DPW potentials for compact symmetric CMC surfaces in the 3-sphere

Abstract

Minimal and CMC immersions of a compact surface M in the 3-sphere can be studied via their associated family of flat SL(2, C)-connections on a rank 2 holomorphic vector bundle E over M. However, for surfaces of genus g greater than 2 it is a difficult task to describe family of holomorphic flat connections. It is easier to consider a related family of meromorphic flat connections and then reconstruct the associated family of the immersion via loop group factorisation methods. The aim of this thesis is to show that it is possible to define a family of meromorphic flat connections on a class of CMC surfaces in S3, from which it is possible to reconstruct the immersion. We consider CMC surfaces M in the 3-sphere having a group of symmetries which is finite and such that the quotient of the surface by the group is the Riemann sphere. We show that the surfaces constructed by Lawson in 1970 and by Karcher, Pinkall and Sterling in 1988, belong to this class of surfaces. We define a holomorphic vector bundle V over the Riemann sphere, equipped with a parabolic structure. We consider a family of logarithmic flat connections on V and we show that such family of logarithmic connections has a prescribed asymptotic at 0 . The main theorem of the thesis shows that the family of logarithmic connections on V can be used to define a DPW potential on the CMC surface satisfying the necessary properties to reconstruct the immersion of the surface M into the 3-sphere via loop group factorisation.