We study the holomorphic sections of the Deligne-Hitchin moduli space of a compact Riemann surface, especially the sections that are invariant under the natural antiholomorphic involutions of the moduli space. Their relationships with the harmonic maps are established. As a by product, a question of Simpson on such sections, posed in [Si4], is answered.The other complex structure J on M SD arises from the identification of M SD with the moduli space of flat complex connections due to a result of Donaldson, [Do], in the case of rank n = 2 and by a result of Corlette in general [Co]. The third complex structure K is of course K = IJ = −JI. The significance of the self-duality equations in mathematics and theoretical physics is primarily due to, to quote Hitchin, the various incarnations of the solutions, bringing together geometry, topology, analysis and algebra in a harmonious way.Every hyper-Kähler manifold M has a twistor space associated to it. Topologically it is just the product of the space M with the 2-sphere:It has a tautological complex structure given by the natural complex structure on the 2sphere and the above complex structure xI p + yJ p + zK p on M × {(x, y, z)}; the natural projection of the twistor space to CP 1 is evidently holomorphic. The importance of the twistor space construction is due to the fact that the hyper-Kähler structure is encoded in holomorphic data on the twistor space. In the case of the moduli space of solutions to the self-duality equations, the corresponding twistor space, which is known as the Deligne-Hitchin moduli space and is denoted by M DH , admits another interpretation which is due to Deligne; see [Si2]. From Deligne's perspective, this Deligne-Hitchin moduli space M DH is considered as the moduli space of λ-connections on the Riemann surface Σ glued with the moduli space of λ-connections on the conjugate Riemann surface Σ via the Riemann-Hilbert isomorphism; for definitions and details see Section 1. As a consequence, integrable surfaces like harmonic maps and minimal surfaces can be described by sections of the twistor space M DH −→ CP 1 satisfying certain reality conditions that depend on what the target space is. We note that for the case of G C = SL(2, C), these surfaces are lying in any of the following: S 3 , H 3 and its quotients, and their Lorentzian counterparts.

Minimal and CMC surfaces in $S^3$ can be treated via their associated family of flat $\SL(2,\C)$-connections. In this the paper we parametrize the moduli space of flat $\SL(2,\C)$-connections on the Lawson minimal surface of genus 2 which are equivariant with respect to certain symmetries of Lawson's geometric construction. The parametrization uses Hitchin's abelianization procedure to write such connections explicitly in terms of flat line bundles on a complex 1-dimensional torus. This description is used to develop a spectral curve theory for the Lawson surface. This theory applies as well to other CMC and minimal surfaces with the same holomorphic symmetries as the Lawson surface but different Riemann surface structure. Additionally, we study the space of isospectral deformations of compact minimal surface of genus $g\geq2$ and prove that it is generated by simple factor dressing.Comment: 39 pages; sections about isospectral deformations and about CMC surfaces have been added; the theorems on the reconstruction of surfaces out of spectral data have been improved; 1 figure adde

In this paper we consider special linear Fuchsian systems of rank 2 on a 4−punctured sphere and the corresponding parabolic structures. Through an explicit abelianization procedure we construct a 2−to−1 correspondence between flat line bundle connections on a torus and these Fuchsian systems. This naturally equips the moduli space of flat SL(2, C) connections on a 4−punctured sphere with a new set of Darboux coordinates. Furthermore, we apply our theory to give a complex analytic proof of Witten's formula for the symplectic volume of the moduli space of unitary flat connections on the 4−punctured sphere.

Mid-dimensional (A, B, A) and (B, B, B)-branes in the moduli space of flat G C -connections appearing from finite group actions on compact Riemann surfaces are studied. The geometry and topology of these spaces is then described via the corresponding Higgs bundles and Hitchin fibrations.Date: March 30, 2018.

For every g 1, we show the existence of a complete and smooth family of closed constant mean curvature surfaces f g ϕ , ϕ ∈ [0, π 2 ], in the round 3-sphere deforming the Lawson surface ξ1,g to a doubly covered geodesic 2-sphere with monotonically increasing Willmore energy. To do so we use an implicit function theorem argument in the parameter s = 1 2(g+1) . This allows us to give an iterative algorithm to compute the power series expansion of the DPW potential and area of f g ϕ at s = 0 explicitly. In particular, we obtain for large genus Lawson surfaces ξ1,g, due to the real analytic dependence of its area and DPW potential on s, a scheme to explicitly compute the coefficients of the power series in s in terms of multilogarithms. Remarkably, the third order coefficient of the area expansion coincides numerically with 9 4 ζ(3), where ζ is the Riemann ζ function (while the first and second order term were shown to be log(2) and 0 respectively in [10]).Figure 1. A cutaway view of the Lawson surface of genus 9, which alludes to the convergence to a doubly covered sphere for g → ∞. Image by Nick Schmitt.