The Standard Solution

Surgery cuts a horn along the central sphere of a strong $\delta$-neck (12B.7 The Singular Time: Horns and Strong Necks) and glues in a cap. What happens next is a Ricci flow starting from a metric that, near the cap, is almost a round cylinder with a cap on the end. To control that flow with constants independent of where and when the surgery happens, Perelman fixes once and for all a model: a rotationally symmetric metric on $\mathbb{R}^3$, a round half-cylinder closed off by a cap, and its Ricci flow, the standard solution. After surgery, the flow near the cap is close to the standard solution for as long as the standard solution is controlled, and its properties become properties of every surgery.

This chapter proves what is needed about the standard solution, following Morgan and Tian's chapter 12, with Perelman's five claims as the outline. Everything here is a one-time computation about one explicit flow; its payoff comes in 12B.10 Surgery on a δ-Neck, where surgery is defined so that the cap is close to the model, and in 12B.11 Noncollapsing After Surgery and 12B.12 Canonical Neighbourhoods After Surgery, where the model's canonical neighbourhoods and its extinction time control the flow after surgery.

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

• construct a standard initial metric as a hypersurface of revolution and compute its curvature;
• state Morgan and Tian's Theorem 12.5 (uniqueness, $T = 1$, positive curvature, rotational symmetry, asymptotics, noncollapsing) and prove the parts that are maximum-principle arguments;
• prove that the flow stays rotationally symmetric by evolving Killing fields;
• explain how uniqueness is reduced to a strictly parabolic problem by the harmonic map flow (DeTurck's trick), and how Perelman's and Kleiner–Lott's arguments differ;
• prove that the standard solution exists exactly on $[0, 1)$, that $R \geq \frac{c}{1 - t}$, and that every point has a canonical neighbourhood of one of three types.

8.1The standard initial metric

Perelman asks only for asymptotic convergence to the cylinder at infinity, and notes that "the most obvious choice, the round hemisphere, does not fit, because the metric on $\mathbb{R}^3$ would not be smooth enough": a hemisphere of radius $\sqrt2$ glued to the cylinder of radius $\sqrt2$ has a metric that is $C^{1,1}$ but whose curvature jumps across the gluing sphere (Exercise 8.11). Any smooth choice will do; what matters is that the choice is fixed.

Fix one standard initial metric $g_0$ once and for all. Outside the ball of radius $A_0$ about the tip it is exactly the half-cylinder, its scalar curvature lies between two positive constants, and the ball has bounded volume (Lemma 12.3).

8.2The main theorem

A standard Ricci flow is a Ricci flow $(\mathbb{R}^3, g(t))$, $0 \leq t < T$, with $g(0) = g_0$, curvature bounded on each compact time interval, and $T$ maximal for this property (Definition 12.4).

The proofs fall into three groups: maximum-principle arguments (existence, positive curvature, symmetry, noncollapsing); a PDE uniqueness argument; and arguments that use the canonical neighbourhood machinery of 12B.7 The Singular Time: Horns and Strong Necks ($T = 1$ and the structure near $t = 1$). Morgan and Tian remark that every property except uniqueness is "fairly straightforward", and that the rest of the argument needs only compactness of the set of standard solutions, not uniqueness.

8.3Existence, completeness and positive curvature

Existence. Double large balls $B_R$ about the tip along their cylindrical boundary to get rotationally symmetric metrics on $S^3$ that converge to $(\mathbb{R}^3, g_0)$ as $R \to \infty$. Their Ricci flows have nonnegative curvature (preserved in dimension three, 11A.7 Maximum Principles under Ricci Flow), and $R_{\max}$ satisfies $\frac{d}{dt}R_{\max} \leq R_{\max}^2$, since $\partial_tR = \Delta R + 2|\operatorname{Ric}|^2 \leq \Delta R + R^2$ under nonnegative curvature ($2|\operatorname{Ric}|^2 \leq R^2$, Exercise 8.12). So the curvature is bounded by a constant $Q_0$ on a time interval $[0, t_0]$ independent of $R$, and Hamilton's compactness for flows gives a limit flow on $\mathbb{R}^3$ on $[0, t_0]$; extend it maximally.

Completeness and positive curvature (Lemma 12.6). On $[0, t_0]$ the curvature is bounded, so $g(0) \leq Cg(t_0)$ and $g(t_0)$ is complete. Nonnegative curvature is preserved, but on a noncompact manifold the tensor maximum principle needs a localising function: Morgan and Tian use a function $\varphi$ with $\varphi \to \infty$ at infinity, small on a given compact set, and $(\partial_t - \Delta)\varphi \geq C\varphi$ (such functions exist on complete flows of bounded curvature; Morgan and Tian cite Hamilton's compactness paper of 1995), and run the argument of 11A.7 Maximum Principles under Ricci Flow on $\operatorname{Rm} + \varphi g$, whose minimum eigenvalue is attained on a compact set. Strict positivity for $t > 0$ follows from the strong maximum principle (11A.8 The Strong Maximum Principle and Splitting): a zero would split the flow locally, which the tip, where the curvature is positive, forbids.

Asymptotics (Proposition 12.7). Along points $y_k \to \infty$ the initial metrics converge to $S^2\times\mathbb{R}$, so, with Shi's estimates on $[0, T']$, the flows converge to a flow on $S^2\times\mathbb{R}$ with nonnegative curvature whose slices have two ends; by the splitting theorem each slice is a product, and by the strong maximum principle the flow is the product of a flow on $S^2$ starting at the round metric with the line, which is the shrinking round cylinder. Hence:

8.4Rotational symmetry

Perelman proves symmetry with a one-line maximum principle (Claim 1), which Morgan and Tian expand. Let $X(t)$ be a vector field evolving by

and let $h = \mathcal L_Xg$, the symmetrised covariant derivative, $h_{ij} = \nabla_iX_j + \nabla_jX_i$. A long computation with the commutation formulas and the contracted Bianchi identity (Lemma 12.9) gives

the Lichnerowicz Laplacian, which is the linearisation of the Ricci flow (11A.2 How Curvature Evolves). Hence $\partial_t|h|^2 \leq \Delta|h|^2 - 2|\nabla h|^2 + C|h|^2$. If $X(0)$ is a Killing field of $g_0$, then $h(0) = 0$; $|h|^2$ decays at spatial infinity, where the flow is asymptotically a product, so its maximum is attained, and $\frac{d}{dt}\max|h|^2 \leq C\max|h|^2$ gives $h \equiv 0$. So $X(t)$ is Killing for $g(t)$. Then (an observation of Bennett Chow) a Killing field satisfies $\Delta X + \operatorname{Ric}(X) = 0$ (Exercise 8.13), so $\partial_tX = 0$: the Killing fields do not move, and $\mathfrak{so}(3)$ acts by isometries for all $t$ (Corollary 12.12).

The meaning is simple: the Ricci flow is diffeomorphism-invariant, so a symmetry of the initial metric should be a symmetry of the flow. For a compact manifold this follows at once from uniqueness. Here uniqueness is harder than symmetry, and Morgan and Tian use symmetry to prove uniqueness, so they need an argument that does not presuppose it.

8.5Noncollapsing

8.6Uniqueness

The standard solution is a Ricci flow on a noncompact manifold, for which uniqueness is not automatic: Hamilton's and DeTurck's arguments (11A.4 Short-Time Existence and Uniqueness) are for compact manifolds, and on noncompact manifolds uniqueness can fail without growth conditions. The relevant general theorem, uniqueness of complete Ricci flows with bounded curvature, was proved by Bing-Long Chen and Xi-Ping Zhu (2006). Morgan and Tian give a self-contained argument for the special case, following Peng Lu and Gang Tian:

1. DeTurck's trick. Given a Ricci flow $g(t)$ on $[t_0, T]$, solve the harmonic map heat flow $\partial_t\psi_t = \Delta_{g(t), g(t_0)}\psi_t$, $\psi_{t_0} = \operatorname{id}$, for maps from $(\mathbb{R}^3, g(t))$ to $(\mathbb{R}^3, g(t_0))$. Then $\hat g(t) = (\psi_t^{-1})^*g(t)$ solves the Ricci–DeTurck flow $\partial_t\hat g_{ij} = -2\widehat{\operatorname{Ric}}_{ij} + \hat\nabla_iW_j + \hat\nabla_jW_i$ with $W_j = \hat g_{jk}\hat g^{pq}(\hat\Gamma^k_{pq} - \Gamma^k_{pq}(t_0))$, which is strictly parabolic, and $\psi_t$ is recovered from $\hat g$ by the ODE $\partial_t\psi_t = -\hat g^{ij}W$ (Lemma 12.15). So two Ricci flows with the same data give two Ricci–DeTurck flows with the same data, and if those agree, so do the Ricci flows (Corollary 12.16).
2. The harmonic map flow exists (Proposition 12.24), by rotational symmetry: the map is radial, $\psi_t(r\theta) = \hat r(r, t)\theta$, and the harmonic map flow becomes a scalar parabolic equation in one space variable, solved with asymptotic estimates at infinity (Proposition 12.25).
3. Bounded Ricci–DeTurck flows with the same asymptotics are equal (Proposition 12.26), by the maximum principle for the strictly parabolic system.
4. Continuity (§6). On the maximal interval $I$ containing $0$ on which two standard solutions agree: if $I = [0, T_1)$ with $T_1 < T_2$, the curvature of the first would be bounded near $T_1$, contradicting maximality; if $I = [0, t_0]$ is closed, steps 1–3 extend the agreement past $t_0$.

Perelman's argument (Claim 3) is the same in spirit and shorter: with rotational symmetry, the linearised Ricci flow becomes a system of two equations in one variable, $f_t = f'' + a_1f' + b_1g' + c_1f + d_1g$, $g_t = a_2f' + b_2g' + c_2f + d_2g$, with bounded coefficients and decay at infinity, and uniqueness follows by estimating $\int_{-A}^A(f^2 + g^2)$ as $A \to \infty$. Kleiner and Lott give their own proof of uniqueness (§66), and pointed out to Morgan and Tian that it can also be derived from the other properties by the compactness arguments used for $\kappa$-solutions; in any case the rest of the proof needs only that the set of standard solutions is compact.

8.7The extinction time and canonical neighbourhoods

The remaining claims use the machinery of 12B.7 The Singular Time: Horns and Strong Necks.

Perelman's proof of Claim 4 is different and uses pseudolocality (12A.9 Pseudolocality): he obtains the standard solution as a limit of flows on $S^3$ built from long capped cylinders, and if the limit went singular at $T < 1$, points far out would sit, at a time $T_1$ slightly before $T$, in balls almost isometric to Euclidean balls of a size $r$ independent of $T_1$, so pseudolocality (I, Theorem 10.1) would bound their curvature up to $T$; the limit at $T$ would then be a positively curved space with a conical point, which a blow-up argument (the cone argument of 12B.6 Bounded Curvature at Bounded Distance) rules out. Morgan and Tian's route avoids pseudolocality entirely, here as everywhere in their book.

The standard solution therefore becomes extinct everywhere at time $1$, at a rate comparable to the cylinder's. This is the property that makes surgery "lose" the cap: a region that looks like the standard solution near a surgery either becomes singular and is removed at a later surgery, or is already far from the cap.

Case 3 is less paradoxical than it looks (Remark 12.33): the neck's time is measured in units of $R(x, t)^{-1}$, which tends to zero as $t \to 1$. Case 2 is the one that is new compared with $\kappa$-solutions: an evolving neck that reaches back to time $0$, the surgery time, rather than for a full rescaled time unit. Morgan and Tian then transfer the theorem to flows that are merely close to the standard solution (Corollary 12.36, by openness of the conditions); this is the form used after surgery in 12B.12 Canonical Neighbourhoods After Surgery.

A distance-decreasing map (Lemma 12.37). Let $h_0$ be the round metric of scalar curvature $1$ on the unit sphere of $T_{p_0}\mathbb{R}^3$ and $\rho : S^2\times[0, \infty) \to \mathbb{R}^3$ send $(x, s)$ to the point at $g_0$-distance $s$ from the tip in direction $x$. Then $\rho^*g_0 \leq h_0 + ds^2$, because each sphere $\{s\}\times S^2$ maps conformally onto a sphere of radius at most $\sqrt2$ (the profile $f$ is at most $\sqrt2$). This harmless-looking inequality is used in the finite-time extinction argument of 12C.3 Finite Extinction: surgery does not increase the areas of the minimal surfaces that measure the manifold's width.

8.8History

Perelman introduced the standard solution in II, §2 (2003), with five claims and short proofs. Uniqueness was the most discussed: Lu and Tian (2005) and Chen and Zhu (2006, for general complete flows of bounded curvature) gave full proofs; Kleiner and Lott (§§60–66) proved all five claims. Morgan and Tian's chapter 12 (2007) gives everything in detail, avoiding pseudolocality in the proof that the extinction time is $1$. The observation that Killing fields are stationary under the flow is due to Bennett Chow, as Morgan and Tian record.

8.9Exercises

Solution

(a) The orbit of $(s, f(s), 0, 0)$ under $SO(3)$ is the sphere of radius $f(s)$ in the hyperplane $x_0 = s$. The profile curve $(s, f(s))$ has tangent $(1, f')$ and normal $(-f', 1)/\sqrt{1 + f'^2}$. (b) The normal curvature of the orbit sphere is its curvature $\frac1f$ times the component of the unit normal in the radial direction, $\frac{1}{\sqrt{1 + f'^2}}$; the meridian is the profile curve, of curvature $\frac{-f''}{(1 + f'^2)^{3/2}}$ (sign chosen so that a concave profile gives a convex surface). (c) $f' = \frac{2 - s}{f}$, $1 + f'^2 = \frac{4}{f^2}$, so $\kappa_1^2 = \frac14$; $f'' = -\frac{4}{f^3}$ (differentiate $ff' = 2 - s$: $f'^2 + ff'' = -1$, so $ff'' = -\frac{4}{f^2}$), so $\kappa_1\kappa_2 = \frac{4/f^4}{16/f^4} = \frac14$. (d) $f' = f'' = 0$: orbit planes have curvature $\frac12$, mixed planes $0$, scalar curvature $2(\frac12 + 0 + 0) = 1$.

Solution

(a) In geodesic polar form the metric is $ds^2 + f(s)^2h_1$ with $h_1$ the unit sphere metric and $f(s) = \sqrt2\sin(s/\sqrt2)$ for $s \leq \frac{\pi}{\sqrt2}$, then $f \equiv \sqrt2$; at the junction $f = \sqrt2$, $f' = 0$ on both sides, so $f$ is $C^1$ with bounded $f''$: $C^{1,1}$. (b) On the hemisphere all sectional curvatures are $\frac12$; on the cylinder the mixed ones are $-\frac{f''}{f} = 0$. The jump is $\frac12$. (c) The curvature of the initial metric is discontinuous, so it is not a smooth initial datum; the flow would smooth it instantly, but the estimates of this chapter (Shi's estimates at time $0$, compactness of standard solutions, closeness after surgery in $C^{[1/\varepsilon]}$) need smooth initial data with bounded derivatives.

Solution

Diagonalise the curvature operator, with $a_1, a_2, a_3 \geq 0$ the sectional curvatures of the three coordinate planes (in dimension three the curvature operator and the Ricci tensor diagonalise together). The Ricci eigenvalues are $a_j + a_k$, so $R = 2(a_1 + a_2 + a_3)$ and $|\operatorname{Ric}|^2 = \sum_{j<k}(a_j + a_k)^2 = 2\sum a_i^2 + 2\sum_{i<j}a_ia_j$. Then $R^2 = 4\sum a_i^2 + 8\sum_{i<j}a_ia_j$ and $R^2 - 2|\operatorname{Ric}|^2 = 4\sum_{i<j}a_ia_j \geq 0$. Equality when at most one $a_i$ is nonzero, as on the cylinder $S^2\times\mathbb{R}$, where indeed $\partial_tR = R^2$ for the spatially constant $R = \frac{1}{1 - t}$.

Solution

(a) Apply $\nabla_j$ to $\nabla_iX_j + \nabla_jX_i = 0$ and sum over $j$: $\nabla_j\nabla_iX_j + \Delta X_i = 0$. Commute: $\nabla_j\nabla_iX_j = \nabla_i\nabla_jX_j + R_{ik}X_k$, and $\nabla_jX_j = 0$ for a Killing field (trace of the Killing equation). So $\Delta X_i + R_{ik}X_k = 0$. (b) The evolution equation's right side vanishes. (c) A one-parameter group of isometries of all $g(t)$ is generated by a fixed vector field; its dual one-form $g(t)(X, \cdot)$ changes because $g(t)$ does.

Solution

$G(t) = \frac{C}{1 - Ct}$ solves $G' = G^2$ with $G(0) = C$; ODE comparison for forward difference quotients (Morgan–Tian, Corollary 2.22) gives $R_{\max} \leq G$. So the solution exists at least until $\frac1C$, with $C \geq \max R(\cdot, 0) \geq 1$ (the cylinder has $R = 1$ and the tip $R = \frac32$ from curvature $\frac14$: $R = 6\cdot\frac14$). So $T \geq \frac1C$, which is at most $\frac23$ (and $\frac12$ with Morgan and Tian's choice $C = \max(2, \max R(\cdot, 0))$). The cylinder alone has $C = 1$ and the bound is sharp for it; the cap is what makes the a priori bound weaker than the truth $T = 1$.