Uniformization theorem

In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. In particular it implies that every Riemann surface admits a Riemannian metric of constant curvature. For compact Riemann surfaces, those with universal cover the unit disk are precisely the hyperbolic surfaces of genus greater than 1, all with non-abelian fundamental group; those with universal cover the complex plane are the Riemann surfaces of genus 1, namely the complex tori or elliptic curves with fundamental group Z2; and those with universal cover the Riemann sphere are those of genus zero, namely the Riemann sphere itself, with trivial fundamental group.

The uniformization theorem is a generalization of the Riemann mapping theorem from proper simply connected open subsets of the plane to arbitrary simply connected Riemann surfaces. The uniformization theorem also has an equivalent statement in terms of closed Riemannian 2-manifolds: each such manifold has a conformally equivalent Riemannian metric with constant curvature.

Many classical proofs of the uniformization theorem rely on constructing a real-valued harmonic function on the simply connected Riemann surface, possibly with a singularity at one or two points and often corresponding to a form of Green's function. Four methods of constructing the harmonic function are widely employed: the Perron method; the Schwarz alternating method; Dirichlet's principle; and Weyl's method of orthogonal projection. In the context of closed Riemannian 2-manifolds, several modern proofs invoke nonlinear differential equations on the space of conformally equivalent metrics. These include the Beltrami equation from Teichmüller theory and an equivalent formulation in terms of harmonic maps; Liouville's equation, already studied by Poincaré; and Ricci flow along with other nonlinear flows.

History

Felix Klein (1883) and Henri Poincaré (1882) conjectured the uniformization theorem for (the Riemann surfaces of) algebraic curves. Henri Poincaré (1883) extended this to arbitrary multivalued analytic functions and gave informal arguments in its favor. The first rigorous proofs of the general uniformization theorem were given by Poincaré (1907) and Paul Koebe (1907a, 1907b, 1907c). Paul Koebe later gave several more proofs and generalizations. The history is described in Gray (1994); a complete account of uniformization up to the 1907 papers of Koebe and Poincaré is given with detailed proofs in de Saint-Gervais (2016) (the Bourbaki-type pseudonym of the group of fifteen mathematicians who jointly produced this publication).

Classification of connected Riemann surfaces

Every Riemann surface is the quotient of free, proper and holomorphic action of a discrete group on its universal covering and this universal covering is holomorphically isomorphic (one also says: "conformally equivalent" or "biholomorphic") to one of the following:

1. the Riemann sphere
2. the complex plane
3. the unit disk in the complex plane.

Rado's theorem shows that every Riemann surface is automatically second-countable. Although Rado's theorem is often used in proofs of the uniformization theorem, some proofs have been formulated so that Rado's theorem becomes a consequence. Second countability is automatic for compact Riemann surfaces.

Classification of closed oriented Riemannian 2-manifolds

On an oriented 2-manifold, a Riemannian metric induces a complex structure using the passage to isothermal coordinates. If the Riemannian metric is given locally as

${\displaystyle ds^{2}=E\,dx^{2}+2F\,dx\,dy+G\,dy^{2},}$

then in the complex coordinate z = x + iy, it takes the form

${\displaystyle ds^{2}=\lambda |dz+\mu \,d{\overline {z}}|^{2},}$

where

${\displaystyle \lambda ={1 \over 4}\left(E+G+2{\sqrt {EG-F^{2}}}\right),\,\,\,\mu =(E-G+2iF)/4\lambda ,}$

so that λ and μ are smooth with λ > 0 and |μ| < 1. In isothermal coordinates (u, v) the metric should take the form

${\displaystyle ds^{2}=\rho (du^{2}+dv^{2})}$

with ρ > 0 smooth. The complex coordinate w = u + i v satisfies

${\displaystyle \rho \,|dw|^{2}=\rho |w_{z}|^{2}\left|dz+{w_{\overline {z}} \over w_{z}}\,d{\overline {z}}\right|^{2},}$

so that the coordinates (u, v) will be isothermal locally provided the Beltrami equation

${\displaystyle {\partial w \over \partial {\overline {z}}}=\mu {\partial w \over \partial z}}$

has a locally diffeomorphic solution, i.e. a solution with non-vanishing Jacobian.

These conditions can be phrased equivalently in terms of the exterior derivative and the Hodge star operator .[1] u and v will be isothermal coordinates if du = dv, where is defined on differentials by ∗(p dx + q dy) = −q dx + p dy. Let ∆ = ∗dd be the Laplace–Beltrami operator. By standard elliptic theory, u can be chosen to be harmonic near a given point, i.e. Δ u = 0, with du non-vanishing. By the Poincaré lemma dv = ∗du has a local solution v exactly when d(∗du) = 0. This condition is equivalent to Δ u = 0, so can always be solved locally. Since du is non-zero and the square of the Hodge star operator is −1 on 1-forms, du and dv must be linearly independent, so that u and v give local isothermal coordinates.

The existence of isothermal coordinates can be proved by other methods, for example using the general theory of the Beltrami equation, as in Ahlfors (2006), or by direct elementary methods, as in Chern (1955) and Jost (2006).

From this correspondence with compact Riemann surfaces, a classification of closed orientable Riemannian 2-manifolds follows. Each such is conformally equivalent to a unique closed 2-manifold of constant curvature, so a quotient of one of the following by a free action of a discrete subgroup of an isometry group:

1. the sphere (curvature +1)
2. the Euclidean plane (curvature 0)
3. the hyperbolic plane (curvature −1).

The first case gives the 2-sphere, the unique 2-manifold with constant positive curvature and hence positive Euler characteristic (equal to 2). The second gives all flat 2-manifolds, i.e. the tori, which have Euler characteristic 0. The third case covers all 2-manifolds of constant negative curvature, i.e. the hyperbolic 2-manifolds all of which have negative Euler characteristic. The classification is consistent with the Gauss–Bonnet theorem, which implies that for a closed surface with constant curvature, the sign of that curvature must match the sign of the Euler characteristic. The Euler characteristic is equal to 2 – 2g, where g is the genus of the 2-manifold, i.e. the number of "holes".

Methods of proof

Hilbert space methods

In 1913 Hermann Weyl published his classic textbook "Die Idee der Riemannschen Fläche" based on his Göttingen lectures from 1911 to 1912. It was the first book to present the theory of Riemann surfaces in a modern setting and through its three editions has remained influential. Dedicated to Felix Klein, the first edition incorporated Hilbert's treatment of the Dirichlet problem using Hilbert space techniques; Brouwer's contributions to topology; and Koebe's proof of the uniformization theorem and its subsequent improvements. Much later Weyl (1940) developed his method of orthogonal projection which gave a streamlined approach to the Dirichlet problem, also based on Hilbert space; that theory, which included Weyl's lemma on elliptic regularity, was related to Hodge's theory of harmonic integrals; and both theories were subsumed into the modern theory of elliptic operators and L2 Sobolev spaces. In the third edition of his book from 1955, translated into English in Weyl (1964), Weyl adopted the modern definition of differential manifold, in preference to triangulations, but decided not to make use of his method of orthogonal projection. Springer (1957) followed Weyl's account of the uniformisation theorem, but used the method of orthogonal projection to treat the Dirichlet problem. This approach will be outlined below. Kodaira (2007) describes the approach in Weyl's book and also how to shorten it using the method of orthogonal projection. A related account can be found in Donaldson (2011).

Nonlinear flows

In introducing the Ricci flow, Richard S. Hamilton showed that the Ricci flow on a closed surface uniformizes the metric (i.e., the flow converges to a constant curvature metric). However, his proof relied on the uniformization theorem. The missing step involved Ricci flow on the 2-sphere: a method for avoiding an appeal to the uniformization theorem (for genus 0) was provided by Chen, Lu & Tian (2006);[2] a short self-contained account of Ricci flow on the 2-sphere was given in Andrews & Bryan (2010).

Generalizations

Koebe proved the general uniformization theorem that if a Riemann surface is homeomorphic to an open subset of the complex sphere (or equivalently if every Jordan curve separates it), then it is conformally equivalent to an open subset of the complex sphere.

In 3 dimensions, there are 8 geometries, called the eight Thurston geometries. Not every 3-manifold admits a geometry, but Thurston's geometrization conjecture proved by Grigori Perelman states that every 3-manifold can be cut into pieces that are geometrizable.

The simultaneous uniformization theorem of Lipman Bers shows that it is possible to simultaneously uniformize two compact Riemann surfaces of the same genus >1 with the same quasi-Fuchsian group.

The measurable Riemann mapping theorem shows more generally that the map to an open subset of the complex sphere in the uniformization theorem can be chosen to be a quasiconformal map with any given bounded measurable Beltrami coefficient.

References

Harmonic functions

Perron's method

• Heins, M. (1949), "The conformal mapping of simply-connected Riemann surfaces", Ann. of Math., 50 (3): 686–690, doi:10.2307/1969555, JSTOR 1969555
• Heins, M. (1951), "Interior mapping of an orientable surface into S2", Proc. Amer. Math. Soc., 2 (6): 951–952, doi:10.1090/s0002-9939-1951-0045221-4
• Heins, M. (1957), "The conformal mapping of simply-connected Riemann surfaces. II", Nagoya Math. J., 12: 139–143, doi:10.1017/s002776300002198x
• Pfluger, Albert (1957), Theorie der Riemannschen Flächen, Springer
• Ahlfors, Lars V. (2010), Conformal invariants: topics in geometric function theory, AMS Chelsea Publishing, ISBN 978-0-8218-5270-5
• Beardon, A. F. (1984), "A primer on Riemann surfaces", London Mathematical Society Lecture Note Series, Cambridge University Press, 78, ISBN 978-0521271042
• Forster, Otto (1991), Lectures on Riemann surfaces, Graduate Texts in Mathematics, 81, translated by Bruce Gilligan, Springer, ISBN 978-0-387-90617-1
• Farkas, Hershel M.; Kra, Irwin (1980), Riemann surfaces (2nd ed.), Springer, ISBN 978-0-387-90465-8
• Gamelin, Theodore W. (2001), Complex analysis, Undergraduate Texts in Mathematics, Springer, ISBN 978-0-387-95069-3
• Hubbard, John H. (2006), Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1. Teichmüller theory, Matrix Editions, ISBN 978-0971576629
• Schlag, Wilhelm (2014), A course in complex analysis and Riemann surfaces., Graduate Studies in Mathematics, 154, American Mathematical Society, ISBN 978-0-8218-9847-5

Schwarz's alternating method

• Nevanlinna, Rolf (1953), Uniformisierung, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, 64, Springer
• Behnke, Heinrich; Sommer, Friedrich (1965), Theorie der analytischen Funktionen einer komplexen Veränderlichen, Die Grundlehren der mathematischen Wissenschaften, 77 (3rd ed.), Springer
• Freitag, Eberhard (2011), Complex analysis. 2. Riemann surfaces, several complex variables, abelian functions, higher modular functions, Springer, ISBN 978-3-642-20553-8

Dirichlet principle

• Weyl, Hermann (1964), The concept of a Riemann surface, translated by Gerald R. MacLane, Addison-Wesley, MR 0069903
• Courant, Richard (1977), Dirichlet's principle, conformal mapping, and minimal surfaces, Springer, ISBN 978-0-387-90246-3
• Siegel, C. L. (1988), Topics in complex function theory. Vol. I. Elliptic functions and uniformization theory, translated by A. Shenitzer; D. Solitar, Wiley, ISBN 978-0471608448

Weyl's method of orthogonal projection

• Springer, George (1957), Introduction to Riemann surfaces, Addison-Wesley, MR 0092855
• Kodaira, Kunihiko (2007), Complex analysis, Cambridge Studies in Advanced Mathematics, 107, Cambridge University Press, ISBN 9780521809375
• Donaldson, Simon (2011), Riemann surfaces, Oxford Graduate Texts in Mathematics, 22, Oxford University Press, ISBN 978-0-19-960674-0

Sario operators

• Sario, Leo (1952), "A linear operator method on arbitrary Riemann surfaces", Trans. Amer. Math. Soc., 72 (2): 281–295, doi:10.1090/s0002-9947-1952-0046442-2
• Ahlfors, Lars V.; Sario, Leo (1960), Riemann surfaces, Princeton Mathematical Series, 26, Princeton University Press

Nonlinear differential equations

Beltrami's equation

Harmonic maps

• Jost, Jürgen (2006), Compact Riemann surfaces: an introduction to contemporary mathematics (3rd ed.), Springer, ISBN 978-3-540-33065-3

Liouville's equation

• Berger, Melvyn S. (1971), "Riemannian structures of prescribed Gaussian curvature for compact 2-manifolds", Journal of Differential Geometry, 5 (3–4): 325–332, doi:10.4310/jdg/1214429996
• Berger, Melvyn S. (1977), Nonlinearity and functional analysis, Academic Press, ISBN 978-0-12-090350-4
• Taylor, Michael E. (2011), Partial differential equations III. Nonlinear equations, Applied Mathematical Sciences, 117 (2nd ed.), Springer, ISBN 978-1-4419-7048-0

Flows on Riemannian metrics