Riemann surfaces which are doubles of plane domains

Weierstrass points of plane domains

“…This definition employs the " Noether mapping " υ of a nonhyperelliptic Riemann surface & of genus p into P^C), the complex protective space of dimension p -1 (see [1]), and can only be employed if & is nonhyperelliptic. The mapping υ is accomplished by selecting a basis of Abelian differentials of the first kind on & and considering them as homogeneous coordinates in P ί?…”

“…There is an anti-analytic involution of & onto itself which maps £& anticonformally onto &* and leaves the boundary curves point wise fixed. All of those Riemann surfaces of genus p which are the doubles of plane domains can be characterized in a way which will prove very useful in this study (see [1]). …”

“…There is a characterization of those Riemann surfaces which are the doubles of plane domains [1] From the illustration of ^R it is clear that it has exactly eight inflection points; and the curvature changes sign at each of these points. Since the curvature does change sign, the order of each inflection point must be odd.…”

“…Now let ^ be a plane domain whose double is &. (How to make an actual determination of Si from a knowledge of & or & is discussed in [1]). Then we have shown that & has 8 Weierstrass points (counted with multiplicity) in its interior.…”