Uniformization theorem

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In mathematics, the uniformization theorem for surfaces says that any surface admits a Riemannian metric of constant Gauss curvature. In fact, one can find a metric with constant Gauss curvature in any given conformal class.

From this, a classification of surfaces follows. A surface is 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 includes all surfaces with positive Euler characteristic: the sphere and the real projective plane. The second includes all surfaces with vanishing Euler characteristic: the Euclidean plane, cylinder, Möbius strip, torus, and Klein bottle. The third case covers all surfaces with negative Euler characteristic: almost all surfaces are hyperbolic. Note that for closed surfaces, this 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.

On an oriented surface a Riemannian metric naturally induces an almost complex structure as follows: For a tangent vector v we define J(v) as the vector of the same length which is orthogonal to v and such that (v, J(v)) is positively oriented. On surfaces any almost complex structure is integrable, thus turns the given surface into a Riemann surface. Therefore the above classification of orientable surfaces of constant Gauss curvature is equivalent to the following classification of Riemann surfaces:

Every Riemann surface is the quotient of a 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") to one of the following:

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