Executive Summary : | The study of geometric structures on surfaces has deep and fascinating connections with complex analysis, differential equations, and differential geometry. The most familiar example is a hyperbolic structure on a closed and oriented surface S_g of genus g at least 2; the uniformization theorem identifies the moduli space of such structures with the moduli space of Riemann surfaces M_g, which is ubiquitous in geometry. The Teichmüller space T_g of "marked" Riemann surfaces is the universal cover of M_g, and is parametrized by holomorphic quadratic differentials on S equipped with a choice of a complex structure. Another example is the space P_g of marked complex projective structures on S_g, which are geometric structures modelled on CP^1 arising from solutions of the Schwarzian differential equation on Riemann surfaces. All these spaces of structures are themselves complex analytic objects, and carry symplectic and Kähler structures, which makes them interesting from the point of view of complex geometry. Some recent work of the PI has introduced spaces of geometric structures on punctured surfaces, where the defining holomorphic objects admit poles of arbitrary order at the punctures, e.g. the Teichmüller space of crowned hyperbolic structures and the deformation space of meromorphic projective structures. The first goal of the current proposal is to develop the theory of the complex and symplectic geometry of these moduli spaces. One way of understanding meromorphic geometric structures is to consider them as structures on a "marked and bordered surface", as introduced in the seminal work of Fock-Goncharov. This project would develop the analogues of the complex-analytic aspects of classical Teichmüller theory, in the setting of marked and bordered surfaces. Fock-Goncharov had introduced the notion of "framed" representations and birational coordinates on their moduli space, which was utilized in a novel way in recent work of the PI, and has led to some breakthrough results. Developments in "higher Teichmüller theory" in the last few decades have shown that Hitchin representations into PSL(n,R) and some of their complex deformations into PSL(n,C) arise as the holonomy of certain geometric structures on surfaces, or certain bundles over surfaces. Hitchin representations correspond to certain holomorphic objects (semistable Higgs bundles) on a Riemann surface via the "nonabelian Hodge correspondence". Although there has been work extending this correspondence to the case of meromorphic Higgs bundles, the study of the corresponding spaces of geometric structures is relatively unexplored. The second goal of the project is to extend the geometric understanding of hyperbolic crowned surfaces and meromorphic projective structures, to the higher-rank case. |