1,809 research outputs found
Abelianization of Fuchsian Systems on a 4-punctured sphere and applications
In this paper we consider special linear Fuchsian systems of rank on a
punctured sphere and the corresponding parabolic structures. Through an
explicit abelianization procedure we obtain a to correspondence between
flat line bundle connections on a torus and these Fuchsian systems. This
naturally equips the moduli space of flat connections on a
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
punctured sphere.Comment: 23 pages, comments are welcom
Lawson's genus two minimal surface and meromorphic connections
We investigate the Lawson genus 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 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 and stable bundles
We consider compact minimal surfaces 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 of spin
bundles by its dual in terms of an associated element of We also consider the family of holomorphic structures associated
to a minimal surface in For surfaces of genus 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 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 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
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