Rotationally Symmetric Flows and the Standard Solution

Symmetry turns the Ricci flow into something one can almost solve. For a metric invariant under the rotations of $S^n$, the whole geometry is two functions of one space variable and time, and the flow is a pair of nonlinear heat equations in one dimension. In this setting one can see exactly why necks pinch, how fast, and what a singularity looks like up close; and the same setting gives the standard solution, the rotationally symmetric flow on $\mathbb{R}^3$ that Perelman glues into the manifold at every surgery.

11B.5 Singularities met the neckpinch through its equations and a computed example. This chapter derives the equations completely, proves the basic estimates with the maximum principle in one space dimension (the gradient bound $|\psi_s| \leq 1$, the rate at which a neck can shrink, and the bound $\psi_{\min}^2 \leq 2(n - 1)(T - t)$ at a pinch), describes Angenent and Knopf's theorem and the degenerate neckpinch, and then turns to the standard solution: its initial metric, its curvature, and the properties that Morgan and Tian's Theorem 12.5 lists, which are exactly what surgery needs.

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

• write a rotationally symmetric metric as a warped product, compute its curvatures, and state the conditions for smoothness at the poles;
• derive the Ricci flow as a system for the warping function and the length element, including the commutator of $\partial_t$ and $\partial_s$;
• prove with the maximum principle that $|\psi_s| \leq 1$ is preserved, that a neck cannot shrink faster than the cylinder, and that at a neckpinch $\psi_{\min}^2 \leq 2(n - 1)(T - t)$;
• state Angenent and Knopf's neckpinch theorem and describe the degenerate neckpinch;
• construct a standard initial metric, verify its curvature, and state the properties of the standard solution used in surgery.

6.1Warped products and their curvature

Let $M$ be $(a, b)\times S^n$, possibly closed off at one or both ends, with a metric

$\varphi, \psi > 0$, $g_{S^n}$ the round unit metric. With arclength $s$ along the $x$-direction, $ds = \varphi\,dx$ and $\partial_s = \frac1\varphi\partial_x$, the metric is $ds^2 + \psi(s)^2g_{S^n}$, the warped product of 9A.5 Computing Curvature, and its curvature is:

• planes containing $\partial_s$ ("radial" planes) have sectional curvature $K_0 = -\frac{\psi_{ss}}{\psi}$;
• planes tangent to the spheres $\{x\}\times S^n$ have $K_1 = \frac{1 - \psi_s^2}{\psi^2}$;
• the Ricci tensor is diagonal, with $\operatorname{Ric}(\partial_s, \partial_s) = nK_0 = -n\frac{\psi_{ss}}{\psi}$ and, on unit vectors $e$ tangent to the spheres, $\operatorname{Ric}(e, e) = K_0 + (n - 1)K_1 = -\frac{\psi_{ss}}{\psi} + (n - 1)\frac{1 - \psi_s^2}{\psi^2}$.

These are the formulas of 9A.5 Computing Curvature with $\varphi$ there renamed $\psi$. Examples: the round unit $S^{n+1}$ is $\psi = \sin s$, $s \in (0, \pi)$, with $K_0 = K_1 = 1$; the cylinder $S^n(\rho)\times\mathbb{R}$ is $\psi = \rho$, with $K_0 = 0$ and $K_1 = \frac{1}{\rho^2}$; flat $\mathbb{R}^{n+1}$ is $\psi = s$.

Smoothness at a pole. If the metric closes off at $x = a$, where $\psi \to 0$, it extends smoothly over the added point exactly when, measured from the pole, $\psi$ is an odd function of the distance $s$ with $\psi_s = 1$ at the pole (no cone angle) and $\varphi$ is even (9A.1 Riemannian Metrics and Model Spaces, warped products at the axis). On $S^{n+1}$ there are two poles, at $s = 0$ and at $s = \ell$, the total length, with $\psi_s = 1$ and $\psi_s = -1$ there.

6.2The flow

The Ricci flow preserves the rotational symmetry (isometries are preserved, 11A.13 Homogeneous Flows), so $g(t) = \varphi(x, t)^2dx^2 + \psi(x, t)^2g_{S^n}$ with the same coordinate $x$. Comparing $\partial_tg = -2\operatorname{Ric}$ component by component:

Here $\partial_t$ is at fixed $x$, and $\psi_s$, $\psi_{ss}$ are derivatives in arclength at the current time. Since $\partial_s = \varphi^{-1}\partial_x$ and $\partial_t(\varphi^{-1}) = -n\frac{\psi_{ss}}{\psi}\varphi^{-1}$, the two derivatives do not commute:

The equation for $\psi$ is a heat equation in $s$, with the reaction $-(n - 1)\frac{1 - \psi_s^2}{\psi}$ that shrinks the spheres. For the cylinder, $\psi_t = -\frac{n - 1}{\psi}$, so $\psi^2 = \rho_0^2 - 2(n - 1)t$, the shrinking neck.

6.3What the maximum principle says

In one space dimension the maximum principle can be applied to $\psi$, to its derivatives and to combinations of them, and each application says something geometric.

So the spheres of symmetry never become "too steep": the profile stays a graph with slope at most $1$, and the tangential curvature $\frac{1 - \psi_s^2}{\psi^2}$ is nonnegative. (Angenent and Knopf work with initial metrics satisfying this, together with positive scalar curvature.)

So a neck cannot shrink faster than the round cylinder of the same radius, and the cylinder is the extreme case: $\psi_{\min}^2 \leq 2(n - 1)(T - t)$, with equality for the cylinder itself. The curvature at the neck is then at least $K_1 = \frac{1}{\psi_{\min}^2} \geq \frac{1}{2(n - 1)(T - t)}$, the rate of a Type I singularity (11B.5 Singularities).

6.4The neckpinch theorem

The proof works in rescaled variables adapted to the expected cylinder: $u = \frac{\psi}{\sqrt{2(n - 1)(T - t)}}$ as a function of $\sigma = \frac{s}{\sqrt{T - t}}$ and $\tau = -\log(T - t)$. The cylinder becomes the stationary solution $u \equiv 1$, and the task is to show that $u \to 1$ on regions $|\sigma| \leq C$, using barriers (sub- and supersolutions) built from the linearised equation around $1$, whose structure resembles the Ornstein–Uhlenbeck operator of a Gaussian. The radial curvature $K_0 = -\frac{\psi_{ss}}{\psi}$ is shown to be small compared with $K_1$ near the neck, so that the neck becomes cylindrical. In later work Angenent and Knopf, and Angenent, Isenberg and Knopf, gave precise asymptotics of the profile near the pinch.

The degenerate neckpinch. If the neck is moved towards one bulb and that bulb is made small, the bulb and the neck can pinch together: the singularity forms at a pole, at the tip of a shrinking cap, rather than in the middle of a neck. Gu and Zhu (2008) proved that such Type II singularities occur for suitable rotationally symmetric initial data; Angenent, Isenberg and Knopf (2015) analysed them in detail, showing that the rescaled limit at the tip, with Hamilton's Type II point picking (11B.5 Singularities), is the Bryant soliton (11B.1 Ricci Solitons). In three dimensions such caps are among the $\kappa$-solutions of Perelman's classification.

6.5The standard solution

At a surgery, Perelman cuts a thin neck along a central sphere and glues in a cap on each side. The cap is not arbitrary: it is a piece of a fixed rotationally symmetric metric on $\mathbb{R}^3$, rescaled, and the flow from that metric, the standard solution, must be well understood, because just after surgery the manifold near the cap looks like it.

The choice of the tip curvature $\frac14$ is a normalisation; what matters is that the cap has positive curvature and joins the cylinder of scalar curvature $1$ smoothly with nonnegative curvature, so that gluing it in does not destroy the pinching estimates.

Most of these follow from what Books 11A and 11B have proved. The flow exists by Shi's theorem (11A.9 Flows on Complete Manifolds); the symmetry is preserved by uniqueness; positive curvature follows from nonnegative curvature by the strong maximum principle (11A.8 The Strong Maximum Principle and Splitting): the metric is not flat, and it does not split, because near the tip it has positive curvature in all directions. Far out the metric is the cylinder, which becomes extinct at $t = 1$ (its scalar curvature $\frac{1}{1 - t}$ blows up there), and a comparison argument shows the whole solution lives exactly that long. Uniqueness is the hard part: Morgan and Tian prove it by solving the harmonic map heat flow in this rotationally symmetric, noncompact setting and passing to the Ricci–DeTurck flow (their §12.4–12.6), following the strategy of 11A.4 Short-Time Existence and Uniqueness. They also prove (Theorem 12.32) that every point of the standard solution has a canonical neighbourhood of the kinds described in 10A.6 Necks, Caps and Their Topology, which is what lets the canonical neighbourhood assumption survive a surgery.

How the standard solution enters surgery. At a surgery, a strong $\delta$-neck of scale $h$ is cut along its central sphere; the neck, rescaled by $h^{-2}$, is close to $S^2\times(-\delta^{-1}, \delta^{-1})$, and a ball around the tip of the standard initial metric, rescaled by $h^2$, is glued onto the cut, with a partition of unity in the overlap and a slight bending of the metric (Hamilton's idea) to keep the curvature pinched toward positive. For a while after surgery, the region near the cap evolves like a rescaled standard solution, by stability of the flow, and the properties above become properties of the flow with surgery (12B.9 Surgery, 12B.13 Ricci Flow with Surgery for All Time).

6.6History

Rotationally symmetric Ricci flows were studied by Ivey (1994) and by Miles Simon (2000), who proved neckpinching for a class of metrics. Angenent and Knopf proved the precise neckpinch theorem in 2004, and with Isenberg analysed the asymptotics and the degenerate neckpinch in the following decade; Gu and Zhu proved that Type II singularities occur in 2008. Perelman introduced the standard solution in §2 of his second preprint (2003), asserting its properties briefly; Morgan and Tian's chapter 12 proves them, including uniqueness, and Chen and Zhu proved uniqueness more generally.

6.7Exercises

Solution

$\psi_s = \cos\frac sr$ and $\psi_{ss} = -\frac1r\sin\frac sr$, so $K_0 = \frac{1}{r^2}$ and $K_1 = \frac{1 - \cos^2(s/r)}{r^2\sin^2(s/r)} = \frac{1}{r^2}$. At $s = 0$ and $s = \pi r$, $\psi = 0$ and $\psi_s = \pm1$: smooth poles. The round $S^{n+1}(r)$ has $\operatorname{Ric} = \frac{n}{r^2}g$, so $r^2$ decreases at rate $2n$. In the equation for $\psi$, at fixed $x$ (that is, fixed angle $\frac sr$), $\partial_t\psi = r'\sin\frac sr$ and $r' = -\frac nr$; while $\psi_{ss} - (n - 1)\frac{1 - \psi_s^2}{\psi} = -\frac1r\sin\frac sr - \frac{n - 1}{r}\sin\frac sr = -\frac nr\sin\frac sr$. They agree.

Solution

$\partial_t\partial_s = \partial_t(\varphi^{-1}\partial_x) = -\varphi^{-2}(\partial_t\varphi)\partial_x + \varphi^{-1}\partial_x\partial_t = -n\frac{\psi_{ss}}{\psi}\partial_s + \partial_s\partial_t$, using $\partial_t\varphi = n\frac{\psi_{ss}}{\psi}\varphi$ and that $\partial_t$, $\partial_x$ commute. Lengths in the $s$-direction change at the rate $\partial_t\log\varphi = n\frac{\psi_{ss}}{\psi} = -\operatorname{Ric}(\partial_s, \partial_s)$: they shrink where the radial Ricci curvature is positive (where the profile is concave) and grow where it is negative (where it is convex, as at a neck).

Solution

$K_1 = \frac{1 - \psi_s^2}{\psi^2} < 0$ when $|\psi_s| > 1$. A profile that rises steeply, faster than $45°$ in the $(s, \psi)$ picture, as in a trumpet's bell, has this property; such a region can be put on a sphere between two caps with $\psi_s = \pm1$. The slope bound gives $K_1 \geq 0$, which with the preserved positivity of scalar curvature controls the reaction terms; it is preserved by Proposition 6.1, so it is a natural class to work in, and it makes the neck's curvature dominated by the tangential direction.

Solution

$f^2 = 4s - s^2$, $2ff' = 4 - 2s$, so $f' = \frac{2 - s}{f}$ and $1 + f'^2 = \frac{f^2 + (2 - s)^2}{f^2} = \frac{4}{f^2}$. The orbital curvature is $\frac{1}{f^2(1 + f'^2)} = \frac14$. Differentiating $ff' = 2 - s$: $f'^2 + ff'' = -1$, so $f'' = -\frac{1 + f'^2}{f} = -\frac{4}{f^3}$, and the profile curvature is $\frac{-f''}{f(1 + f'^2)^2} = \frac{4/f^3}{f\cdot16/f^4} = \frac14$. Where $f = \sqrt2$, $f' = f'' = 0$: orbital curvature $\frac12$, profile curvature $0$, the cylinder of scalar curvature $2\cdot\frac12 = 1$.

Solution

For $S^2(\rho)\times\mathbb{R}$ in dimension three, $n = 2$ and $\rho^2 = 2 - 2t$, so the cylinder pinches at $t = 1$, where its scalar curvature $\frac{1}{1 - t}$ blows up. The standard solution is asymptotic to this cylinder at infinity on every interval $[0, t_0]$, $t_0 < 1$ (property 5); its scalar curvature far out is therefore at least about $\frac{1}{1 - t}$, which tends to infinity as $t \to 1$. A complete flow with bounded curvature on $[0, t_0]$ for every $t_0 < 1$ but curvature unbounded as $t \to 1$ cannot be continued past $t = 1$ as a flow of bounded curvature. So the maximal time is at most $1$. (Morgan and Tian's Corollary 12.8 records $T \leq 1$ in this way.)