Double of a Riemann surface

A two-sheeted covering surface of a finite Riemann surface . Each interior point is brought into correspondence with a pair of points and of the double ; in other words, two conjugate points and are situated over . Each point of the boundary of is brought into correspondence with a point . Moreover, two non-intersecting neighbourhoods of the points are situated over each neighbourhood of an interior point . If is a local uniformizing parameter in a neighbourhood of the interior point , it will also be a local uniformizing parameter in a -neighbourhood of one out of the two conjugate points of lying over , say in a -neighbourhood of the point ; then, in a -neighbourhood of the conjugate point , the complex conjugate of the variable will be a local uniformizing parameter. If is a local uniformizing parameter at a point of the boundary of , then the variable which is equal to on one sheet of and to on the other will be a local uniformizing parameter at the point lying over it.

In the case of a compact orientable Riemann surface , the double simply consists of two compact orientable Riemann surfaces, and its study is accordingly of no interest. In all other cases the double of the Riemann surface is a compact orientable Riemann surface. This fact permits one to simplify the study of certain problems in the theory of functions on by reducing them to the study of functions on . The genus (cf. Genus of a surface) of is , where is the genus of and is the number of components of the boundary of , which are assumed to be non-degenerate. For instance, the double of a simply-connected plane domain is a sphere, while the double of an -connected plane domain is a sphere with handles.

Analytic differentials on a Riemann surface (cf. Differential on a Riemann surface) are analytic differentials on the double characterized by the fact that they assume conjugate values at conjugate points of and take real values at the points lying over points of the boundary of .

References

 [1] M. Schiffer, D.C. Spencer, "Functionals of finite Riemann surfaces" , Princeton Univ. Press (1954) [2] E. Picard, "Traité d'analyse" , 2 , Gauthier-Villars (1926)