On ovals on Riemann surfaces

Abstract

We prove that $k$ ($k\ge 9$) non-conjugate symmetries of a Riemann surface of genus $g$ have at most $2g-2+2^{r-3}(9-k)$ ovals in total, where $r$ is the smallest positive integer for which $k\le 2^{r-1}$. Furthermore we prove that for arbitrary $k\ge 9$ this bound is sharp for infinitely many values of $g$.