Ricci Solitons

The heat equation has a solution that keeps its shape: the Gaussian, which spreads out by rescaling, $(4\pi t)^{-n/2}e^{-|x|^2/4t}$. The Ricci flow has such solutions too. A Ricci soliton is a metric that evolves only by rescaling and diffeomorphisms, so that it is the same geometry at every time, just larger, smaller or moved around. Solitons are the fixed points of the flow once its symmetries are factored out. When the flow develops a singularity and you zoom in, the rescaled picture often settles down to a soliton (11B.4 Singularities). So solitons are the expected models of singularities, and the main characters of Books 11B and 12B: the shrinking sphere and cylinder, the steady cigar and Bryant soliton, and the expanders that come out of cones.

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

• derive the soliton equation $\operatorname{Ric} + \frac12\mathcal L_Xg = \lambda g$ from self-similarity, and classify solitons as shrinking, steady or expanding;
• prove the identities satisfied by gradient solitons;
• verify that the Gaussian, round cylinders and the cigar are solitons;
• describe the Bryant soliton through its ODE, and compute its curvature decay;
• explain why solitons are the natural singularity models.

1.1Self-similar shapes

1.2The soliton equation

Suppose $g(t) = \sigma(t)\phi_t^*g_0$, with $\sigma(t) > 0$ and diffeomorphisms $\phi_t$ generated by vector fields $Y_t$. Differentiating at $t = 0$ (11A.3 Short-Time Existence and Uniqueness): $-2\operatorname{Ric}(g_0) = \sigma'(0)g_0 + \mathcal L_{Y_0}g_0$. Writing $\lambda = -\frac12\sigma'(0)$ and $X = Y_0$, this is the soliton equation

Conversely, a metric satisfying it generates the self-similar flow $g(t) = (1 - 2\lambda t)\phi_t^*g$ (Exercise 1.2). The soliton is shrinking if $\lambda > 0$, steady if $\lambda = 0$, expanding if $\lambda < 0$. It is a gradient soliton if $X = \nabla f$, when $\frac12\mathcal L_{\nabla f}g = \nabla^2f$ and the equation reads

An Einstein metric is a soliton with $X = 0$, and on a closed manifold steady and expanding solitons are Einstein (and, in dimension three, shrinking ones have constant curvature: Ivey's theorem, the fourth exercise of 11A.5 Hamilton–Ivey Pinching). Interesting solitons are noncompact.

For a steady soliton, $R + |\nabla f|^2$ is constant: as $R$ decays at infinity, $|\nabla f|$ approaches a constant, the speed at which the soliton is "pushed along" by its diffeomorphisms.

1.3Examples

The Gaussian shrinker. On $\mathbb{R}^n$ with the flat metric, $f = \frac{|x|^2}{4}$ has $\nabla^2f = \frac12g$, so $\operatorname{Ric} + \nabla^2f = \frac12g$: a shrinking soliton with $\lambda = \frac12$, the flat metric seen as shrinking toward the origin. It is the Ricci flow version of the Gaussian, and it is the equality case of Perelman's reduced volume (12A.5 Reduced Distance and Reduced Volume).

Round spheres and cylinders. The round $S^n$ is a shrinker with $f = 0$. The cylinder $S^{k}\times\mathbb{R}^{n-k}$ with sphere radius $\sqrt{2(k - 1)}$ is a gradient shrinker with $\lambda = \frac12$ and $f = \frac{|z|^2}{4}$ in the Euclidean factor (Exercise 1.3). For $k = 2$, $n = 3$, this is the neck $S^2\times\mathbb{R}$, the main singularity model of the three-dimensional flow.

The cigar. On $\mathbb{R}^2$,

satisfies $\operatorname{Ric} + \nabla^2f = 0$ (checked numerically in the figure script, and in Exercise 1.4). It is Hamilton's cigar, a steady gradient soliton with $R = \frac{4}{1 + x^2 + y^2}$, positive curvature, a rounded tip at the origin and an end asymptotic to a cylinder of circumference $2\pi$ (11A.7 Ricci Flow on Surfaces, 9B.3 Collapsing and Noncollapsing). In physics it is Witten's two-dimensional black hole. The product $\text{cigar}\times\mathbb{R}$ is a three-dimensional steady soliton; whether it could appear as a singularity model was Hamilton's "cigar problem" (11B.4 Singularities).

The Bryant soliton. In dimension $n \geq 3$ there is a rotationally symmetric steady gradient soliton on $\mathbb{R}^n$, found by Robert Bryant, with positive curvature. Write $g = ds^2 + \psi(s)^2g_{S^{n-1}}$ and $f = f(s)$. The soliton equation becomes two ODEs, from the curvatures of warped products (9A.5 Computing Curvature):

with $\psi(0) = 0$, $\psi'(0) = 1$, $f'(0) = 0$ (Exercise 1.5). Integrating (Figure 1.2), $\psi(s)$ grows like $\sqrt s$ and the curvature decays like $\frac1s$. That is slower than the cigar's exponential decay, and it means the Bryant soliton is not collapsed at large scales. At distance $s$ the curvature is about $\frac1s$, so the natural length scale there is $\sqrt s$, and the spheres of symmetry have radius about $\sqrt s$ as well; balls of that radius therefore have volume comparable to the cube of the radius, unlike the cigar's thin end. It is the model of a degenerate neckpinch (11B.4 Singularities) and one of Perelman's $\kappa$-solutions (12B.1 κ-Solutions).

Expanders. Expanding gradient solitons with positive curvature exist on $\mathbb{R}^n$ that are asymptotic to cones over spheres. They describe how the flow smooths out a cone point instantly: the solution coming out of a cone is an expander.

1.4Why solitons model singularities

A soliton is a fixed point of the flow in the space of metrics modulo diffeomorphisms and scaling. When the flow becomes singular, the blow-up procedure (11B.4 Singularities) rescales near points of high curvature, and in favourable cases the rescaled flows converge to a limit that is itself self-similar. For Type I singularities, Natasa Šešum (2006) and Carlo Mantegazza and Reto Müller (2015), building on Perelman's reduced volume, proved that suitable blow-up limits are gradient shrinking solitons. In dimension three the shrinking ones are classified: by work of Perelman, Ni–Wallach and Cao–Chen–Zhu, the complete nonflat three-dimensional gradient shrinkers with bounded curvature are the quotients of $S^3$ and of $S^2\times\mathbb{R}$. This is the precise sense in which singularities of the three-dimensional flow look like round points and necks.

1.5History

Hamilton introduced Ricci solitons and the cigar in "The Ricci flow on surfaces" (1988). Robert Bryant constructed his soliton in unpublished work; its properties are written out in Chow–Lu–Ni. Ivey proved the three-dimensional compact shrinker theorem in 1993. Perelman classified three-dimensional $\kappa$-noncollapsed shrinkers with bounded positive curvature in 2003; Ni and Wallach (2008) and Cao, Chen and Zhu (2008) completed the classification of three-dimensional gradient shrinkers.

1.6Exercises

Solution

$\partial_tg(t) = -2\lambda\phi_t^*g + (1 - 2\lambda t)\phi_t^*\mathcal L_{X/(1 - 2\lambda t)}g = \phi_t^*(-2\lambda g + \mathcal L_Xg) = \phi_t^*(-2\operatorname{Ric}(g)) = -2\operatorname{Ric}(\phi_t^*g) = -2\operatorname{Ric}(g(t))$, the last step because $\operatorname{Ric}$ is invariant under scaling.

Solution

$\operatorname{Ric} = \frac{k - 1}{\rho^2}g_{S^k(\rho)}\oplus0 = \frac12g_{S^k(\rho)}\oplus0$, and $\nabla^2f = 0\oplus\frac12g_{\mathbb{R}^{n-k}}$. Sum: $\frac12g$. $R = \frac{k(k - 1)}{\rho^2} = \frac k2$, $|\nabla f|^2 = \frac{|z|^2}{4} = f$, so $R + |\nabla f|^2 - f = \frac k2$.

Solution

With $f = 2u$: $\nabla_i\nabla_j(2u) = 2u_{ij} - (4u_iu_j - 2\delta_{ij}|\nabla u|^2)$. For $u = -\frac12\log(1 + r^2)$: $u_i = -\frac{x_i}{1 + r^2}$, $u_{ij} = -\frac{\delta_{ij}}{1 + r^2} + \frac{2x_ix_j}{(1 + r^2)^2}$. Then $2u_{ij} - 4u_iu_j = -\frac{2\delta_{ij}}{1 + r^2} + \frac{4x_ix_j - 4x_ix_j}{(1 + r^2)^2} = -\frac{2\delta_{ij}}{1 + r^2}$, and $2\delta_{ij}|\nabla u|^2 = \frac{2r^2\delta_{ij}}{(1 + r^2)^2}$. Total: $-\frac{2\delta_{ij}}{1 + r^2} + \frac{2r^2\delta_{ij}}{(1 + r^2)^2} = -\frac{2\delta_{ij}}{(1 + r^2)^2}$. And $\operatorname{Ric} = Kg = \frac{2}{1 + r^2}\cdot\frac{\delta}{1 + r^2}$.

Solution

Radial: $-(n - 1)\frac{\psi''}{\psi} + f'' = 0$. Tangential (on unit vectors): $-\frac{\psi''}{\psi} + (n - 2)\frac{1 - \psi'^2}{\psi^2} + \frac{f'\psi'}{\psi} = 0$, giving $\psi'' = (n - 2)\frac{1 - \psi'^2}{\psi} + f'\psi'$. With $n = 3$: $\frac{\psi''}{\psi} \to -K_0$, so $f''(0) = -2K_0$; tangential: $K_0 + K_0 - 2K_0 = 0$ ✓. At the tip the metric has constant curvature $K_0$ to leading order, so $R(0) = 6K_0$.

Solution

$\nabla f$ has Euclidean gradient $-\frac{2x}{1 + r^2}$, and $|df|^2_g = e^{-2u}|df|^2_E = (1 + r^2)\frac{4r^2}{(1 + r^2)^2} = \frac{4r^2}{1 + r^2}$. Then $R + |\nabla f|^2 = \frac{4 + 4r^2}{1 + r^2} = 4$. At infinity $|\nabla f| \to 2$: the diffeomorphisms that move the cigar flow along $\nabla f$, which far out slides points along the cylinder at speed $2$, while the tip stays fixed.