Adjoint Transform of Willmore Surfaces in $n$-sphere

Details Adjoint Transform of Willmore Surfaces in $n$-sphere Adjoint Transform of Willmore Surfaces in $n$-sphere

2005-08-08

arxiv.org/abs/math/0508139v1

manuscripta mathematica 120(2006), no.2, 163--179

Mathematics

Differential Geometry

22 pages; based on part of the author's PhD dissertation

Scientific paper

10.1007/s00229-006-0635-0

After the surface theory of M\"obius geometry, this study concerns a pair of conformally immersed surfaces in $n$-sphere. Two new invariants $\theta$ and $\rho$ associated with them are introduced as well as the notion of touch and co-touch. This approach is helpful in research about transforms of certain surface classes. As an application, we define adjoint transform for any given Willmore surface in $n$-sphere. It always exists locally (yet not unique in general) and generalizes known duality theorems of Willmore surfaces. This theory on surface pairs reaches its high point by a characterization of adjoint Willmore surfaces in terms of harmonic maps.

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