Shi's Derivative Estimates

A bounded solution of the heat equation becomes smooth instantly: on $\mathbb{R}^n$, if $|u| \leq K$ then $|\nabla^ku(\cdot, t)| \leq C_kKt^{-k/2}$, whatever the initial data. Wan-Xiong Shi proved in 1989 that the Ricci flow does the same to its curvature. If $|\operatorname{Rm}| \leq K$ on a time interval, every covariant derivative of $\operatorname{Rm}$ is bounded after any positive time, with the parabolic rate $t^{-k/2}$; and the statement localises, so that a curvature bound on a ball, for a time, controls the derivatives on the half ball.

These estimates are the quiet workhorse of the whole subject. They are why a flow can be continued as long as its curvature is bounded (11A.4 Short-Time Existence and Uniqueness); why a sequence of flows with bounded curvature has smooth limits (11B.3 Compactness of Ricci Flows); why in Perelman's canonical neighbourhoods and in Morgan and Tian's definitions a bound on curvature at one scale controls the geometry "in the $C^{[1/\varepsilon]}$ topology"; and why scalar curvature bounds near a point bound $|\nabla R|$ and $\partial_tR$ there, as in the definition of $(C, \varepsilon)$-caps (10A.6 Necks, Caps and Their Topology). 11A.4 Short-Time Existence and Uniqueness proved the global estimate for the first derivative. This chapter proves the global estimates for every $k$, gives Shi's product trick, constructs the cut-off functions that localise maximum principle arguments under a Ricci flow, and proves Hamilton's local estimate for the first derivative in full.

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

• write the evolution inequality for $|\nabla^k\operatorname{Rm}|^2$ and say where each term comes from;
• prove the global estimates $|\nabla^k\operatorname{Rm}| \leq C_kKt^{-k/2}$ for all $k$ by a weighted-sum maximum principle;
• use Shi's product quantity $(BK^2 + |\operatorname{Rm}|^2)|\nabla\operatorname{Rm}|^2$, which satisfies a differential inequality with a quadratic absorption term;
• construct cut-off functions from the time-dependent distance and bound their heat operator using Laplacian comparison and distance distortion;
• prove the local estimate $|\nabla\operatorname{Rm}|^2 \leq CK^2\big(\frac{1}{r^2} + \frac1t + K\big)$;
• explain the consequences: continuation of the flow, bounds on time derivatives, and smooth control of blow-up sequences.

5.1The heat equation as a model

On $\mathbb{R}^n$, let $u$ solve $\partial_tu = \Delta u$ with $|u| \leq K$. Then $|\nabla u|^2$ satisfies $(\partial_t - \Delta)|\nabla u|^2 = -2|\nabla^2u|^2 \leq 0$, and $u^2$ satisfies $(\partial_t - \Delta)u^2 = -2|\nabla u|^2$. The function $F = t|\nabla u|^2 + u^2$ therefore has

so $F \leq \max F(\cdot, 0) = \max u_0^2 \leq K^2$ by the maximum principle, and $|\nabla u| \leq \frac{K}{\sqrt t}$. This is Bernstein's trick (6A.7 Parabolic Regularity): the bad term $|\nabla u|^2$ created by differentiating the weight $t$ is paid for by the good term $-2|\nabla u|^2$ in the evolution of the lower-order quantity $u^2$. Shi's estimates are this argument for the curvature tensor, with two new features: the equations are nonlinear (the reaction terms $\operatorname{Rm}*\operatorname{Rm}$), and the metric moves. The reaction terms are harmless for a short time $t \leq \frac{1}{K}$, because $|\operatorname{Rm}| \leq K$ and time is measured in units of $\frac1K$.

5.2The evolution of the derivatives

Write $A*B$ for any tensor obtained from $A$ and $B$ by products, contractions with $g(t)$ and constant coefficients (11A.2 How Curvature Evolves), and $\nabla^k$ for the $k$-th covariant derivative of the current metric.

The good term $-2|\nabla^{k+1}\operatorname{Rm}|^2$ is the parabolic smoothing; the sum is the price of nonlinearity. In the sum, the terms with $i = 0$ and $i = k$ are $|\operatorname{Rm}||\nabla^k\operatorname{Rm}|^2$, linear in the quantity being estimated with the bounded coefficient $|\operatorname{Rm}|$; the terms with $1 \leq i \leq k - 1$ involve only lower derivatives besides one factor of $\nabla^k\operatorname{Rm}$. This triangular structure makes induction possible.

5.3The global estimates

Two remarks.

• Scaling. The constants depend only on $n$ and $k$, and the estimate is invariant under parabolic rescaling (11A.1 The Equation and Its First Solutions): replacing $g(t)$ by $\lambda g(t/\lambda)$ replaces $K$ by $\frac K\lambda$, $t$ by $\lambda t$ and $|\nabla^k\operatorname{Rm}|$ by $\lambda^{-1-k/2}|\nabla^k\operatorname{Rm}|$, and both sides scale the same way. So it is enough to prove it for $K = 1$, and this is how one remembers its form: on the parabolic scale $\sqrt t$, the $k$-th derivative is the curvature divided by $(\sqrt t)^k$.
• Initial data. If the initial metric already has bounded derivatives, $|\nabla^j\operatorname{Rm}(\cdot, 0)| \leq K$ for $j \leq l$, the same argument without the weights $t^j$ for $j \leq l$ gives bounds $|\nabla^k\operatorname{Rm}| \leq C_{k,l}t^{-\max\{k - l, 0\}/2}$: no blow-up at $t = 0$ in the derivatives that were initially controlled. This is Morgan and Tian's Theorem 3.29, which they need because surgery produces metrics whose initial derivatives are controlled but whose curvature is not small.

5.4Shi's product trick

Shi's original argument for $k = 1$ used a different quantity, whose advantage is that its evolution contains a quadratic absorption term. That is exactly what is needed to localise, because a cut-off function produces linear error terms that a quadratic term can dominate.

For a closed manifold this already gives the global $k = 1$ estimate in a sharper form, with no use of the time weight: comparing $\max F$ with the solution of $y' = -\frac{\delta}{K^4}y^2 + CKy$, which comes down from $+\infty$ at $t = 0$, gives $F \leq CK^4(\frac1t + K)$, that is $|\nabla\operatorname{Rm}|^2 \leq CK^2(\frac1t + K)$ (Exercise 5.8). The extra term $K$ inside the bracket is the honest form for all times, not only $t \leq \frac1K$.

5.5Cut-off functions under the flow

To localise, we need functions supported in a ball, equal to $1$ on a smaller ball, whose gradient and heat operator are controlled. Under a Ricci flow the natural choice is a function of the time-dependent distance, and its heat operator is controlled by two facts proved earlier: Laplacian comparison (9B.1 Laplacian Comparison) and the distortion of distances by bounded curvature (11A.2 How Curvature Evolves).

The function $\eta$ is only Lipschitz, because $d_t$ is; but it is smooth wherever $d_t$ is, and at other points it has smooth upper barriers with the stated bounds (Calabi's trick, 9B.1 Laplacian Comparison), which is all that the maximum principle uses. A smooth $\eta$ with the same properties exists too (Morgan and Tian's Proposition 3.33, citing Chow and Knopf), but its construction is longer.

5.6The local estimate

The three terms in the bracket each have a meaning. $\frac{1}{r^2}$ is the price of localising to a ball of radius $r$; $\frac1t$ is the price of having no information on the initial derivatives; and $K$ is the natural scale of a flow with curvature $K$, the only term left on a complete flow that has had a long time to smooth out. On a ball of radius $r = K^{-1/2}$, after time $t = K^{-1}$, all three are of size $K$, and the estimate says $|\nabla\operatorname{Rm}| \leq CK^{3/2}$: the gradient of curvature at its own scale. This is the scale-invariant bound $|\nabla R| \leq CR^{3/2}$ that appears in the definition of a $(C, \varepsilon)$-cap (10A.6 Necks, Caps and Their Topology).

Higher derivatives localise in the same way, using the weighted sums of the global proof multiplied by a cut-off: for every $k$ there is $C_k$ with $|\nabla^k\operatorname{Rm}| \leq C_kK\big(\frac{1}{r^2} + \frac1t + K\big)^{k/2}$ on the half ball (statement here; Morgan and Tian's Theorem 3.28 and its strengthening 3.29 are of this kind, with constants depending also on $\alpha$ for times $t \leq \frac\alpha K$).

5.7Consequences

Continuation. If $|\operatorname{Rm}| \leq K$ on $M\times[0, T)$ for a closed $M$, the global estimates on intervals $[t - \frac1K, t]$ bound all derivatives of curvature on $[\frac{1}{K}, T)$ independently of $t$. As in 11A.4 Short-Time Existence and Uniqueness, the metrics then converge smoothly as $t \to T$, and the flow continues. So the only obstruction to continuing a flow is the blow-up of curvature.

Time derivatives. The Ricci flow equation turns space derivatives into time derivatives: $\partial_tg = -2\operatorname{Ric}$, $\partial_t\operatorname{Rm} = \Delta\operatorname{Rm} + \operatorname{Rm}*\operatorname{Rm}$, and by Proposition 5.1 $\partial_t^j\nabla^k\operatorname{Rm}$ is a polynomial in $\nabla^i\operatorname{Rm}$ with $i \leq k + 2j$. So bounds on $|\nabla^i\operatorname{Rm}|$ give bounds on all mixed space-time derivatives, with the parabolic counting: one time derivative is worth two space derivatives. For example, wherever $|\operatorname{Rm}| \leq K$ on a parabolic neighbourhood of size $K^{-1/2}$ in space and $K^{-1}$ in time, $|\partial_tR| \leq CK^2$ at its centre, which is the second condition in (8) of the definition of a $(C, \varepsilon)$-cap.

Smooth limits. In a blow-up sequence, the rescaled flows have $|\operatorname{Rm}| \leq C$ on growing parabolic neighbourhoods of the base points. By the local estimates all derivatives of curvature are then bounded on slightly smaller neighbourhoods, uniformly in the sequence. That is the input to the compactness theorem for flows (11B.3 Compactness of Ricci Flows), which converts derivative bounds into smooth convergence. It is also why closeness "in $C^{[1/\varepsilon]}$" is not more demanding than closeness in $C^0$ for flows: once the curvature is controlled for a definite time, the higher derivatives follow.

5.8History

Wan-Xiong Shi proved his global and local derivative estimates in "Deforming the metric on complete Riemannian manifolds" (Journal of Differential Geometry, 1989), together with short-time existence on complete noncompact manifolds of bounded curvature, for which the local estimates are essential. Hamilton gave the local first-derivative estimate the form used here in "The formation of singularities in the Ricci flow" (1995). The version with bounds on the initial derivatives is folklore; Morgan and Tian credit the proof they give to Lu Peng.

5.9Exercises

Solution

Bochner's formula (9A.6 The Laplacian and the Bochner Formula) gives $\square|\nabla u|^2 = -2|\nabla^2u|^2 - 2\operatorname{Ric}(\nabla u, \nabla u) \leq -2|\nabla^2u|^2$ when $\operatorname{Ric} \geq 0$, and $\square u^2 = -2|\nabla u|^2$. So $\square F \leq |\nabla u|^2 - 2t|\nabla^2u|^2 - |\nabla u|^2 \leq 0$. Then $F \leq \max F(0) \leq \frac12K^2$, so $t|\nabla u|^2 \leq \frac12K^2$. The Ricci term enters through Bochner: with negative Ricci curvature one would need an extra exponential factor, exactly as the reaction terms of the Ricci flow need the restriction $tK \leq 1$.

Solution

$\tilde g$ is a Ricci flow (11A.1 The Equation and Its First Solutions) with $|\widetilde{\operatorname{Rm}}|_{\tilde g} = K^{-1}|\operatorname{Rm}|_g \leq 1$ on $s \in [0, 1]$. Covariant derivatives are unchanged as operators, while each extra index adds a factor $K^{-1/2}$ to the norm: $|\tilde\nabla^k\widetilde{\operatorname{Rm}}|_{\tilde g} = K^{-1-k/2}|\nabla^k\operatorname{Rm}|_g$. The estimate $|\tilde\nabla^k\widetilde{\operatorname{Rm}}| \leq C_ks^{-k/2}$ becomes $K^{-1-k/2}|\nabla^k\operatorname{Rm}| \leq C_k(Kt)^{-k/2}$, that is $|\nabla^k\operatorname{Rm}| \leq C_kKt^{-k/2}$.

Solution

Where $y \geq \frac{2b}{a}$, $by \leq \frac a2y^2$, so $y' \leq -\frac a2y^2$, and $\frac{d}{dt}\frac1y \geq \frac a2$. Suppose $y(t_1) > \frac{2}{at_1} + \frac{2b}{a}$. Going backwards from $t_1$, as long as $y \geq \frac{2b}{a}$, $\frac1y$ decreases at rate at least $\frac a2$, so $\frac{1}{y(t)} \leq \frac{1}{y(t_1)} - \frac a2(t_1 - t)$, which becomes $\leq 0$ before $t = 0$ (since $\frac{1}{y(t_1)} < \frac{at_1}{2}$): impossible for a positive $y$. So $y \leq \frac{2}{at} + \frac{2b}{a}$. With $a = \frac{\delta}{K^4}$, $b = CK$: $\max F \leq \frac{2K^4}{\delta t} + \frac{2CK^5}{\delta}$, and $|\nabla\operatorname{Rm}|^2 \leq \frac{F}{BK^2} \leq CK^2(\frac1t + K)$.

Solution

Let $\chi$ be any smooth nonincreasing function with $\chi = 1$ on $(-\infty, \frac12]$ and $\chi = 0$ on $[1, \infty)$, for instance built from $e^{-1/x}$ (8A.2 Partitions of Unity). Then $\Psi = \chi^2$ is smooth and nonincreasing with the same boundary values, $\Psi' = 2\chi\chi'$, and $\Psi'^2 = 4\chi^2\chi'^2 \leq 4\max|\chi'|^2\cdot\Psi$. Also $|\Psi''| = |2\chi'^2 + 2\chi\chi''| \leq C$. The point of squaring: near the edge of the support, where $\Psi \to 0$, its derivative goes to zero at least as fast as $\sqrt\Psi$, which makes $\frac{|\nabla\eta|^2}{\eta}$ bounded.

Solution

Shift time so that the neighbourhood starts at $0$ and ends at $Q^{-1}$, and apply the theorem with $K = 4Q$, $r = Q^{-1/2}$, $t = Q^{-1}$: $|\nabla\operatorname{Rm}|^2 \leq C(4Q)^2(Q + Q + 4Q) = CQ^3$. Since $R$ is a contraction of $\operatorname{Rm}$, $|\nabla R| \leq C|\nabla\operatorname{Rm}| \leq C'Q^{3/2}$. Under the rescaling $\tilde g = Qg$ that makes $R(x) = 1$, $|\nabla R|$ scales by $Q^{-3/2}$, so the bound reads $|\tilde\nabla\tilde R| \leq C'$: a bound independent of scale. Condition (8) in the definition of a $(C, \varepsilon)$-cap (10A.6 Necks, Caps and Their Topology; Morgan and Tian's Definition 9.72) asks exactly for $\frac{|\nabla R|}{R^{3/2}} < C$, and canonical neighbourhoods of a flow satisfy it for this reason.