2,038 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
Constrained Willmore Tori and Elastic Curves in 2-Dimensional Space Forms
In this paper we consider two special classes of constrained Willmore tori in
the 3-sphere. The first class is given by the rotation of closed elastic curves
in the upper half plane - viewed as the hyperbolic plane - around the x-axis.
The second is given as the preimage of closed constrained elastic curves, i.e.,
elastic curve with enclosed area constraint, in the round 2-sphere under the
Hopf fibration. We show that all conformal types can be isometrically immersed
into S^3 as constrained Willmore (Hopf) tori and write down all constrained
elastic curves in H^2 and S^2 in terms of the Weierstrass elliptic functions.
Further, we determine the closing condition for the curves and compute the
Willmore energy and the conformal type of the resulting tori.Comment: 23 pages, 2 figure
Towards a constrained Willmore conjecture
We give an overview of the constrained Willmore problem and address some
conjectures arising from partial results and numerical experiments.
Ramifications of these conjectures would lead to a deeper understanding of the
Willmore functional over conformal immersions from compact surfaces.Comment: 17page
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