The Equation and Its First Solutions

This is the equation the whole Path has been preparing:

A Riemannian metric $g(t)$ moves in the direction of minus twice its Ricci curvature. Richard Hamilton introduced it in 1982 and used it at once to prove a theorem about 3-manifolds (11A.6 Hamilton’s 1982 Theorem). Twenty years later Perelman completed Hamilton's program and proved the Poincaré and geometrization conjectures with it.

This first chapter of Book 11A explains why the equation is a heat equation for the metric, what its symmetries are, and how to solve it exactly when the metric is symmetric enough: Einstein metrics shrink or expand homothetically, products flow factor by factor, and the round cylinder shrinks into a line. These exact solutions are the reference points for everything later. Their curvature evolutions check the formulas of 11A.2 How Curvature Evolves, and their shapes are the models of the singularities of 11B.4 Singularities.

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

• explain, through harmonic coordinates, why the Ricci flow is a nonlinear heat equation for the metric, and why the constant is $-2$;
• use the scaling and diffeomorphism symmetries of the equation;
• write the exact solutions for Einstein metrics, products, cylinders and flat metrics;
• define the volume-normalised flow and relate it to the unnormalised one;
• say where Ricci flow appears in physics and computing, and how those uses differ from the smooth flow.

1.1Where the equation also appears

1.2A heat equation for the metric

Why should $-2\operatorname{Ric}$ be a Laplacian? In harmonic coordinates, coordinates $x^k$ with $\Delta x^k = 0$, which exist near any point, the Ricci tensor has the form

where $Q$ is quadratic in the first derivatives of $g$ (Petersen; Topping, chapter 1). So, in such coordinates,

each component of the metric satisfies a heat equation, coupled to the others through lower-order terms. The factor $2$ makes the leading coefficient exactly $1$, and the minus sign makes it a forward heat equation, well-posed forward in time (6A.1 What a PDE Is). The catch, that the coordinates themselves depend on the metric, is the subject of 11A.3 Short-Time Existence and Uniqueness.

The sign also explains the geometry. Where $\operatorname{Ric}$ is positive, as on a sphere, the metric shrinks; where it is negative, as on a hyperbolic manifold, it expands. Since Ricci curvature measures how volume of small balls falls short of Euclidean (9A.4 Curvature and What It Means), the flow shrinks directions that are "too curved" and expands those that are not curved enough.

1.3Symmetries

Scaling. If $g(t)$ is a Ricci flow, so is

because $\operatorname{Ric}$ is unchanged by constant scaling (9A.4 Curvature and What It Means) while $\partial_t$ picks up a factor $\frac1\lambda$. Distances scale by $\sqrt\lambda$ and time by $\lambda$: parabolic scaling, as for the heat equation (6A.1 What a PDE Is). Curvature scales by $\lambda^{-1}$. Blowing up a flow near a singularity, with $\lambda = |\operatorname{Rm}|$ at a point of large curvature, uses exactly this (11B.4 Singularities).

Diffeomorphism invariance. If $\phi$ is a diffeomorphism, then $\phi^*g(t)$ is also a Ricci flow, because $\operatorname{Ric}(\phi^*g) = \phi^*\operatorname{Ric}(g)$ (8A.6 Flows and the Lie Derivative). Two consequences come later. The equation is not strictly parabolic, which DeTurck's trick repairs (11A.3 Short-Time Existence and Uniqueness). And solutions that change only by diffeomorphisms and scaling, the Ricci solitons, are the natural self-similar solutions (11B.1 Ricci Solitons).

1.4Exact solutions

Einstein metrics. If $\operatorname{Ric}(g_0) = \lambda g_0$, then $g(t) = (1 - 2\lambda t)g_0$ (9A.4 Curvature and What It Means).

• The round sphere of radius $r_0$ in dimension $n$ has $\lambda = \frac{n - 1}{r_0^2}$, so $r(t)^2 = r_0^2 - 2(n - 1)t$. It shrinks to a point at $T = \frac{r_0^2}{2(n - 1)}$, staying round.
• Hyperbolic metrics ($\lambda = -(n - 1)$) expand forever: $g(t) = (1 + 2(n - 1)t)g_0$.
• Flat metrics do not move.

Products. The Ricci tensor of a product is the product of the Ricci tensors (9A.5 Computing Curvature), so each factor flows independently. The round cylinder $S^{n-1}\times\mathbb{R}$ with sphere radius $\rho_0$ evolves by

pinching everywhere at once at $T = \frac{\rho_0^2}{2(n - 2)}$, while the line factor does not move (Figure 1.1, Figure 1.2). For $n = 3$: $S^2(\sqrt{\rho_0^2 - 2t})\times\mathbb{R}$. This is the neck, the most important singularity model of the three-dimensional flow.

1.5The normalised flow

On a closed manifold, the volume changes by $\frac{d}{dt}\operatorname{Vol} = -\int R\,dV$ (8A.7 Tensors and Index Notation), so a sphere shrinks to nothing. Hamilton also used the normalised Ricci flow

which preserves volume (Exercise 1.3). It is the unnormalised flow, rescaled at each time to keep volume fixed, with a reparametrised time. So the two have the same geometry up to scaling. On the round sphere the normalised flow is stationary, which is why Hamilton's theorem (11A.6 Hamilton’s 1982 Theorem) is stated as "the normalised flow converges to a round metric".

1.6History

Hamilton introduced the Ricci flow in "Three-manifolds with positive Ricci curvature" (Journal of Differential Geometry, 1982). Friedan's paper on nonlinear sigma models appeared in Physical Review Letters in 1980, with a fuller account in 1985. DeTurck's simplified existence proof followed in 1983. Discrete Ricci flows for surfaces were introduced by Bennett Chow and Feng Luo in 2003, and graph versions in the 2010s.

1.7Exercises

Solution

$\partial_t((r_0^2 - 2(n - 1)t)g_{S^n}) = -2(n - 1)g_{S^n} = -2\operatorname{Ric}(g_{S^n})$, and $\operatorname{Ric}$ of a rescaled metric is unchanged. Likewise for the cylinder, with $m = n - 1$. Scalar curvature: $R = \frac{n(n - 1)}{r^2} = \frac{n(n - 1)}{2(n - 1)(T - t)} = \frac{n}{2(T - t)}$ for the sphere, and $R = \frac{(n - 1)(n - 2)}{\rho^2} = \frac{n - 1}{2(T - t)}$ for the cylinder.

Solution

$\partial_tg_\lambda(t) = \lambda\cdot\frac1\lambda(\partial_tg)(t/\lambda) = -2\operatorname{Ric}(g(t/\lambda)) = -2\operatorname{Ric}(g_\lambda(t))$. The $(1, 3)$ curvature tensor is scale invariant and its norm involves the metric so that $|\operatorname{Rm}_{\lambda g}|_{\lambda g} = \lambda^{-1}|\operatorname{Rm}_g|_g$. So $g_\lambda$ has $|\operatorname{Rm}| \leq \frac K\lambda$ on $[0, \lambda T)$: choosing $\lambda = K$ normalises the curvature bound to $1$.

Solution

$\frac12\operatorname{tr}(-2\operatorname{Ric} + \frac2nrg) = -R + r$, so $\frac{d}{dt}\operatorname{Vol} = \int(r - R)\,dV = r\operatorname{Vol} - \int R\,dV = 0$ by the definition of $r$.

Solution

$\operatorname{Ric} = -(n - 1)g_0$ gives $g(t) = (1 + 2(n - 1)t)g_0$. For an Einstein metric, $\operatorname{Ric} = \lambda g$, $R = n\lambda$, $r = n\lambda$, and $-2\operatorname{Ric} + \frac2nrg = -2\lambda g + 2\lambda g = 0$.

Solution

$\frac{d}{dt}\log\operatorname{Vol} = -r$, so $\frac{d}{dt}\log\psi = -\frac2n\frac{d}{dt}\log\operatorname{Vol} = \frac2nr$. Then $\frac{\partial\tilde g}{\partial\tilde t} = \frac{1}{\psi}\frac{d}{dt}(\psi g) = \frac{\psi'}{\psi}g + \partial_tg = \frac2nrg - 2\operatorname{Ric}(g)$. Now $\operatorname{Ric}(\tilde g) = \operatorname{Ric}(g)$ and $\tilde r = \psi^{-1}r$, so $\frac2nrg = \frac2n\tilde r\psi g = \frac2n\tilde r\tilde g$. Hence $\partial_{\tilde t}\tilde g = -2\operatorname{Ric}(\tilde g) + \frac2n\tilde r\tilde g$.