proof of the uniformization theorem
We will merely use the fact that . If is compact, then is a complex curve of genus , so . On the other hand, the elementary Riemann mapping theorem says that an open set with is either equal to or biholomorphic to the unit disk. Thus, all we have to show is that a non compact Riemann surface with can be embedded in the complex plane .
Let be an exhausting sequence of relatively compact connected open sets with smooth boundary in . We may assume that has no relatively compact connected components, otherwise we “fill the holes” of by taking the union with all such components. We let be the double of the manifold with boundary , i.e. the union of two copies of with opposite orientations and the boundaries identified. Then is a compact oriented surface without boundary.
Fact: we have . We postpone the proof of this fact to the end of the present paragraph and we continue with the proof of the uniformization theorem.
Extend the almost complex structure of in an arbitrary way to , e.g. by an extension of a Riemmanian metric. Then becomes a compact Riemann surface of genus , thus and we obtain in particular a holomorphic embedding . Fix a point and a non zero linear form . We can take the composition of with an affine linear map so that and . By the well-known properties of injective holomorphic maps, is then uniformly bounded on every small disk centered at , thus also on every compact subset of by a connectedness argument. Hence has a subsequence converging towards an injective holomorphic map .
Proof of the ”fact”: Let us first compute the cohomology with compact support . Let be a closed -form with compact support in . By Poincaré duality , so for some ”test” function . As on a neighborhood of and as all connected components of this set are non compact, must be equal to the constant zero near . Hence is the zero class in and we get . The exact sequence of the pair yelds
thus . Finally, the Mayer-Vietoris sequence applied to small neighborhoods of the two copies of in gives an exact sequence
where the first map is onto. Hence .
|Title||proof of the uniformization theorem|
|Date of creation||2013-03-22 15:37:50|
|Last modified on||2013-03-22 15:37:50|
|Last modified by||Simone (5904)|