Transformations and singularities of isothermic surfaces

Abstract

We determine the limiting behaviour of Darboux and Calapso transforms of polarized curves, where the polarization has a pole of first or second order. We then study the analogous problem for isothermic surfaces. We consider those isothermic surfaces for which their Hopf differential factorizes into a real function and a meromorphic quadratic differential. Upon restriction to a simply connected patch, away from the zeros and poles of this differential, the Darboux and Calapso transformations yield new isothermic surfaces. We investigate the limiting behaviour of these transformed patches as the zeros and poles of the meromorphic quadratic differential are approached and determine whether they are continuous around those points.