Conformal Geometry of Surfaces in the 4-Sphere and Quaternions

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dc.creatorBurstall, F.
dc.creatorFerus, D.
dc.creatorLeschke, K.
dc.creatorPedit, F.
dc.creatorPinkall, U.
dc.date2000-02-10
dc.date.accessioned2026-07-07T07:35:19Z
dc.date.available2026-07-07T07:35:19Z
dc.descriptionThis is the first comprehensive introduction to the authors' recent attempts toward a better understanding of the global concepts behind spinor representations of surfaces in 3-space. The important new aspect is a quaternionic-valued function theory, whose "meromorphic functions" are conformal maps into quaternions, which extends the classical complex function theory on Riemann surfaces. The first results along these lines were presented at the ICM 98 in Berlin. An important new invariant of the quaternionic holomorphic theory is the Willmore energy. For quaternionic holomorphic curves in the quaternionic projective line, i.e., the conformal 4-sphere, this energy is the classical Willmore energy of conformal surfaces. Using these new techniques, the article discusses the mean curvature sphere of conformal surfaces in the 4-sphere, Willmore surfaces in the 4-sphere, Baecklund tranformations of Willmore surfaces, super-conformal surfaces and twistor projections, and a duality between Willmore surfaces in the 3-sphere and minimal surfaces in hyperbolic 3-space. Finally, a new proof of Montiel's recent classification of Willmore 2-spheres in the 4-sphere is given.
dc.identifierhttps://arxiv.org/abs/math/0002075
dc.identifierhttp://arxiv.org/abs/math/0002075
dc.identifierLecture Notes in Mathematics vol 1772, Springer-Verlag, 2002
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/120051
dc.subjectDifferential Geometry
dc.subjectAlgebraic Geometry
dc.subject53C; 14H
dc.titleConformal Geometry of Surfaces in the 4-Sphere and Quaternions
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