Abelianization of Fuchsian Systems on a 4-punctured sphere and applications

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 obtain 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 $SL(2,\mathbb C)-$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.Comment: 23 pages, comments are welcom

Lawson's genus two minimal surface and meromorphic connections

We investigate the Lawson genus $2$ surface by methods from integrable system theory. We prove that the associated family of flat connections comes from a family of flat connections on a $4-$punctured sphere. We describe the symmetries of the holonomy and show that it is already determined by the holonomy around one of the punctures. We show the existence of a meromorphic DPW potential for the Lawson surface which is globally defined on the surface. We determine this potential explicitly up to two unknown functions depending only on the spectral parameter

Higher genus minimal surfaces in S3S^3 and stable bundles

We consider compact minimal surfaces $f\colon M\to S^3$ of genus 2 which are homotopic to an embedding. We assume that the associated holomorphic bundle is stable. We prove that these surfaces can be constructed from a globally defined family of meromorphic connections by the DPW method. The poles of the meromorphic connections are at the Weierstrass points of the Riemann surface of order at most 2. For the existence proof of the DPW potential we give a characterization of stable extensions $0\to S^{-1}\to V\to S\to 0$ of spin bundles $S$ by its dual $S^{-1}$ in terms of an associated element of $P H^0(M;K^2).$ We also consider the family of holomorphic structures associated to a minimal surface in $S^3.$ For surfaces of genus $g\geq2$ the holonomy of the connections is generically non-abelian and therefore the holomorphic structures are generically stable

A spectral curve approach to Lawson symmetric CMC surfaces of genus 2

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