Abstract
An isothermic immersion of a surface in a space form, which is then embedded equivariantly into Möbius space, becomes an isothermic immersion of a surface in Möbius space. Thus, an isothermic immersion in a space form remains isothermic under conformal transformations. An isothermic immersion into Möbius space is special if it comes from a CMC immersion into a space form. By a theorem of Voss, the Bryant quartic differential form of an umbilic free conformal immersion into Möbius space is holomorphic if and only if it is Willmore or special isothermic. A minimal immersion into a space form followed by the equivariant embedding of the space form into Möbius space becomes a Willmore immersion into Möbius space. Moreover, it is isothermic, with isolated umbilics, since this is true for minimal immersions in space forms and these properties are preserved by the embeddings into Möbius space. The theorem of Thomsen states that up to Möbius transformation, all isothermic Willmore immersions with isolated umbilics arise in this way.
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Jensen, G.R., Musso, E., Nicolodi, L. (2016). Isothermic Immersions into Möbius Space. In: Surfaces in Classical Geometries. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-27076-0_14
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