Let S$S$ be an oriented surface of genus g$g$ and n$n$ punctures. The periods of any meromorphic differential on S$S$, with respect to a choice of complex structure, determine a representation χ:Γg,n→double-struckC$\chi :\Gamma _{g,n} \rightarrow \mathbb {C}$ where normalΓg,n$\Gamma _{g,n}$ is the first homology group of S$S$. We characterise the representations that thus arise, that is, lie in the image of the period map sans-serifPer:normalΩMg,n→sans-serifHom(normalΓg,n,C)$\textsf {Per}:\Omega \mathcal {M}_{g,n}\rightarrow \textsf {Hom}(\Gamma _{g,n}, {\mathbb {C}})$. This generalises a classical result of Haupt in the holomorphic case. Moreover, we determine the image of this period map when restricted to any stratum of meromorphic differentials, having prescribed orders of zeros and poles. Our proofs are geometric, as they aim to construct a translation structure on S$S$ with the prescribed holonomy χ$\chi$. Along the way, we describe a connection with the Hurwitz problem concerning the existence of branched covers with prescribed branching data.

We consider complex projective structures on Riemann surfaces and their groups of projective automorphisms. We show that the structures achieving the maximal possible number of projective automorphisms allowed by their genus are precisely the Fuchsian uniformizations of Hurwitz surfaces by hyperbolic metrics. More generally we show that Galois Belyĭ curves are precisely those Riemann surfaces for which the Fuchsian uniformization is the unique complex projective structure invariant under the full group of biholomorphisms.

Let S be a closed and oriented surface of genus g at least 2. In this (mostly expository) article, the object of study is the space P(S) of marked isomorphism classes of projective structures on S. We show that P(S), endowed with the canonical complex structure, carries exotic hermitian structures that extend the classical ones on the Teichmüller space T (S) of S. We shall notice also that the Kobayashi and Carathéodory pseudodistances, which can be defined for any complex manifold, can not be upgraded to a distance. We finally show that P(S) does not carry any Bergman pseudometric.

Let S be a punctured surface of finite type and negative Euler characteristic. We determine all possible representations ρ : π 1 (S ) → PSL 2 (C) that arise as the monodromy of the Schwarzian equation on S with regular singularities at the punctures. Equivalently, we determine the holonomy representations of complex projective structures on S , whose Schwarzian derivatives (with respect to some uniformizing structure) have poles of order at most two at the punctures. Following earlier work that dealt with the case when there are no apparent singularities, our proof reduces to the case of realizing a degenerate representation with apparent singularities. This mainly involves explicit constructions of complex affine structures on punctured surfaces, with prescribed holonomy. As a corollary, we determine the representations that arise as the holonomy of spherical metrics on S with cone-points at the punctures.