The Singular Time: Horns and Strong Necks

Proof. In space. Along a unit-speed path $\gamma$ from $z$ of length $s_0 < r$, let $R(s) = R(\gamma(s))$. Where $R(s) \geq r_0^{-2}$ the gradient bound gives $|R'| \leq CR^{3/2}$, that is, $|\frac{d}{ds}R^{-1/2}| \leq \frac C2$. If $R \geq r_0^{-2}$ along the whole path, $R(s)^{-1/2} \geq R(z)^{-1/2} - \frac C2s \geq R(z)^{-1/2}(1 - \frac14) = \frac34R(z)^{-1/2}$, so $R \leq \frac{16}{9}R(z)$. If the path dips below $r_0^{-2}$, apply the same argument from the last point where $R = r_0^{-2}$, over a length at most $\frac{r_0}{2C}$: $R \leq \frac{16}{9}r_0^{-2}$. Either way $R \leq \frac{16}{9}(R(z) + r_0^{-2})$ on the ball.

In time. Along a flow line, $|\frac{d}{dt}R| \leq CR^2$ where $R \geq r_0^{-2}$, that is, $|\frac{d}{dt}R^{-1}| \leq C$. Starting from at most $\frac{16}{9}(R(z) + r_0^{-2})$, over time $\Delta t$ the reciprocal $R^{-1}$ decreases by at most $C\Delta t = \frac{1}{16(R(z) + r_0^{-2})}$, so $R^{-1} \geq \frac{9}{16}(R(z) + r_0^{-2})^{-1} - \frac{1}{16}(R(z) + r_0^{-2})^{-1} = \frac12(R(z) + r_0^{-2})^{-1}$. So $R \leq 2(R(z) + r_0^{-2})$. Intervals where $R$ dips below $r_0^{-2}$ are handled as before.

Proof. By 12B.6 Bounded Curvature at Bounded Distance (whose hypotheses are 1 and 2), the rescaled curvature is bounded on rescaled balls $B(x_n, 0, A)$ by a constant $Q(A)$, for large $n$. Hypothesis 5 and Lemma 7.1 then give a backward time $t_0'(A)$ and a bound $Q'(A)$ on the rescaled curvature on $B(x_n, 0, A)\times(-t_0'(A), 0]$ (Corollary 11.3). With pinching, a bound on $R$ bounds $|\operatorname{Rm}|$; Shi's estimates (11A.5 Shi's Derivative Estimates) bound all derivatives (Corollary 11.4). With noncollapsing, which after rescaling holds on scales $\leq rQ_n^{1/2} \to \infty$, the injectivity radius at $x_n$ is bounded below (9B.3 Collapsing and Noncollapsing), and Hamilton's compactness theorem (11B.3 Compactness of Ricci Flows) gives a subsequence converging on every ball, hence a complete limit $(M_\infty, g_\infty)$, of nonnegative curvature by pinching. Its curvature is bounded: points of rescaled curvature at least $4$ have $(2C, 2\varepsilon)$-canonical neighbourhoods in the limit, by openness, so if the curvature were unbounded there would be $2\varepsilon$-necks of arbitrarily small scale in a complete nonnegatively curved three-manifold, contradicting Morgan and Tian's Proposition 2.19 (12B.2 The Asymptotic Soliton and Three-Dimensional Shrinkers). With a global curvature bound, Lemma 7.1 gives a uniform backward time $t_0$ on all balls, and Hamilton's compactness for flows gives the limit flow on $[-t_0, 0]$.

Proof. If the curvature of the limit on $(-T, 0]$ is bounded, then near time $-T$ the flows have bounded curvature on large balls, Lemma 7.1 gives uniform parabolic neighbourhoods reaching past $-T$, and the limit extends: a contradiction (Proposition 11.10). So it suffices to bound the curvature.

Distances in the past (Claim 11.12). Let $Q$ bound $R(\cdot, 0)$ on the limit. Hamilton's Harnack inequality (11B.2 Ancient Solutions and the Harnack Inequality), on the limit, which has nonnegative curvature operator and is defined on $(-T, 0]$, gives $\partial_tR \geq -\frac{R}{t + T}$, so $R(x, -t_0) \leq Q\frac{T}{T - t_0}$, and with nonnegative curvature $\operatorname{Ric} \leq \frac{QT}{2(T - t_0)}$. Integrating the distance-change estimate of 12A.8 Local Noncollapsing from the Reduced Volume (Lemma 8.3) in time,

since $\int_{-t_0}^0\sqrt{\frac{QT}{3(T + t)}}\,dt \leq 2\sqrt{\frac{QT}{3}}\sqrt T$. So compact sets have bounded diameter at all times.

Curvature on compact sets (Lemma 11.11). If $X$ is compact and connected and $\min_XR(\cdot, t)$ is bounded for all $t$, then $R$ is bounded on $X\times(-T, 0]$. Otherwise there are times $t_m \to -T$ with a point $y_m$ of curvature between $2$ and $C'$ and a point $z_m$ of curvature tending to infinity, at bounded distance; transfer to the approximating flows, where $y_m$ plays the role of $x$ in 12B.6 Bounded Curvature at Bounded Distance (points of curvature at least $4R(y_m)$ before $t_m$ have canonical neighbourhoods, by hypothesis 2 since $R(y_m) \geq 1 = R(x_n)$), and the unbounded curvature at bounded distance contradicts bounded curvature at bounded distance.

Compact limits. $\min R(\cdot, t)$ is nondecreasing in $t$ (maximum principle), and is at most $1$ at $t = 0$, so it is at most $1$ for all earlier $t$, and Lemma 11.11 applies to $X = M_\infty$.

Noncompact limits (Lemma 11.13, Claims 11.14–11.15). Far from $x_\infty$, the curvature is bounded for all times, for a geometric reason. In a complete noncompact nonnegatively curved manifold, every point $y$ far from $x_\infty$, at distance $d$, has a point $z$ at distance $d$ from $y$ with $d(x_\infty, z) > \frac32d$ (follow a ray: by Toponogov, $y$ is close in angle to a ray from $x_\infty$, and points further along the ray serve). If $R(y, t)$ were huge for some $t$, $y$ would have a canonical neighbourhood of tiny scale at time $t$, a cap or a neck. Not a cap, and not a neck from one end of which both geodesics leave: minimizing $g(t)$-geodesics from $y$ to $x_\infty$ and to $z$ would then leave the small neighbourhood in nearly the same direction, and Toponogov's comparison would give $d_t(x_\infty, z)^2 \leq (a_0 - a_1)^2 + (4\pi\varepsilon)^2a_0a_1$ with $a_0 = d_t(x_\infty, y)$ and $a_1 = d_t(y, z)$. Both $a_0$ and $a_1$ lie between $d$ and $d + 16\sqrt{Q/3}\,T \leq \frac32d$ (choosing $D \geq 32\sqrt{Q/3}\,T$), so the right side is less than $d^2$ when $\varepsilon < \frac{1}{8\pi}$. But distances do not increase forward in time under nonnegative Ricci curvature, so $d_t(x_\infty, z) \geq d_0(x_\infty, z) > \frac32d$. Contradiction. So $y$ is the centre of a neck from whose opposite ends the geodesics to $x_\infty$ and $z$ leave. Then a ball of radius $4\pi R(y, t)^{-1/2}$ about $y$ separates $x_\infty$ from $z$ at time $t$, hence at time $0$ (distances shrink forward). If $R(y, t)$ were unbounded as $t \to -T$, arbitrarily small $g(0)$-balls about $y$ would separate $x_\infty$ from $z$, which is absurd. With the curvature bounded on the sphere $\partial B(x_\infty, 0, D)$ at all times, Lemma 11.11 bounds it on $B(x_\infty, 0, D)$ and then on every ball (Corollary 11.16).

Extension (Claim 11.17). With the curvature bounded on $B(x_\infty, 0, 2A)\times(-T, 0]$, Lemma 7.1 bounds it on $B(x_n, 0, A)\times[-(T + \delta), 0]$ for the approximating flows, so the time $-T$ slices converge, to a complete manifold with bounded curvature (again by Proposition 2.19), and the limit extends past $-T$; Hamilton's Harnack inequality bounds the curvature on $[-T, 0]$ by its bound at $-T$.

$T_0 = \infty$. The limit is ancient, complete, nonnegatively curved, with curvature bounded on each slice (Proposition 2.19) and hence everywhere (Harnack: $\partial_tR \geq 0$ for ancient solutions), non-flat ($R(x_\infty, 0) = 1$), and $\kappa$-noncollapsed on scales $\leq rQ_n^{1/2} \to \infty$, so on all scales. It is a $\kappa$-solution.

Proof. 1 and 2 (Lemma 11.21). If $R(x, t_n) \leq Q$ along some $t_n \to T$, then once $T - t_n \leq \Delta t$ of Lemma 7.1, applied at $(x, t_n)$ forward in time (the lemma's proof works in either time direction), the curvature is bounded on a ball $B(x, t_n, r)\times[t_n, T)$. So that ball lies in $\Omega$, and $\Omega$ is open; incidentally $\liminf$ can be replaced by $\lim$, and $R(x, t) \to \infty$ for $x \notin \Omega$. On such a region, pinching bounds $|\operatorname{Rm}|$, Shi's estimates bound all its derivatives, so $\partial_tg = -2\operatorname{Ric}$ is bounded in every $C^k$ and $g(t)$ converges smoothly. $R(g(T))$ is bounded below by pinching ($R \geq -6/(1 + 4T)$). It is proper: if $x_n \in \Omega$ converge to $x \notin \Omega$ with $R(x_n, T)$ bounded, Lemma 7.1 (backward and then in space) bounds $R$ near $x_n$ on a definite time interval before $T$ and a ball of definite size, which eventually contains $x$, so $x \in \Omega$, a contradiction.

3 is formal (Definition 11.22): glue $\Omega\times[T^-, T]$ to the space-time along $\Omega\times[T^-, T)$. Note that $(\Omega, g(T))$ need not be complete.

5 (Lemma 11.23, Claim 11.24). For $x \in \Omega$ with $R(x, T) > r_0^{-2}$, the points $(x, t)$ for $t$ near $T$ have canonical neighbourhoods, and these lie in a fixed compact subset of $\Omega$ (their curvature is comparable to $R(x, T)$, by the lemma). The metrics converge smoothly there, so by openness (12B.4 Compactness and Universal Estimates for κ-Solutions, Proposition 9.79) the neighbourhoods pass to the limit with constants doubled. For strong necks one must also pass the backward time interval to the limit, which works because the necks at $(x, t_n)$ extend backward for rescaled time $1$ and $t_n \to T$.

4 is the analysis of ends that follows.

Proof. $\Omega_\rho$ is compact, so it meets only finitely many components. Let $E$ be an end of such a component $\Omega_0$, and $X_0$ the component of $\{R \geq 2C^2\rho^{-2}\}$ containing a neighbourhood of $E$. Every point of $X_0$ has a canonical neighbourhood, not a compact one (since $X_0$ contains an end). If one were the core of a cap, the capped tube it lies in would be all of $\Omega_0$, and the cap, whose curvature is within a factor $C$ of a point of $X_0$, would have curvature above $2C\rho^{-2} > \rho^{-2}$, as would the necks; then $\Omega_0$ would miss $\Omega_\rho$. So $X_0$ lies in a strong $2\varepsilon$-tube of necks centred in $X_0$. At a frontier point $y$ of $X_0$, $R(y) = 2C^2\rho^{-2}$; the central sphere of its neck has $R \leq 4C^2\rho^{-2}$, so lies in $\Omega_{\rho/2C}$, and the part of the tube beyond it is the horn. Finiteness of the number of ends: horn boundaries that are homotopically nontrivial and pairwise non-parallel are disjoint spheres in the compact set $\Omega_{\rho/2C}$, of which there are finitely many; the degenerate cases (trivial or parallel spheres) force $\Omega_0$ to have at most two ends.

Proof. Suppose not, for some $\rho$ and $\delta$: there are flows and points $x_n$ at time $T_n$ in horns $\mathcal H_n$, with $Q_n = R(x_n) \to \infty$, none the centre of a strong $\delta$-neck.

Hypotheses of Theorem 7.2 (Claims 11.32–11.33). Pinching and canonical neighbourhoods are assumed. The boundary of $\mathcal H_n$ has curvature at most $16C^2\rho^{-2}$, so its distance from $x_n$ is at least of order $\varepsilon^{-1}\rho/C$, which in the rescaled metric tends to infinity; so rescaled balls of any fixed radius about $x_n$ lie in $\mathcal H_n$, away from its boundary. They also stay away from the end of the horn: if the end were at bounded rescaled distance, the curvature would be unbounded at bounded distance, contradicting 12B.6 Bounded Curvature at Bounded Distance. Points of the horn are centres of strong necks, which supplies noncollapsing on their scale and the backward time $R(y)^{-1}$.

The limit is a cylinder (Claim 11.34). So a subsequence of the rescaled flows converges on a short time interval to a complete nonnegatively curved limit with $R = 1$ at the base point. A minimizing ray from $x_n$ to the end of the horn and a minimizing geodesic from $x_n$ to the boundary have lengths tending to infinity, and in the limit together form a line crossing the central sphere of the neck at $x_\infty$ once. A complete nonnegatively curved manifold with a line splits isometrically (9B.5 Splitting and Soul Theorems, the Cheeger–Gromoll splitting theorem), so the limit is $(S^2, h)\times\mathbb{R}$ with $h$ of nonnegative curvature on $S^2$. By the strong maximum principle the flow is a product of a positively curved flow on $S^2$ with the static line.

Extending backward (Claim 11.35). On this product limit no point can be in the core of a cap, because the central sphere of the cap's neck bounds a ball in the cap but is isotopic to the $S^2$ factor of the product, which does not bound. So the points of the approximating flows over compact sets of the limit, at all times in the window, are centres of strong necks, which extend the flow backward by rescaled time comparable to $1$ with curvature not increasing. Theorem 7.2 can be reapplied at the earlier time, with a backward step depending only on the curvature bound $Q_0$ of the limit at time $0$, which bounds it at all earlier times. Repeating, and taking a diagonal subsequence, the limit exists on $(-\infty, 0]$.

Conclusion. The limit is the product of an ancient positively curved flow on $S^2$ with $\mathbb{R}$. Ancient $\kappa$-noncollapsed flows on $S^2$ with positive curvature are round (Hamilton; Morgan–Tian, Corollary 9.50, which 12B.2 The Asymptotic Soliton and Three-Dimensional Shrinkers used), so the limit is the round shrinking cylinder, and for large $n$ the point $x_n$ is the centre of a strong $\delta$-neck. Contradiction.

The last statement. Points of $\mathcal H$ with $R \leq \rho^{-2}$ form a compact set, at distance from any $\delta$-neck of scale $h$ since $h < \rho/2C$. A minimizing geodesic from such a point to the end of the horn passes a point $y$ with $R(y) = h^{-2}$, whose $\delta$-neck separates it from the end.