Short-Time Existence and Uniqueness

Does the Ricci flow have a solution? For a heat equation the answer is standard: linearise, check that the linearisation is uniformly parabolic, and solve by a contraction or an implicit function argument (6A.7 Nonlinear Parabolic Equations). The Ricci flow fails the second step. Its diffeomorphism invariance (11A.1 The Equation and Its First Solutions) makes the linearised operator degenerate: in the directions of infinitesimal diffeomorphisms it does no smoothing at all. Hamilton's 1982 existence proof had to use the Nash–Moser inverse function theorem. A year later Dennis DeTurck found a short way around the degeneracy: add a carefully chosen diffeomorphism term to make the equation strictly parabolic, solve it, and then undo the diffeomorphisms.

This chapter proves short-time existence and uniqueness on closed manifolds. It then proves the derivative estimates of Wan-Xiong Shi, which show that a curvature bound controls all derivatives of curvature. From them follows the basic fact about singularities: the flow continues as long as the curvature stays bounded.

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

• compute the principal symbol of $-2\operatorname{Ric}$ and identify its kernel;
• carry out DeTurck's trick and prove short-time existence on a closed manifold;
• explain uniqueness through the harmonic map heat flow;
• state and prove the first of Shi's derivative estimates;
• prove that at a finite maximal existence time the curvature is unbounded.

3.1Gauge fixing

3.2Why the flow is only weakly parabolic

The linearisation of $g \mapsto -2\operatorname{Ric}(g)$ at $g$, from the variation formula of 11A.2 How Curvature Evolves, is

Its principal symbol, obtained by replacing each $\nabla$ by $i\xi$ for a covector $\xi$ and keeping the second-order part, is

where $h(\xi)_j = h_{jm}\xi^m$. For a strictly parabolic equation the symbol would be negative definite, like $-|\xi|^2h$ for the heat equation. Here it is not. It vanishes on every $h = \xi\otimes X + X\otimes\xi$ (Exercise 3.4), the symbols of Lie derivatives $\mathcal L_Xg$. Those are exactly the directions in which a metric moves when you pull it back by a diffeomorphism, and the Ricci flow cannot smooth them, because it does not see them.

3.3DeTurck's trick

Fix a background metric $\tilde g$, for example the initial metric, and define the vector field

which is a vector field because a difference of connections is a tensor (9A.2 Connections). It vanishes when $g = \tilde g$. The Ricci–DeTurck flow is

The added term contributes exactly $-\xi\otimes h(\xi) - h(\xi)\otimes\xi + (\operatorname{tr}h)\xi\otimes\xi$ to the symbol (Exercise 3.5). It cancels the bad part, leaving $-|\xi|^2h$, and in coordinates the equation becomes

a strictly parabolic quasilinear system, which has a unique smooth solution for a short time on a closed manifold (6A.7 Nonlinear Parabolic Equations).

Uniqueness. Given two Ricci flows $g_1(t)$, $g_2(t)$ with the same initial metric, one wants to reverse the construction: find diffeomorphisms turning each into a Ricci–DeTurck flow, and then use uniqueness for that strictly parabolic system. The needed diffeomorphisms solve the harmonic map heat flow from $(M, g_i(t))$ to $(M, \tilde g)$, $\partial_t\phi = \Delta_{g_i(t), \tilde g}\phi$, which is itself strictly parabolic (Eells and Sampson, 1964; 6A.9 Calculus of Variations and Gradient Flows). Its solutions starting at the identity push $g_i(t)$ forward to Ricci–DeTurck flows with the same initial data, which must coincide, and the $g_i$ are then recovered from them. Hamilton gave this argument for closed manifolds. Bing-Long Chen and Xi-Ping Zhu (2006) proved uniqueness among complete flows with bounded curvature, which matters for the noncompact limits of 11B.3 Compactness of Ricci Flows.

3.4Shi's derivative estimates

A bound on curvature controls all its derivatives after any positive time: the flow is smoothing, like a heat equation.

The proof is Bernstein's method (6A.6 Parabolic Regularity): build a quantity from $|\nabla^k\operatorname{Rm}|^2$ and lower derivatives whose heat-operator image has a good sign, and apply the maximum principle. For $k = 1$ (Exercise 3.7), the evolution equations give

The function $F = t|\nabla\operatorname{Rm}|^2 + \beta|\operatorname{Rm}|^2$, with $\beta$ large, satisfies $(\partial_t - \Delta)F \leq C\beta K^3$, so $F \leq \beta K^2 + C\beta K^2$, which gives $|\nabla\operatorname{Rm}|^2 \leq \frac{CK^2}{t}$. Higher $k$ follow by induction, each step using the bounds already proved (2A.1 The Natural Numbers). Shi (1989) also proved local versions, on balls, which are what blow-up arguments use (11B.3 Compactness of Ricci Flows). Figure 3.2 shows the same $t^{-1/2}$ smoothing for the ordinary heat equation.

3.5How long the flow lasts

Nataša Šešum (2005) proved more: the Ricci curvature must blow up at a finite singular time. The sphere's curvature $\frac{1}{2(n - 1)(T - t)}$ shows the rate $\frac{1}{T - t}$ is attained; singularities at this rate are called Type I (11B.4 Singularities).

3.6History

Hamilton proved short-time existence in 1982 with the Nash–Moser implicit function theorem. DeTurck's trick appeared in 1983, with an improved version in 2003. Eells and Sampson introduced the harmonic map heat flow in 1964. Wan-Xiong Shi proved his derivative estimates, global and local, in 1989, and existence on complete noncompact manifolds with bounded curvature the same year. Chen and Zhu's uniqueness theorem dates from 2006, and Šešum's Ricci blow-up theorem from 2005. Choquet-Bruhat's harmonic-gauge existence theorem for Einstein's equations is from 1952, and Pretorius's binary black hole simulation from 2005.

3.7Exercises

Solution

$h(\xi) = \xi\langle X, \xi\rangle + X|\xi|^2$ and $\operatorname{tr}h = 2\langle X, \xi\rangle$. Then $\sigma(\xi)h = -|\xi|^2(\xi\otimes X + X\otimes\xi) + \xi\otimes(\xi\langle X, \xi\rangle + X|\xi|^2) + (\xi\langle X, \xi\rangle + X|\xi|^2)\otimes\xi - 2\langle X, \xi\rangle\xi\otimes\xi = 0$. If $h(\xi) = 0$ and $\operatorname{tr}h = 0$, then $\sigma(\xi)h = -|\xi|^2h$, negative definite on such $h$.

Solution

$g_{lk}g^{ij}\Gamma_{ij}^l = \frac12g^{ij}(\partial_ig_{jk} + \partial_jg_{ik} - \partial_kg_{ij}) = g^{ij}\partial_ig_{jk} - \frac12\partial_k(g^{ij}g_{ij})$-type terms, giving the stated form. Linearising in $h$: $W_k \approx \nabla^ih_{ik} - \frac12\nabla_k\operatorname{tr}h$, and $\mathcal L_Wg = \nabla_jW_k + \nabla_kW_j \approx \nabla_j\nabla^ih_{ik} + \nabla_k\nabla^ih_{ij} - \nabla_j\nabla_k\operatorname{tr}h$. Replacing $\nabla$ by $i\xi$: $-\xi_j\xi^ih_{ik} - \xi_k\xi^ih_{ij} + \xi_j\xi_k\operatorname{tr}h$, the stated symbol. Adding it to $\sigma(\xi)$ leaves $-|\xi|^2h$.

Solution

Write $\phi_{t + s} = \psi_{t, s}\circ\phi_t$ with $\psi_{t, s}$ the flow from time $t$ to $t + s$ of $V$, so $\frac{d}{ds}\big|_0\psi_{t, s}^* = \mathcal L_{V_t}$. Then $\phi_{t + s}^*\hat g(t + s) = \phi_t^*\psi_{t, s}^*\hat g(t + s)$, and differentiating at $s = 0$ gives $\phi_t^*(\mathcal L_{V_t}\hat g(t) + \partial_t\hat g(t))$.

Solution

$(\partial_t - \Delta)(t|\nabla\operatorname{Rm}|^2) = |\nabla\operatorname{Rm}|^2 + t(\partial_t - \Delta)|\nabla\operatorname{Rm}|^2 \leq |\nabla\operatorname{Rm}|^2 + Ct|\operatorname{Rm}||\nabla\operatorname{Rm}|^2$, dropping $-2t|\nabla^2\operatorname{Rm}|^2$; and $\beta(\partial_t - \Delta)|\operatorname{Rm}|^2 \leq -2\beta|\nabla\operatorname{Rm}|^2 + C\beta|\operatorname{Rm}|^3$. With $t|\operatorname{Rm}| \leq tK \leq 1$, choosing $2\beta \geq 1 + C$ makes the gradient terms nonpositive, so $(\partial_t - \Delta)F \leq C\beta K^3$. By the maximum principle, $F(t) \leq F(0) + C\beta K^3t \leq \beta K^2 + C\beta K^2$. So $t|\nabla\operatorname{Rm}|^2 \leq \beta(1 + C)K^2$.

Solution

For constant curvature $\kappa = \frac{1}{r^2}$, $R_{ijkl} = \kappa(g_{il}g_{jk} - g_{ik}g_{jl})$ and $|\operatorname{Rm}|^2 = 2n(n - 1)\kappa^2$, so $|\operatorname{Rm}| = \sqrt{2n(n - 1)}\,\kappa = \frac{\sqrt{2n(n - 1)}}{2(n - 1)(T - t)}$, using $r^2 = 2(n - 1)(T - t)$. The product with $T - t$ is constant, matching the lower bound's rate. The theorem does not rule out faster blow-up: it gives only a lower bound, and singularities that blow up faster than $\frac{C}{T - t}$ are called Type II (11B.4 Singularities).