Flows on Complete Manifolds

So far the maximum principles of Book 11A have been proved on closed manifolds, where a function attains its maximum. But most of the flows that matter in Perelman's argument are not compact. $\kappa$-solutions may be noncompact ($S^2\times\mathbb{R}$, the Bryant soliton, the cross-cap); the standard solution of surgery lives on $\mathbb{R}^3$; and blow-up limits are complete noncompact flows. And many arguments are local: they concern a region of a manifold, with information on its boundary, as in 11A.8 The Strong Maximum Principle and Splitting.

This chapter supplies the versions of the theory that work there. First the maximum principle on a compact domain with boundary, which needs nothing new. Then the maximum principle on a complete manifold with bounded curvature, where the difficulty is that a supremum need not be attained; the remedy is to subtract a small multiple of a function that grows at infinity and is a strict supersolution of the heat equation, built from the distance. Then Shi's existence theorem and the uniqueness theorem of Chen and Zhu, stated. Finally the normalisation of initial metrics that Morgan and Tian impose at the start of the flow with surgery, and what it buys for the first stretch of time.

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

• prove the maximum principle for systems on a compact domain with boundary, given control on the boundary;
• explain why the maximum principle can fail on noncompact manifolds without growth conditions, with the classical example of Tychonoff;
• prove the scalar maximum principle on a complete manifold with bounded curvature for functions bounded above, with an exhaustion barrier built from the distance;
• state Shi's existence theorem, the uniqueness of Chen and Zhu, and the preservation of curvature conditions for complete flows of bounded curvature;
• define normalised initial conditions and prove that they give curvature and volume control for a definite time.

9.1Domains with boundary

This is Morgan and Tian's Theorem 4.9, and it is what 11A.8 The Strong Maximum Principle and Splitting used, with a Dirichlet solution of the heat equation as the second component. The role of the boundary condition is simply to keep the maximum away from the boundary. On a noncompact manifold there is no boundary, but there is infinity, and the maximum can escape to it.

9.2Why noncompactness is a problem

On $\mathbb{R}$ the heat equation has nonzero solutions with zero initial data. Andrey Tychonoff's example (1935) is

which converges, solves $u_t = u_{xx}$, vanishes at $t = 0$, and is not identically zero; it grows faster than $e^{cx^2}$ for every $c$ at positive times. So uniqueness, and with it the maximum principle, fails without a growth condition. The classical theorem (Tychonoff, Täcklind) is that uniqueness holds among solutions with $|u| \leq Ce^{c|x|^2}$. For the Ricci flow one assumes bounded curvature, which makes all geometric quantities that appear bounded, and then the maximum principle holds with room to spare.

9.3The maximum principle on complete manifolds

The hypothesis "bounded above" can be weakened to growth like $e^{cd^2}$, as in the classical theorem, but bounded is what is used: the quantities to which the principle is applied, such as curvature or $R_{\min}$-comparisons, are bounded on flows of bounded curvature. With it, every scalar consequence of 11A.7 Maximum Principles under Ricci Flow holds on complete flows of bounded curvature: for instance, $R \geq -\frac{n}{2t}$, and $R_{\min}$ (now an infimum) is nondecreasing.

The tensor and system versions extend in the same way (statement here; the barrier is added to the distance from the convex set): on a complete Ricci flow of bounded curvature, closed convex parallel-invariant sets preserved by the ODE are preserved by the PDE. In particular nonnegative curvature operator, and in dimension three nonnegative sectional and Ricci curvature, and the Hamilton–Ivey pinching (11A.10 Hamilton–Ivey Pinching), are preserved for complete flows of bounded curvature. This is the form in which they apply to $\kappa$-solutions and to the standard solution.

9.4Existence and uniqueness on complete manifolds

The proof solves the flow on an exhaustion of $M$ by compact domains, with Dirichlet boundary conditions for the Ricci–DeTurck flow, and uses the local derivative estimates of 11A.5 Shi's Derivative Estimates to obtain bounds independent of the domain, so that the solutions converge. This is the reason Shi proved local estimates at all: on a noncompact manifold there is no global maximum to work with.

The proof follows the strategy of 11A.4 Short-Time Existence and Uniqueness, harmonic map heat flow to the Ricci–DeTurck flow, but on a noncompact manifold the harmonic map heat flow must be solved with control at infinity, which is where the work lies. Morgan and Tian use a version of this argument to prove the uniqueness of the standard solution of surgery (their chapter 12, §4–6), a rotationally symmetric flow on $\mathbb{R}^3$ whose uniqueness matters because surgery must glue in a canonical cap.

Without bounded curvature both theorems can fail: there are complete metrics with unbounded curvature on which solutions are not unique, and the restriction is not a technicality.

9.5Normalised initial conditions

The flow with surgery is started from a metric that is normalised, so that all constants can be taken universal.

Every metric on a compact manifold becomes normalised after multiplication by a large constant: scaling by $\lambda^2$ divides $|\operatorname{Rm}|$ by $\lambda^2$, and for $\lambda$ large, balls of radius $1$ are tiny balls of the original metric, whose volume ratio is close to Euclidean (9B.2 Volume Comparison, Bishop–Gromov and the expansion of the volume of small balls). Since scaling does not change topology, nothing is lost for the Poincaré conjecture.

The proposition is the starting point of the inductive construction of the flow with surgery (12B.13 Ricci Flow with Surgery for All Time): it provides curvature and noncollapsing control on an initial time interval, before any singularity can form, and all later constants are determined from these.

9.6History

Tychonoff's example of nonuniqueness for the heat equation dates from 1935. Shi proved existence on complete manifolds with bounded curvature, with his local derivative estimates, in 1989. Bing-Long Chen and Xi-Ping Zhu proved uniqueness among complete flows of bounded curvature in 2006, in time for the expositions of Perelman's work. Maximum principles on complete manifolds with barrier functions of this kind go back to work on parabolic equations on complete manifolds in the 1980s, by Karp and Li among others.

9.7Exercises

Solution

$u(x, t) = t + \frac{x^2}{2} - 1$ has $u_t = 1 = u_{xx}$, and $u(x, 0) = \frac{x^2}{2} - 1 \leq 0$ on $[0, 1]$, but $u(x, t) > 0$ for $t > 1$. The boundary values $u(0, t) = t - 1$ and $u(1, t) = t - \frac12$ become positive: heat flows in through the boundary. The proposition requires the boundary values to stay in the set (here, $u \leq 0$ on $\partial U$).

Solution

$\Delta(1 + |x|^2) = 2n$, so $(\partial_t - \Delta)\psi = e^{At}\big(A(1 + |x|^2) - 2n\big)$. At $x = 0$ this is $e^{At}(A - 2n)$, so $A \geq 2n + 1$ suffices for the bound $\geq 1$ everywhere (as $1 + |x|^2 \geq 1$ and $e^{At} \geq 1$). In the lemma the constant $C$ plays the role of $2n$, enlarged by the curvature.

Solution

For the unit $S^3$, $R_{ijkl} = \delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk}$, whose nonzero components are $\pm1$ for $i \neq j$, $\{k, l\} = \{i, j\}$: $3\cdot2\cdot2 = 12$ ordered index choices, so $|\operatorname{Rm}|^2 = 12$. After scaling by $\lambda^2$, $|\operatorname{Rm}| = \frac{\sqrt{12}}{\lambda^2} \leq 1$ needs $\lambda \geq 12^{1/4} \approx 1.86$. For the volume, a ball of radius $r \leq 1$ in $\lambda^2g$ is a ball of radius $\frac r\lambda$ in $g$, of $g$-volume $\pi(\frac{2r}{\lambda} - \sin\frac{2r}{\lambda})$, so its $\lambda^2g$-volume is $\lambda^3\pi(\frac{2r}{\lambda} - \sin\frac{2r}{\lambda}) \approx \frac43\pi r^3(1 - \frac{r^2}{5\lambda^2})$. With $\omega_3 = \frac43\pi$, the condition $\geq \frac{\omega_3}{2}r^3$ holds easily for $\lambda \geq 1.86$. So $\lambda = 12^{1/4}$.

Solution

Static $\mathbb{R}$ is a complete Ricci flow with zero curvature. If $u$ were bounded on $\mathbb{R}\times[0, T]$, then $u$ and $-u$ would both be bounded above, solve the heat equation, and vanish at $t = 0$; the theorem would give $u \leq 0$ and $-u \leq 0$, so $u \equiv 0$, contradicting the construction. So $u$ is unbounded at every positive time, and in fact it must grow faster than any $Ce^{cx^2}$, by the sharper classical uniqueness theorem.