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 **Z**^{2}; and those with universal cover the Riemann sphere are those of genus zero, namely the Riemann sphere itself, with trivial fundamental group.

- History
- Classification of connected Riemann surfaces
- Classification of closed oriented Riemannian 2-manifolds
- Methods of proof
- Hilbert space methods
- Nonlinear flows
- Generalizations
- See also
- Notes
- References
- Historic references
- Historical surveys
- Harmonic functions
- Nonlinear differential equations
- General references
- External links

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.

Felix Klein ( 1883 ) and Henri Poincaré ( 1882 ) conjectured the uniformization theorem for (the Riemann surfaces of) algebraic curves. HenriPoincaré ( 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 PaulKoebe ( 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).

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:

- the Riemann sphere
- the complex plane
- 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.

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

then in the complex coordinate *z* = *x* + i*y*, 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 constant curvature, so a quotient of one of the following by a free action of a discrete subgroup of an isometry group:

- 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 – 2*g*, where *g* is the genus of the 2-manifold, i.e. the number of "holes".

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 *L*^{2} 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).

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).

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.

- ↑ DeTurck & Kazdan 1981; Taylor 1996a , pp. 377–378
- ↑ Brendle 2010

In mathematics, the **Poincaré conjecture** is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space.

In the part of mathematics referred to as topology, a **surface** is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space.

In mathematics, a **conformal map** is a function that locally preserves angles, but not necessarily lengths.

In mathematics, particularly in complex analysis, a **Riemann surface** is a one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.

**Riemannian geometry** is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a *Riemannian metric*, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.

**Grigori Yakovlevich Perelman** is a Russian mathematician who is known for his contributions to the fields of geometric analysis, Riemannian geometry, and geometric topology.

In differential geometry, the **Ricci curvature tensor**, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space.

In mathematics, a **minimal surface** is a surface that locally minimizes its area. This is equivalent to having zero mean curvature.

In Riemannian geometry, the **scalar curvature** is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point. Specifically, the scalar curvature represents the amount by which the volume of a small geodesic ball in a Riemannian manifold deviates from that of the standard ball in Euclidean space. In two dimensions, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In more than two dimensions, however, the curvature of Riemannian manifolds involves more than one functionally independent quantity.

In the mathematical field of differential geometry, the **Ricci flow**, sometimes also referred to as **Hamilton's Ricci flow**, is a certain partial differential equation for a Riemannian metric. It is often said to be analogous to the diffusion of heat and the heat equation, due to formal similarities in the mathematical structure of the equation; however, it exhibits many phenomena not present in the study of the heat equation. Many results for Ricci flow have also been shown for the mean curvature flow of hypersurfaces.

**Shing-Tung Yau** is an American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University.

In mathematics, **low-dimensional topology** is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.

**Richard Streit Hamilton** is Davies Professor of Mathematics at Columbia University. He is known for contributions to geometric analysis and partial differential equations. He made foundational contributions to the theory of the Ricci flow and its use in the resolution of the Poincaré conjecture and geometrization conjecture in the field of geometric topology.

In differential geometry, the **Weyl curvature tensor**, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic. The Weyl tensor differs from the Riemann curvature tensor in that it does not convey information on how the volume of the body changes, but rather only how the shape of the body is distorted by the tidal force. The Ricci curvature, or trace component of the Riemann tensor contains precisely the information about how volumes change in the presence of tidal forces, so the Weyl tensor is the traceless component of the Riemann tensor. It is a tensor that has the same symmetries as the Riemann tensor with the extra condition that it be trace-free: metric contraction on any pair of indices yields zero.

**Geometric analysis** is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations are used to establish new results in differential geometry and differential topology. The use of linear elliptic PDEs dates at least as far back as Hodge theory. More recently, it refers largely to the use of nonlinear partial differential equations to study geometric and topological properties of spaces, such as submanifolds of Euclidean space, Riemannian manifolds, and symplectic manifolds. This approach dates back to the work by Tibor Radó and Jesse Douglas on minimal surfaces, John Forbes Nash Jr. on isometric embeddings of Riemannian manifolds into Euclidean space, work by Louis Nirenberg on the Minkowski problem and the Weyl problem, and work by Aleksandr Danilovich Aleksandrov and Aleksei Pogorelov on convex hypersurfaces. In the 1980s fundamental contributions by Karen Uhlenbeck, Clifford Taubes, Shing-Tung Yau, Richard Schoen, and Richard Hamilton launched a particularly exciting and productive era of geometric analysis that continues to this day. A celebrated achievement was the solution to the Poincaré conjecture by Grigori Perelman, completing a program initiated and largely carried out by Richard Hamilton.

In mathematics, specifically in differential geometry, **isothermal coordinates** on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric locally has the form

In mathematics, the **differential geometry of surfaces** deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: *extrinsically*, relating to their embedding in Euclidean space and *intrinsically*, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.

The **Geometry Festival** is an annual mathematics conference held in the United States.

In mathematics, a **planar Riemann surface** is a Riemann surface sharing the topological properties of a connected open subset of the Riemann sphere. They are characterized by the topological property that the complement of every closed Jordan curve in the Riemann surface has two connected components. An equivalent characterization is the differential geometric property that every closed differential 1-form of compact support is exact. Every simply connected Riemann surface is planar. The class of planar Riemann surfaces was studied by Koebe who proved in 1910 as a generalization of the uniformization theorem that every such surface is conformally equivalent to either the Riemann sphere or the complex plane with slits parallel to the real axis removed.

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