## Conformal actions with prescribed periods on Riemann surfaces

### Volume 213 / 2011

Fundamenta Mathematicae 213 (2011), 169-190
MSC: Primary 30F10; Secondary 30F35, 37E30, 14H37.
DOI: 10.4064/fm213-2-3

#### Abstract

It is a natural question what is the set of minimal periods of a holomorphic maps on a Riemann surface of negative Euler characteristic. Sierakowski studied ordinary holomorphic periods on classical Riemann surfaces. Here we study orientation reversing automorphisms acting on classical Riemann surfaces, and also automorphisms of non-orientable unbordered Klein surfaces to which, following Singerman, we shall refer to as non-orientable Riemann surfaces. We get a complete set of conditions for the existence of conformal actions with a prescribed order and a prescribed set of periods together with multiplicities. This lets us determine the minimal genus of a surface which admits such an action.