Double of a Riemann surface

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

In the case of a compact orientable Riemann surface $ R $, 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 $ R $ by reducing them to the study of functions on $ W $. The genus (cf. Genus of a surface) of $ W $ is $ g + m - 1 $, where $ g $ is the genus of $ R $ and $ m $ is the number of components of the boundary of $ R $, 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 $ m $- connected plane domain is a sphere with $ m - 1 $ handles.

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

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)

Comments

The process of constructing the double of a Riemann surface is called duplication. This process can be applied to any connected two-dimensional manifold $ M $ with boundary (cf. Boundary (of a manifold)) to yield a regular imbedding of $ M $ in a connected two-dimensional manifold (cf. [a1], § 13.H).

References

[a1]	L.V. Ahlfors, L. Sario, "Riemann surfaces" , Princeton Univ. Press (1974)
[a2]	H.M. Farkas, I. Kra, "Riemann surfaces" , Springer (1980)
[a3]	G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10