This paper is motivated by
the observation that Noether’s theorem for quadratic differentials fails for
hyperelliptic Riemann surfaces. In this paper we provide an appropriate substitute
for Noether’s theorem which is valid for plane domains with hyperelliptic double. Our
result is somewhat more explicit than Noether’s, and, in contrast with the case of
nonhyperelliptic surfaces, it provides a basis for the (even) quadratic differentials
which holds globally for all domains with hyperelliptic double. An important fact
which plays a significant role in these considerations is that no two normal
differentials of the first kind can have a common zero on a domain with hyperelliptic
double.
