# Darboux and Calapso transforms of meromorphically isothermic surfaces

## TL;DR

This paper studies how Darboux and Calapso transformations affect certain special isothermic surfaces with meromorphic Hopf differentials, focusing on their behavior near singularities and the continuity of the transformations.

## Contribution

It characterizes the limiting behavior of Darboux and Calapso transforms of meromorphically isothermic surfaces near singularities, providing insights into their continuity and geometric properties.

## Key findings

- Transformations yield new isothermic surfaces away from singularities.
- Limiting behavior near zeros or poles is explicitly characterized.
- Continuity of transformed surfaces around singular points is analyzed.

## Abstract

We consider those simply connected isothermic surfaces for which their Hopf differential factorizes into a real function and a meromorphic quadratic differential that has a zero or pole at some point, but is nowhere zero and holomorphic otherwise. Upon restriction to a simply connected patch that does not contain the zero or pole, the Darboux and Calapso transformations yield new isothermic surfaces. We determine the limiting behaviour of these transformed patches as the zero or pole of the meromorphic quadratic differential is approached and investigate whether they are continuous around that point.