This project concerns the investigation of new hyper-Kähler manifolds which are given by spaces of singular solutions to Hitchin's equation.
Hitchin's self-duality equations over compact Riemann surfaces are two-dimensional reductions of the Yang-Mills equations. These equations and their solutions provide deep links between various mathematical subjects like algebraic geometry and integrable systems, topology and representations of fundamental groups of surfaces, and differential geometric structures and harmonic maps. The moduli space of solutions of the self-duality equations carries a variety of interesting geometric structures, e.g., a completely integrable system and a hyper-Kähler structure induced by a natural $L^2$-metric. (Irreducible) solutions are parametrized by means of the Hitchin-Kobayashi correspondence either by (stable) Higgs pairs or by conjugacy classes of (irreducible) representations, and yield harmonic maps in corresponding non-compact symmetrical spaces.
The twistor space of the moduli space of solutions of the Hitchin equations has a complex analytical reincarnation as the Deligne-Hitchin moduli space of $\lambda$-connections. In this project we investigate special classes of singular solutions of Hitchin's equations over compact Riemannian surfaces which are given by real holomorphic sections of the Deligne-Hitchin moduli space. Global solutions of the Hitchin equations correspond to so-called twistor lines. It was shown recently that other components of real holomorphic sections besides twistor lines do exists, and that some of these components of real holomorphic sections carry at least locally an induced hyper-Kähler metric. We will examine questions regarding the completeness of these metrics and investigate new methods for constructing real holomorphic sections.
In particular, we will answer the following questions and problems:
- detecting different components of real holomorphic sections;
- construction of new real holomorphic sections via the analytic construction of special singular harmonic mappings;
- gluing of local prototypes of singular solutions of the Hitchin equations;
- a first investigation of the hyper-Kähler metric near infinity;
- generalisation of previous results to the higher rank $ \geq2$ case;
- applications in the AdS / CFT correspondence: Construction of examples of minimal surfaces with non-trivial topology in anti-deSitter space-time; examination of the renormalised area in terms of a natural energy functional on the space of holomorphic sections.