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Bäcklund transforms of conformal maps into the 4-sphere. (English) Zbl 1092.53048
Opozda, Barbara (ed.) et al., PDEs, submanifolds and affine differential geometry. Proceedings of the conference and autumn school, Bȩdlewo, Poland, September 23–27, 2003. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 69, 103-118 (2005).
Authors’ abstract: Transformations which preserve special surface classes in 3- and 4-space play an important role in surface geometry. One of the motivations to study those transformations comes from the fact that they allow to construct more complicated surfaces from given simple ones. Historical examples include the Bäcklund transformation on surfaces of constant Gaussian curvature and the Darboux transformation on isothermic surfaces. More recently, also a Bäcklund transformation on Willmore surfaces has been studied. We consider a general Bäcklund transformation on conformal surfaces $f:M\to {S}^{4}$ where $M$ is a Riemann surface. This will allow us to explicitly construct new conformal immersions of a given Riemann surface into 4-space from a given one by solving abelian integrals.
##### MSC:
 53C42 Immersions (differential geometry) 53Axx Classical differential geometry