An overview of the various transformations of isothermic surfaces and their interrelations is given using aquaternionic formalism. Applications to the theory of cmc-1 surfaces inhyperbolic space are given and relations between the two theories are discussed. Within this context, we give Möbius geometric characterizations for cmc-1 surfaces in hyperbolic space and theirminimal cousins.
Hertrich-Jeromin, U., Musso, E., & Nicolodi, L. (2001). Möbius geometry of surfaces of constant mean curvature 1 in hyperbolic space. Annals of Global Analysis and Geometry, 19(2), 185-205. https://doi.org/10.1023/A:1010738712475