Related papers: Meromorphic projective structures, grafting and th…

This is a survey of the theory of complex projective (CP^1) structures on compact surfaces. After some preliminary discussion and definitions, we concentrate on three main topics: (1) Using the Schwarzian derivative to parameterize the…

We prove that for any orientable connected surface of finite type which is not a a sphere with at most four punctures or a torus with at most two punctures, any homeomorphism of the space of geodesic laminations of this surface, equipped…

Let $\mathcal{G}^*(S,\rho)$ be the graph whose vertices are marked complex projective structures with holonomy $\rho$ and whose edges are graftings from one vertex to another. If $\rho$ is quasi-Fuchsian, a theorem of Goldman implies that…

We compare two relationships between quadratic differentials and measured geodesic laminations on hyperbolic Riemann surfaces (by foliations or complex projective structures). Each yields a homeomorphism $\ML(S) \to Q(X)$ for any conformal…

We prove the existence of "half-plane differentials" with prescribed local data on any Riemann surface. These are meromorphic quadratic differentials with higher-order poles which have an associated singular flat metric isometric to a…

Let $(\Sigma,p)$ be a pointed Riemann surface of genus $g\geq 1$. For any integer $k\geq 1$, we parametrize the space of meromorphic quadratic differentials on $\Sigma$ with a pole of order $(k+2)$ at $p$, having a connected critical graph…

We compare some natural triangulations of the Teichm\"uller space of hyperbolic surfaces with geodesic boundary and of some bordifications. We adapt Scannell-Wolf's proof to show that grafting semi-infinite cylinders at the ends of…

We describe the space of measured foliations induced on a compact Riemann surface by meromorphic quadratic differentials. We prove that any such foliation is realized by a unique such differential $q$ if we prescribe, in addition, the…

Every compact Riemann surface $X$ admits a natural projective structure $p_u$ as a consequence of the uniformization theorem. In this work we describe the construction of another natural projective structure on $X$, namely the Hodge…

Let \theta:\pi_1(R) \to \PSL(2,\C) be a homomorphism of the fundamental group of an oriented, closed surface R of genus exceeding one. We will establish the following theorem. Theorem. Necessary and sufficient for \theta to be the monodromy…

Given a compact connected Riemann surface $X$ equipped with an antiholomorphic involution $\tau$, we consider the projective structures on $X$ satisfying a compatibility condition with respect to $\tau$. For a projective structure $P$ on…

Grafting a measured lamination on a hyperbolic surface defines a self-map of Teichmuller space, which is a homeomorphism by a result of Scannell and Wolf. In this paper we study the large-scale behavior of pruning, which is the inverse of…

The Riemann -Rock theorem plays a central role in the theory of Riemann surfaces with applications to several branches in Mathematics and Physics. Suppose $X$ ia a compact Riemann surface of genus $g$ and $P \in X$. By the Riemann-Roch…

Let $S$ be a closed orientable surface of genus at least two. We introduce a bordification of the moduli space $\mathcal{PT}(S)$ of complex projective structures, with a boundary consisting of projective classes of half-translation…

In this paper, we prove a general second main theorem for meromorphic mappings into a subvariety $V$ of $\mathbb P^N(\mathbb C)$ with an arbitrary family of moving hypersurfaces. Our second main theorem generalizes and improves all previous…

In this article, to each generic real meromorphic function (i.e., having only simple branch points in the appropriate sense) we associate a certain combinatorial gadget which we call the park of a function. We show that the park determines…

We define graftable curves on real projective surfaces. In particular, we construct graftable ones in Hitchin case and show that real projective structures with the same Hitchin holonomy, carrying the same weight type, are related to each…

Motivated by W. P. Thurston, we ask: What is the shape of a meromorphic function on a simply connected Riemann surface $\Omega_z$? We consider Speiser functions, i.e. meromorphic functions on a simply connected Riemann surface, that have a…

Given a connection on a meromorphic vector bundle over a compact Riemann surface with reductive Galois group, we associate to it a projective variety. Connections such that their associated projective variety are curves can be classified,…

A (meromorphic) quadratic differential is a (meromorphic) section of the tensor square of the canonical bundle of a Riemann surface. They arose in the study of quasiconformal mappings in the works of Oswald Teichm\"uller, and have played a…