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)

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).