Extremal length and Dirichlet problem on Klein surfaces

The object of this paper is to extend the method of extremal length to Klein surfaces by solving conformally invariant extremal problems on the complex double. Within this method we define the extremal length, the extremal distance, the conjugate extremal distance, the modulus, the reduced extremal distance on a Klein surface and we study their dependences on arcs.

On the connectedness of the branch locus of moduli space of hyperelliptic Klein surfaces with one boundary

In this work, we prove that the hyperelliptic branch locus of orientable Klein surfaces of genus [Formula: see text] with one boundary component is connected and in the case of non-orientable Klein surfaces it has [Formula: see text] components, if [Formula: see text] is odd, and [Formula: see text] components for even [Formula: see text]. We notice that, for non-orientable Klein surfaces with two boundary components, the hyperelliptic branch loci are connected for all genera.