Uniformization theorem
In mathematics, the uniformization theorem states that every
Since every Riemann surface has a
The uniformization theorem also yields a similar classification of closed
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, being a simply connected Riemann surface, is holomorphically isomorphic (one also says: "conformally equivalent" or "biholomorphic") to one of the following:
- the Riemann sphere
- the complex plane
- the unit disk in the complex plane.
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.
Classification of closed oriented Riemannian 2-manifolds
On an oriented 2-manifold, a
then in the complex coordinate z = x + iy, it takes the form
where
so that λ and μ are smooth with λ > 0 and |μ| < 1. In isothermal coordinates (u, v) the metric should take the form
with ρ > 0 smooth. The complex coordinate w = u + i v satisfies
so that the coordinates (u, v) will be isothermal locally provided the Beltrami equation
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 ∆ = ∗d∗d 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
- the sphere (curvature +1)
- the Euclidean plane (curvature 0)
- the hyperbolic plane (curvature −1).
-
genus 0
-
genus 1
-
genus 2
-
genus 3
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
Many classical proofs of the uniformization theorem rely on constructing a real-valued
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.
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
Nonlinear flows
Richard S. Hamilton showed that the normalized 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
See also
Notes
- ^ DeTurck & Kazdan 1981; Taylor 1996a, pp. 377–378
- ^ Brendle 2010
References
Historic references
- JFM 02.0214.02.
- Klein, Felix (1883), "Neue Beiträge zur Riemann'schen Functionentheorie", S2CID 120465625
- Koebe, P. (1907a), "Über die Uniformisierung reeller analytischer Kurven", Göttinger Nachrichten: 177–190, JFM 38.0453.01
- Koebe, P. (1907b), "Über die Uniformisierung beliebiger analytischer Kurven", Göttinger Nachrichten: 191–210, JFM 38.0454.01
- Koebe, P. (1907c), "Über die Uniformisierung beliebiger analytischer Kurven (Zweite Mitteilung)", Göttinger Nachrichten: 633–669, JFM 38.0455.02
- Koebe, Paul (1910a), "Über die Uniformisierung beliebiger analytischer Kurven", Journal für die Reine und Angewandte Mathematik, 138: 192–253, S2CID 120198686
- Koebe, Paul (1910b), "Über die Hilbertsche Uniformlsierungsmethode" (PDF), Göttinger Nachrichten: 61–65
- Poincaré, H. (1882), "Mémoire sur les fonctions fuchsiennes", JFM 15.0342.01
- JFM 15.0348.01
- JFM 38.0452.02
- Hilbert, David (1909), "Zur Theorie der konformen Abbildung" (PDF), Göttinger Nachrichten: 314–323
- S2CID 122843531
- Weyl, Hermann (1913), Die Idee der Riemannschen Fläche (1997 reprint of the 1913 German original), Teubner, ISBN 978-3-8154-2096-6
- Weyl, Hermann (1940), "The method of orthogonal projections in potential theory", Duke Math. J., 7: 411–444,
Historical surveys
- Abikoff, William (1981), "The uniformization theorem", Amer. Math. Monthly, 88 (8): 574–592, JSTOR 2320507
- Gray, Jeremy (1994), "On the history of the Riemann mapping theorem" (PDF), Rendiconti del Circolo Matematico di Palermo. Serie II. Supplemento (34): 47–94, MR 1295591
- Bottazzini, Umberto; Gray, Jeremy (2013), Hidden Harmony—Geometric Fantasies: The Rise of Complex Function Theory, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, ISBN 978-1461457251
- de Saint-Gervais, Henri Paul (2016), Uniformization of Riemann Surfaces: revisiting a hundred-year-old theorem, translated by Robert G. Burns, European Mathematical Society, ISBN 978-3-03719-145-3, translation of French text(prepared in 2007 during centenary of 1907 papers of Koebe and Poincaré)
Harmonic functions
Perron's method
- Heins, M. (1949), "The conformal mapping of simply-connected Riemann surfaces", Ann. of Math., 50 (3): 686–690, JSTOR 1969555
- Heins, M. (1951), "Interior mapping of an orientable surface into S2", Proc. Amer. Math. Soc., 2 (6): 951–952,
- Heins, M. (1957), "The conformal mapping of simply-connected Riemann surfaces. II" (PDF), Nagoya Math. J., 12: 139–143,
- 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, 78, Cambridge University Press, ISBN 978-0521271042
- Forster, Otto (1991), Lectures on Riemann surfaces, Graduate Texts in Mathematics, vol. 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, vol. 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, vol. 64, Springer, ISBN 978-3-642-52802-6
- Behnke, Heinrich; Sommer, Friedrich (1965), Theorie der analytischen Funktionen einer komplexen Veränderlichen, Die Grundlehren der mathematischen Wissenschaften, vol. 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, vol. 107, Cambridge University Press, ISBN 9780521809375
- Donaldson, Simon (2011), Riemann surfaces, Oxford Graduate Texts in Mathematics, vol. 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,
- Ahlfors, Lars V.; Sario, Leo (1960), Riemann surfaces, Princeton Mathematical Series, vol. 26, Princeton University Press
Nonlinear differential equations
Beltrami's equation
- Ahlfors, Lars V. (2006), Lectures on quasiconformal mappings, University Lecture Series, vol. 38 (2nd ed.), American Mathematical Society, ISBN 978-0-8218-3644-6
- Ahlfors, Lars V.; Bers, Lipman (1960), "Riemann's mapping theorem for variable metrics", Ann. of Math., 72 (2): 385–404, JSTOR 1970141
- Bers, Lipman (1960), "Simultaneous uniformization" (PDF), Bull. Amer. Math. Soc., 66 (2): 94–97,
- Bers, Lipman (1961), "Uniformization by Beltrami equations", Comm. Pure Appl. Math., 14 (3): 215–228,
- MR 0348097
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,
- 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, vol. 117 (2nd ed.), Springer, ISBN 978-1-4419-7048-0
Flows on Riemannian metrics
- Hamilton, Richard S. (1988), "The Ricci flow on surfaces", Mathematics and general relativity (Santa Cruz, CA, 1986), Contemp. Math., vol. 71, American Mathematical Society, pp. 237–262
- Chow, Bennett (1991), "The Ricci flow on the 2-sphere", J. Differential Geom., 33 (2): 325–334,
- Osgood, B.; Phillips, R.; Sarnak, P. (1988), "Extremals of determinants of Laplacians", J. Funct. Anal., 80: 148–211,
- Chrusciel, P. (1991), "Semi-global existence and convergence of solutions of the Robinson-Trautman (2-dimensional Calabi) equation", Communications in Mathematical Physics, 137 (2): 289–313, S2CID 53641998
- Chang, Shu-Cheng (2000), "Global existence and convergence of solutions of Calabi flow on surfaces of genus h ≥ 2", J. Math. Kyoto Univ., 40 (2): 363–377,
- Brendle, Simon (2010), Ricci flow and the sphere theorem, Graduate Studies in Mathematics, vol. 111, American Mathematical Society, ISBN 978-0-8218-4938-5
- Chen, Xiuxiong; Lu, Peng; MR 2231924
- Andrews, Ben; Bryan, Paul (2010), "Curvature bounds by isoperimetric comparison for normalized Ricci flow on the two-sphere", Calc. Var. Partial Differential Equations, 39 (3–4): 419–428, S2CID 1095459
- Mazzeo, Rafe; Taylor, Michael (2002), "Curvature and uniformization", S2CID 7192529
- Struwe, Michael (2002), "Curvature flows on surfaces", Ann. Sc. Norm. Super. Pisa Cl. Sci., 1: 247–274
General references
- JSTOR 2032933
- DeTurck, Dennis M.; MR 0644518.
- Gusevskii, N.A. (2001) [1994], "Uniformization", Encyclopedia of Mathematics, EMS Press
- Krushkal, S. L.; Apanasov, B. N.; Gusevskiĭ, N. A. (1986) [1981], Kleinian groups and uniformization in examples and problems, Translations of Mathematical Monographs, vol. 62, Providence, R.I.: MR 0647770
- ISBN 978-0-387-94654-2
- ISBN 978-0-387-94651-1
- Bers, Lipman; John, Fritz; Schechter, Martin (1979), Partial differential equations (reprint of the 1964 original), Lectures in Applied Mathematics, vol. 3A, American Mathematical Society, ISBN 978-0-8218-0049-2
- Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley, ISBN 978-0-471-05059-9
- Warner, Frank W. (1983), Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, vol. 94, Springer, ISBN 978-0-387-90894-6