorous, including two estimates that Perelman asserted with a reference to the literature (a local curvature bound for curve shortening, and a lower bound for the length of a curve that bounds a thin minimal annulus with a well-behaved curve). This chapter follows their structure and proves the parts that carry the ideas.

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

• identify $\pi_3(M)$ with $\pi_2$ of the space of contractible loops when $\pi_2(M) = 0$, and define the width $W_\xi$;
• prove that the forward difference quotient bound $\frac{d}{dt}W_\xi \leq -2\pi - \frac12R_{\min}W_\xi$ implies extinction, and reduce it to a statement about deforming families of loops (Proposition 18.24);
• derive the area estimate for a minimal disk whose boundary moves by curve shortening;
• derive the evolution equations of curve shortening in a Ricci flow ($\partial_t|X|^2$, $[H, S]$, $\partial_tk$, $\frac{d}{dt}L$, $\frac{d}{dt}\Theta$);
• explain ramps in $M\times S^1_\lambda$, prove that they stay ramps, and outline the estimates that transfer their behaviour back to $M$ as $\lambda \to 0$.

4.1The third homotopy group as spheres of loops

Let $\Lambda M$ be the space of $C^1$ loops $S^1 \to M$, with the $C^1$ topology. Its components correspond to conjugacy classes in $\pi_1(M)$; the component of contractible loops contains the constant loops.

For a contractible loop $\gamma$ let $A(\gamma)$ be the infimum of the areas of Lipschitz disks spanning it: a continuous function on $\Lambda M$, invariant under reparametrisation. For a family $\Gamma : S^2 \to \Lambda M$ of contractible loops let $W(\Gamma) = \max_cA(\Gamma(c))$, and for $\xi \in \pi_3(M)$,

(Definition 18.17). Perelman writes the same quantity as $A(\alpha, g)$ for $\alpha \in \pi_*(\Lambda M, M)$. If $\pi_1(M)$ is finite and $\pi_2(M) = 0$, the universal cover is a homotopy sphere and $\pi_3(M) \cong \mathbb{Z}$, so there is a nontrivial $\xi$ (Claim 18.19); in Perelman's language, a non-aspherical irreducible manifold has nontrivial relative homotopy of $(\Lambda M, M)$ by a result of Serre.

A path of components. Along a path of components $\mathcal X$ of the flow with surgery, after $\pi_2$ has died (12C.2 Two-Spheres Disappear), surgeries are along inessential spheres, so the surgery map $\mathcal X(t^-) \to \mathcal X(t)$ is a homotopy equivalence or a degree-one map onto a homotopy sphere, and is $(1 + \eta)$-Lipschitz (12B.10 Surgery on a δ-Neck). Either way it is injective on $\pi_3$, so a nonzero $\xi$ propagates to nonzero classes $\xi(t)$, and $W_\xi(t) = W(\xi(t))$ is defined.

Extinction from the estimate (§3.2). With $R_{\min} \geq -\frac{6}{1 + 4t}$, $W_\xi$ lies below the solution of $w' = -2\pi + \frac{3w}{1 + 4t}$, which is

and becomes negative at a time depending only on $T_1$ and $W_\xi(T_1)$, as in 12C.2 Two-Spheres Disappear. Since $W_\xi \geq 0$, the path of components must end: the component is removed by surgery. There are finitely many components at time $T_1$, and components created later by surgery are homotopy spheres or have the same homotopy type as the one they came from, so the whole flow becomes extinct. This proves the finite-time extinction theorem of 12C.2 Two-Spheres Disappear, given the proposition.

The continuity statements are as for $W_2$: upper semicontinuity because a fixed family's disks vary continuously; lower semicontinuity because areas change by a factor close to $1$ over short times; and the surgery inequality because surgery maps are almost distance non-increasing (Claims 18.21–18.22).

4.2Reduction to deforming families

The forward difference quotient bound needs, for any family $\Gamma$ nearly realising $W_\xi(t_0)$, a deformation $\tilde\Gamma(t)$ of it whose disks' areas obey the ODE. The deformation is allowed to make loops short instead, because short loops carry no topology:

Let $w_a(t)$ solve $w' = -2\pi - \frac12R_{\min}w$ with $w(t_0) = a$.

Why this suffices. If $w(t_1) \geq 0$ for $w$ starting at $W_\xi(t_0)$, every loop of $\tilde\Gamma(t_1)$ either is short and bounds a disk of tiny area (Corollary 18.28), or has $A \leq w(t_1) + \eta$; so $W_\xi(t_1) \leq w(t_1) + \eta$ for every $\eta$. If $w(t_1) < 0$, every loop must be short, so by Lemma 4.3 the class is trivial, a contradiction. Letting $t_1 \to t_0$ gives the forward difference quotient.

4.3The minimal disk computation

Suppose for a moment that the loops are embedded and move by a smooth curve-shortening flow, $\partial_tc = H$, the curvature vector in $g(t)$.

This is the heart of the matter, and it is Perelman's computation: "the rate of change of the area of $D_c$ can be computed as $\int_{D_c}(-\operatorname{Tr}(\operatorname{Ric}^T)) + \int_c(-k_g)$". The curve-shortening flow is exactly the motion of the boundary that turns the boundary term into the geodesic curvature term of Gauss–Bonnet. In dimension at least three, embeddedness is not essential: a generic small perturbation of an immersed family is embedded except at finitely many times, and the estimate holds up to an error $2\delta L$ that vanishes with the perturbation (Lemma 18.32).

4.4Curve shortening in a Ricci flow

For a family $c(x, t)$ with $\partial_tc = H$, write $X = \partial_x$, $S = X/|X|$, $H = \nabla_SS$, $k = |H|$, and $s$ for arc length. Because both the curve and the metric move, derivatives pick up Ricci terms.

The equations are those of the curve-shortening flow in Euclidean space ($\partial_tk = k'' + k^3$, $\frac{d}{dt}L = -\int k^2$) with lower-order corrections from the ambient curvature. Total curvature is controlled, but there is no pointwise bound on $k$: curvature can concentrate. For a plane curve this is how a figure-eight pinches; in three dimensions a curve can develop a singularity where a small loop forms and shrinks, a one-dimensional analogue of the neckpinch of 11B.5 Singularities. Perelman: "in general the curvature of $c^t$ may concentrate near certain points, creating singularities."

4.5Ramps

Let $(M, g(t))\times(S^1_\lambda, ds^2)$ be the product with a circle of length $\lambda$, still a Ricci flow, with the same curvature bounds as $M$ for every $\lambda$. Let $U$ be the unit field along the circle and $u = \langle S, U\rangle$, the "slope" of the curve in the circle direction. A curve is a ramp if $u > 0$ everywhere (Definition 18.40): it winds monotonically around the circle, so in particular it is embedded.

The cancellation in 3 is the point of the construction. Geometrically, a ramp is a graph over the circle direction, and graphs cannot fold over on themselves, so the curvature cannot concentrate.

4.6Back to $M$

Given $\Gamma$, first replace each loop by a regular polygonal approximation, made $C^2$ by a standard reparametrisation (Definition 18.46, Lemma 18.45), so that lengths and total curvatures are uniformly bounded and areas of spanning disks change by less than $\zeta$. For $\lambda \in (0, 1)$ let $c_\lambda(x) = (c(x), \lambda x)$, the loop lifted to a ramp winding once around $S^1_\lambda$, run the curve-shortening flow of the ramps in $M\times S^1_\lambda$, and project: $\tilde\Gamma(t)(c) = p_1(c_\lambda(t))$. As $\lambda \to 0$ the circle factor disappears and the projected flows should approach the curve-shortening flow in $M$ wherever the latter is regular. Making "should approach" precise takes the following steps.

1. Uniform bounds (Corollary 18.38, Claim 18.51). Lengths and total curvatures of the ramps stay bounded, uniformly in $\lambda$ and $c$, so the area swept by $c_\lambda(t)$ over $[t', t'']$ is at most $C(t'' - t')$, and the minimal disk areas $A(p_1c_\lambda(t))$ change at most linearly: $A(t'') - A(t') \leq C(t'' - t')$.
2. Most times are good (18.13). Integrating the length equation, $\int_{t_0}^{t_1}\int k^2\,ds\,dt \leq C$. So for $B$ large, outside a set of times of measure $CB^{-1}$, $\int k^2\,ds \leq B$, and by Cauchy–Schwarz every arc of length $\varepsilon^2B^{-1}$ has total curvature at most $\varepsilon$.
3. Local regularity (Lemma 18.52; Perelman III, §2.2). There are $\delta$ and $C_i$ such that if at time $t'$ a curve-shortening flow has length at least $r$ and total curvature at most $\delta$ on every arc of length $r$, then for $t \in (t', t' + \delta r^2)$, $k^2 \leq C_0(t - t')^{-1}$ and $|\nabla^i_SH|^2 \leq C_i(t - t')^{-(i + 1)}$. This is the curve-shortening analogue of Shi's estimates (11A.5 Shi's Derivative Estimates) and of Ecker and Huisken's interior estimates for mean curvature flow. Perelman cites Ecker–Huisken and Altschuler–Grayson for it; Morgan and Tian give a full proof in their §7.
4. A good set of intervals. Steps 2 and 3 give, for each $c$ and $\lambda$, a finite union $J_B(c, \lambda)$ of intervals covering all but $CB^{-1}$ of $[t_0, t_1]$, on which the projected curves are either short or have uniformly bounded curvature and derivatives. Passing to a subsequence $\lambda_n \to 0$, these intervals stabilise, and on them the projected flows converge smoothly to genuine curve-shortening flows in $M$. On those intervals the disk estimate of Lemma 4.5 applies; on the complementary set of measure $CB^{-1}$ the crude linear bound of step 1 costs at most $CB^{-1}$. Choosing $B > C\zeta^{-1}$ gives the conclusion of Proposition 4.4 for each single loop $c$, for small $\lambda$ (Lemma 18.53).
5. From finitely many loops to the family (Lemmas 18.58–18.59, Proposition 18.63; Perelman III, §§2.4–2.5, 3.5). Take a fine finite net in $S^2$. For two ramps that start close, the area $\mu$ of a minimal annulus between them grows at most exponentially: $\frac{d}{dt}\mu \leq (2n - 1)|\operatorname{Rm}|\mu$, by the same Gauss–Bonnet argument for an annulus ($\chi = 0$, so no $2\pi$, and the extrinsic curvature of a minimal surface is nonpositive). So nearby loops stay close in the sense of annulus area, which controls the disk areas. To control lengths (the "short loop" alternative), one needs: if a curve of length at least $r$, with total curvature at most $\varepsilon$ on arcs of length $r$, bounds a minimal annulus of small area with another curve, then the other curve has length at least $(1 - 100\varepsilon)$ times its length. Perelman calls the argument "elementary"; Morgan and Tian found it "while elementary, quite intricate", and their §6 proves it with an analysis of geodesics in the annulus and "focusing triangles".

Morgan and Tian explain their choice between the two available proofs: Colding and Minicozzi's approach with sweepouts by spheres (12C.3 Finite Extinction) needs paths of spheres whose near-maximal members are close to (usually non-minimal) critical points of energy, while Perelman's approach deals only with area-minimizing disks, "at the expense of dealing with the technical issues related to the boundary. Since the latter are one-dimensional in nature, they are much easier to handle."

4.7History

Altschuler and Grayson introduced the ramp construction in 1992 to define curve shortening "through singularities" for plane curves. Perelman's third preprint (2003) adapted it to Ricci flow in three pages, citing Ecker and Huisken for the local estimate and calling the annulus estimate elementary. Colding and Minicozzi (2005) gave the alternative proof with sweepouts and harmonic spheres. Morgan and Tian (2007, chapter 18) followed Perelman and supplied full proofs of both estimates.

4.8Exercises

Solution

With $\operatorname{Ric} = 0$ and no ambient curvature ($C_1 = 0$): $\frac{d}{dt}L = -\int k^2$, and $\frac{d}{dt}\Theta \leq \int k''\,ds = 0$. For the circle, $H$ points inward with $k = \frac1r$, so $\dot r = -\frac1r$ and $r^2 = r_0^2 - 2t$. Area $\pi r^2$ has derivative $-2\pi$. This is the case of equality in Lemma 4.5: the flat disk has $K = 0$, $R = 0$, and $\int k_{\rm geod} = 2\pi$.

Solution

(a) A totally geodesic hemisphere of a great $S^2$, area $2\pi\rho^2$. (b) No: a great circle is a geodesic, $k = 0$. (c) Under the Ricci flow the hemisphere has area $2\pi\rho(t)^2 = 2\pi(\rho_0^2 - 4t)$, so $\frac{d}{dt}A = -8\pi$. The right side is $-2\pi - \frac12\cdot\frac{6}{\rho^2}\cdot2\pi\rho^2 = -2\pi - 6\pi = -8\pi$: equality, since the hemisphere is totally geodesic and its boundary is a geodesic.

Solution

$\partial_t\frac ku = \frac{\partial_tk}{u} - \frac{k\partial_tu}{u^2} \leq \frac{k'' + k^3 + C_1k}{u} - \frac{k(u'' + k^2u + \operatorname{Ric}(S, S)u)}{u^2} = \frac{k''}{u} - \frac{ku''}{u^2} + \frac{(C_1 - \operatorname{Ric}(S, S))k}{u}$. The $\frac{k^3}{u}$ terms cancel. Now $\big(\frac ku\big)'' = \frac{k''}{u} - \frac{ku''}{u^2} - \frac{2u'}{u}\big(\frac ku\big)'$, so $\frac{k''}{u} - \frac{ku''}{u^2} = \big(\frac ku\big)'' + \frac{2u'}{u}\big(\frac ku\big)'$. At a spatial maximum of $\frac ku$ the first derivative vanishes and the second is nonpositive, so $\frac{d}{dt}\max\frac ku \leq C\max\frac ku$.

Solution

Cone the loop from one of its points by minimizing geodesics, which are unique and vary smoothly since all points are within $\frac\ell2 < i_0$. The cone is parametrised by $(s, r) \mapsto \exp_{c(0)}(r\cdot v(s))$ with $|v(s)| \leq \frac\ell2$, and with curvature bounded by $K$ the Jacobi field estimates give area at most $\int_0^\ell\frac\ell2\cdot C\,ds = C\ell^2$. In flat space the cone over a loop of length $\ell$ has area at most $\frac{\ell^2}{4}$ (the isoperimetric inequality gives $\frac{\ell^2}{4\pi}$ for the minimal disk). Area scales like length squared.

Solution

(a) Gauss–Bonnet on an annulus: $\int_{\mathcal A}K + \int_{\partial\mathcal A}k_g = 2\pi\chi(\mathcal A) = 0$. So the boundary and intrinsic terms combine to $0$ (up to branch points, which only help), leaving the term from the metric change, bounded by a multiple of $|\operatorname{Rm}|\mu$, and the extrinsic curvature term, which is nonpositive for a minimal surface. (b) A linear differential inequality gives $\mu(t) \leq \mu(t_0)e^{C(t - t_0)}$; if $\mu(t_0)$ is small enough the annulus stays thin on the whole finite interval $[t_0, t_1]$. So the minimal disk of one loop plus the annulus is a competitor for the other loop, and their disk areas differ by at most $\mu$.