Uniformization and the Two-Dimensional Ricci Flow

The Riemann mapping theorem said that every simply connected proper region of the plane is conformally a disc. This chapter takes the same idea to surfaces. A surface with a notion of angle, a Riemann surface, can be studied entirely through conformal metrics $e^{2u}|dz|^2$, and the Gaussian curvature of such a metric is a single expression in $u$: $K = -e^{-2u}\Delta u$. So the question "does this surface carry a metric of constant curvature?" becomes a nonlinear PDE for one function.

The answer is the uniformization theorem: every closed orientable surface carries a metric of constant curvature $+1$, $0$ or $-1$, in any given conformal class, and which one is decided by its topology. This is the two-dimensional geometrization theorem, the model for Thurston's conjecture in dimension three (10A.5 Thurston’s Eight Geometries). And there is a second proof of it, by a heat equation: the two-dimensional Ricci flow keeps a metric in its conformal class and evolves the conformal factor by $\partial_tu = e^{-2u}\Delta u$, a nonlinear diffusion that carries any metric to one of constant curvature. Here, for the first time in the guide, the Ricci flow appears as what it is: a heat equation for geometry.

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

• compute the curvature of a conformal metric, and recognise the sphere, the plane and the hyperbolic disc as the three model geometries;
• describe flat tori as quotients $\mathbb{C}/\Lambda$, and higher-genus surfaces as quotients of the hyperbolic disc;
• state the uniformization theorem and explain its consequence for closed surfaces;
• write the two-dimensional Ricci flow as a PDE for the conformal factor, and solve it for the shrinking sphere and the cigar;
• see the heat kernel on the circle as a theta function;
• explain what discrete surface Ricci flow computes in geometry processing.

5.1Conformal metrics and their curvature

On a region of the plane, a conformal metric is $g = e^{2u}|dz|^2$ with $u$ smooth and real: lengths are measured by $\int e^{u}|dz|$ and areas by $\int e^{2u}\,dx\,dy$. Angles are the Euclidean ones, since the metric is a multiple of the Euclidean one at each point. A holomorphic change of coordinate $z = \phi(w)$ turns $e^{2u}|dz|^2$ into $e^{2u\circ\phi}|\phi'(w)|^2|dw|^2$, again a conformal metric (5A.1 Holomorphic Functions Are Conformal). So conformal metrics make sense on any surface built from pieces of the plane glued by holomorphic maps, which is what a Riemann surface is.

We take this formula as a definition here; 8A.9 The Curvature of Surfaces defines Gaussian curvature intrinsically and shows that for conformal metrics it is given by this expression, and Exercise 5.8 derives it from the Gauss–Bonnet formula. Before using it, check that it behaves as a curvature should.

It doesn't depend on the coordinate. Under $z = \phi(w)$ the new conformal factor is $u\circ\phi + \log|\phi'|$; the second term is harmonic (5A.1 Holomorphic Functions Are Conformal, last exercise), and $\Delta_w(u\circ\phi) = |\phi'|^2(\Delta u)\circ\phi$ (5A.4 Harmonic Functions and Conformal Mapping). The factors of $|\phi'|^2$ cancel and $K$ is unchanged.

Scaling. Replacing $g$ by $\lambda^2g$ (adding $\log\lambda$ to $u$) divides $K$ by $\lambda^2$: a sphere of radius $\lambda$ has curvature $1/\lambda^2$.

Flat means harmonic. $K = 0$ exactly when $u$ is harmonic, and the pullback of the flat metric by a holomorphic map has $u = \log|f'|$, harmonic, so it is flat, as it must be.

5.2Riemann surfaces: tori and higher genus

Tori. Let $\Lambda = \{m\omega_1 + n\omega_2\}$ be a lattice in $\mathbb{C}$, with $\omega_1$, $\omega_2$ linearly independent over $\mathbb{R}$. The quotient $\mathbb{C}/\Lambda$ (7A.1 Topological Spaces and Quotients) is a torus, and since translations are holomorphic it is a Riemann surface. Translations are also isometries of $|dz|^2$, so the flat metric descends: every lattice gives a flat torus (Figure 5.2). Different lattices can give different Riemann surfaces. Scaling and rotating the lattice changes nothing conformally, so one may take $\omega_1 = 1$ and $\omega_2 = \tau$ in the upper half-plane, and two values of $\tau$ give conformally equivalent tori exactly when they differ by a transformation $\tau \mapsto \frac{a\tau + b}{c\tau + d}$ with integer entries and $ad - bc = 1$ (Exercise 5.10 proves the easy direction). So there is a whole two-real-parameter family of conformally different tori, the first example of a moduli space. Topologically they are all the same; as Riemann surfaces they are not.

Higher genus. A closed orientable surface of genus $g \geq 2$ cannot be flat: by the Gauss–Bonnet theorem (8A.9 The Curvature of Surfaces),

so its curvature must be negative somewhere. The model is the hyperbolic disc. Cut a genus-2 surface along four loops into an octagon, whose sides are glued in the pattern $a\,b\,a^{-1}b^{-1}c\,d\,c^{-1}d^{-1}$ (7A.5 Computing π₁). All eight vertices are glued to a single point, so the angles there must add up to $2\pi$, which needs eight angles of $\frac{\pi}{4}$. A Euclidean octagon has angles adding up to $6\pi$; but in the hyperbolic disc a regular octagon with angles $\frac\pi4$ exists, and its sides can be glued by hyperbolic isometries (Figure 5.3). The result is a genus-2 surface with a metric of curvature $-1$. Gauss–Bonnet confirms the count: the area of a hyperbolic octagon is $6\pi$ minus its angle sum, $6\pi - 2\pi = 4\pi$, and $-4\pi = 2\pi\chi$ with $\chi = -2$.

5.3The uniformization theorem

The proof is beyond this book; Donaldson's Riemann Surfaces gives one using harmonic functions and Hilbert-space methods of the kind developed in Book 4A. The Riemann mapping theorem is its special case for regions of the plane.

Combined with covering spaces (7A.6 Covering Spaces), the theorem classifies the geometry of every surface. A closed Riemann surface $\Sigma$ has a simply connected universal cover $\tilde\Sigma$, which is a Riemann surface, and $\Sigma = \tilde\Sigma/\Gamma$ for a group $\Gamma$ of conformal automorphisms acting freely. The automorphisms of $\hat{\mathbb{C}}$, $\mathbb{C}$ and $\mathbb{D}$ are known: Möbius maps, maps $az + b$, and disc automorphisms. Checking which of them can act freely with compact quotient gives:

The sign of the curvature is decided by topology, through Gauss–Bonnet: $\chi > 0$, $\chi = 0$, $\chi < 0$. Every closed surface is, conformally, a quotient of one of three model geometries. This is the statement that Thurston's geometrization conjecture extends to three-manifolds, with eight model geometries instead of three, and with the extra complication that a three-manifold must first be cut into pieces (10A.5 Thurston’s Eight Geometries). Perelman's proof of geometrization by Ricci flow (12C.4 Geometrization) is the three-dimensional version of the flow proof below.

5.4The two-dimensional Ricci flow

In two dimensions the Ricci curvature of a metric is $\operatorname{Ric} = Kg$, so Hamilton's Ricci flow $\partial_tg = -2\operatorname{Ric}$ (11A.1 The Equation and Its First Solutions) becomes

The right side is a multiple of $g$, so the flow stays in the conformal class: if $g(0) = e^{2u_0}|dz|^2$, then $g(t) = e^{2u(t)}|dz|^2$. Substituting $K = -e^{-2u}\Delta u$:

The two-dimensional Ricci flow is a heat equation for the conformal factor, with diffusion coefficient $e^{-2u}$, which is $\Delta_g$, the Laplacian of the metric itself. Writing $v = e^{2u}$ for the area density, it becomes $\partial_tv = \Delta\log v$, an equation known in other contexts as logarithmic diffusion. It is a nonlinear parabolic equation of the kind treated in 6A.7 Nonlinear Parabolic Equations. Where $K > 0$ the metric shrinks, and where $K < 0$ it expands: the flow moves curvature around the way heat flows from hot to cold.

To see convergence one normalises: rescale so the area stays fixed, giving the normalised flow $\partial_tg = (\bar r - 2K)g$, with $\bar r$ the average of $2K$. Then:

Richard Hamilton proved this in 1988 ("The Ricci flow on surfaces") when $\chi \leq 0$, and for the sphere under the extra assumption that the initial curvature is positive everywhere; Bennett Chow removed that assumption in 1991 ("The Ricci flow on the 2-sphere") by showing that the curvature becomes positive in finite time. Hamilton's argument for the sphere used the uniformization theorem along the way, so as a proof of uniformization it was circular; Xiuxiong Chen, Peng Lu and Gang Tian (2006) showed how to remove that dependence, so that the Ricci flow gives an independent proof of uniformization for the sphere. The proofs, including Hamilton's entropy for surfaces, are in 11A.7 Ricci Flow on Surfaces.

5.5Theta functions are periodic heat kernels

In 2B.7 Fourier Series and the First Heat Equation the heat equation $u_t = u_{xx}$ on a circle of length $1$ was solved by Fourier series, with kernel

and we promised that $H_t$ is a theta function. Jacobi's theta function is

holomorphic in both variables, and $H_t(x) = \vartheta(x \mid 4\pi it)$. The heat kernel on the circle is a theta function evaluated at an imaginary "time".

On the other hand, the circle is $\mathbb{R}/\mathbb{Z}$, and the heat kernel on a quotient is the sum of the heat kernel of the line over all translates:

That the two expressions agree is the Poisson summation formula applied to a Gaussian, whose Fourier transform we computed by contour shifting in 5A.3 Residues and Fourier Transforms (Exercise 5.12). The first series converges fast for large $t$ (only the lowest frequencies matter), the second for small $t$ (only the nearest copies matter). In terms of $\vartheta$ the identity is Jacobi's transformation law relating $\tau$ and $-1/\tau$. The same construction on a flat torus $\mathbb{C}/\Lambda$, summing the planar heat kernel over the lattice, gives its heat kernel, and theta functions of two variables are the building blocks of meromorphic functions on tori (Stein–Shakarchi, chapter 10). Heat kernels on general manifolds, which have no such formula, are the subject of 9B.7 The Heat Equation on a Manifold.

5.6History

Bernhard Riemann introduced Riemann surfaces in his 1851 thesis and his 1857 work on abelian functions. Henri Poincaré and Paul Koebe independently proved the uniformization theorem in 1907, after partial results by Felix Klein and Poincaré in the 1880s. The hyperbolic disc model is due to Eugenio Beltrami (1868) and Poincaré (1882). Carl Gustav Jacob Jacobi introduced theta functions in his 1829 work on elliptic functions; his transformation law is equivalent to Poisson summation for a Gaussian, that is, to the two formulas for the heat kernel on a circle agreeing. Richard Hamilton's paper "The Ricci flow on surfaces" (1988) and Bennett Chow's "The Ricci flow on the 2-sphere" (1991) proved that the two-dimensional flow uniformizes; Chen, Lu and Tian's note on avoiding circularity appeared in 2006. Chow and Luo's combinatorial Ricci flow dates from 2003, and Jin, Kim, Luo and Gu's discrete surface Ricci flow from 2008.

5.7Exercises

Solution

(a) In polar coordinates, $\Delta f(r) = f'' + f'/r$; for $f = \log(1 + r^2)$, $f' = \frac{2r}{1 + r^2}$, $f'' = \frac{2(1 - r^2)}{(1 + r^2)^2}$, and $f'' + f'/r = \frac{2(1 - r^2) + 2(1 + r^2)}{(1 + r^2)^2} = \frac{4}{(1 + r^2)^2}$. So $\Delta u = -\frac{4}{(1 + r^2)^2}$ and $K = \frac{(1 + r^2)^2}{4}\cdot\frac{4}{(1 + r^2)^2} = 1$. (b) For $\log(1 - r^2)$ the same computation gives $-\frac{4}{(1 - r^2)^2}$, so $\Delta u = \frac{4}{(1 - r^2)^2}$ and $K = -1$. For $u = -\log y$, $\Delta u = \frac{1}{y^2}$ and $K = -y^2\cdot\frac{1}{y^2} = -1$.

Solution

$\tilde ds = e^u\,ds$, so $\oint\tilde\kappa_g\,\tilde ds = \oint(\frac1\rho + \partial_\nu u)\,ds = 2\pi + \int_{B_\rho}\Delta u\,dA$ by the divergence theorem. Gauss–Bonnet for $\tilde g$ gives $\int_{B_\rho}K\,e^{2u}\,dA = 2\pi - 2\pi - \int_{B_\rho}\Delta u\,dA$. Divide by the area $\pi\rho^2$ and let $\rho \to 0$: $Ke^{2u} = -\Delta u$ at the centre.

Solution

On the unit sphere $ds^2 = d\phi^2 + \cos^2\phi\,d\lambda^2$, and $dY = \sec\phi\,d\phi$ (5A.1 Holomorphic Functions Are Conformal), so $ds^2 = \cos^2\phi\,(dY^2 + dX^2)$. If $\tanh Y = \sin\phi$ then $\operatorname{sech}^2Y = 1 - \sin^2\phi = \cos^2\phi$; and indeed $\frac{d}{d\phi}\tanh Y = \operatorname{sech}^2Y\sec\phi = \cos\phi$. With $u = -\log\cosh Y$, $u'' = -\operatorname{sech}^2Y$, so $K = \cosh^2Y\operatorname{sech}^2Y = 1$. Inflation $\sec^2\phi$: $\sec^260° = 4$, $\sec^280° \approx 33.2$.

Solution

$\frac{d}{dt}\int e^{2u}\,dA = \int2\Delta u\,dA = -2\int Ke^{2u}\,dA = -4\pi\chi$ by Gauss–Bonnet. On the sphere $\chi = 2$, so the area is $A - 8\pi t$, zero at $t = A/8\pi$ (for the unit sphere, $4\pi/8\pi = \frac12$). For genus $g \geq 2$, $-4\pi\chi = 8\pi(g - 1) > 0$.

Solution

(a) $u = -\frac12\log(1 + r^2)$, $\Delta u = -\frac12\cdot\frac{4}{(1 + r^2)^2}$, $K = (1 + r^2)\cdot\frac{2}{(1 + r^2)^2} = \frac{2}{1 + r^2}$. The circumference is $\int_0^{2\pi}e^ur\,d\theta = \frac{2\pi r}{\sqrt{1 + r^2}}$. (b) $\partial_t(e^{2u}) = -\frac{a'}{(a + r^2)^2}$, while $2\Delta u = -\Delta\log(a + r^2) = -\frac{4a}{(a + r^2)^2}$ by the computation of Exercise 5.7 scaled. So $a' = 4a$. (c) The pullback of $\frac{|dw|^2}{1 + |w|^2}$ by $w = \lambda z$ is $\frac{\lambda^2|dz|^2}{1 + \lambda^2|z|^2} = \frac{|dz|^2}{\lambda^{-2} + |z|^2}$; take $\lambda = e^{-2t}$.