北京雁栖湖应用数学研究院

Solutions to integrable PDEs on Riemann surfaces like the sinh-Gordon equation or Hitchin's self-duality equations [5] can be described in terms of families of flat connections.
On the one hand, this easily shows that locally, e.g. on the disc, there are (too) many solutions. On the other hand, one can deduce many interesting and deep results for solutions over compact surfaces:
・ for surfaces of genus 0 all solutions can be expressed by holomorphic functions (1980-1990, e.g. Uhlenbeck [11]);
・ for tori, i.e. oriented surfaces of genus 1, one obtains all solutions by means of their algebraic-geometric spectral data (1990-early 2000s, e.g. Hitchin [6]), or again directly by holomorphic functions.
In this course I will explain how to describe solutions of integrable PDEs on compact surfaces of higher genus by means of complex curves into Deligne-Hitchin moduli spaces [2]. In doing so, we will first construct the Deligne-Hitchin moduli space [9,10], and then identify it with the twistor space [7] of solutions of Hitchin's self-duality equations [9]. As a byproduct, we will naturally derive the non-abelian Hodge correspondence [5], see also [4]. After that, we will discuss other equations and their solutions like minimal surfaces in spheres or constant mean curvature surfaces [3]. If time permits, we will discuss certain additional structures on the Deligne-Hitchin moduli space related to hyper-Kähler geometry and to the harmonic map energy [1].