Uniformization_theorem

Uniformization theorem

Simply connected Riemann surface is equivalent to an open disk, complex plane, or sphere

In mathematics, the uniformization theorem states 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. The theorem is a generalization of the Riemann mapping theorem from simply connected open subsets of the plane to arbitrary simply connected Riemann surfaces.

Since every Riemann surface has a universal cover which is a simply connected Riemann surface, the uniformization theorem leads to a classification of Riemann surfaces into three types: those that have the Riemann sphere as universal cover ("elliptic"), those with the plane as universal cover ("parabolic") and those with the unit disk as universal cover ("hyperbolic"). It further follows that every Riemann surface admits a Riemannian metric of constant curvature, where the curvature can be taken to be 1 in the elliptic, 0 in the parabolic and -1 in the hyperbolic case.

The uniformization theorem also yields a similar classification of closed orientable Riemannian 2-manifolds into elliptic/parabolic/hyperbolic cases. Each such manifold has a conformally equivalent Riemannian metric with constant curvature, where the curvature can be taken to be 1 in the elliptic, 0 in the parabolic and -1 in the hyperbolic case.

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

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

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

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

where

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

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

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

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

\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

{\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 ∆ = ∗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 – 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 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.

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.

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[1] [1]
DeTurck & Kazdan 1981; Taylor 1996a, pp. 377–378

[2] [2]
Brendle 2010

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Uniformization_theorem

Uniformization theorem

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

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